Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Numerical methods for nonlinear partial differential equations
Bartels, Sören
2015-01-01
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.
Auxiliary equation method for solving nonlinear partial differential equations
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Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
Bifurcation and stability for a nonlinear parabolic partial differential equation
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
Techniques in Linear and Nonlinear Partial Differential Equations
1991-10-21
nonlinear partial differential equations , elliptic 15. NUMBER OF PAGES hyperbolic and parabolic. Variational methods. Vibration problems. Ordinary Five...NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS FINAL TECHNICAL REPORT PROFESSOR LOUIS NIRENBERG OCTOBER 21, 1991 NT)S CRA&I D FIC ,- U.S. ARMY RESEARCH OFFICE...Analysis and partial differential equations . ed. C. Sadowsky. Marcel Dekker (1990) 567-619. [7] Lin, Fanghua, Asymptotic behavior of area-minimizing
Exact solutions for some nonlinear systems of partial differential equations
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Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)], E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)], E-mail: aramady@yahoo.com
2009-04-30
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear systems of partial differential equations (PDEs) is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDEs) are obtained. Graphs of the solutions are displayed.
Exact periodic wave solutions for some nonlinear partial differential equations
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El-Wakil, S.A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt); Elgarayhi, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: elgarayhi@yahoo.com; Elhanbaly, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)
2006-08-15
The periodic wave solutions for some nonlinear partial differential equations, including generalized Klein-Gordon equation, Kadomtsev-Petviashvili (KP) equation and Boussinesq equations, are obtained by using the solutions of Jacobi elliptic equation. Under limit conditions, exact solitary wave solutions, shock wave solutions and triangular periodic wave solutions have been recovered.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Exact solutions for some nonlinear partial differential equations
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Peng, Yan-Ze
2003-08-11
Exact solutions to some nonlinear partial differential equations, including (2+1)-dimensional breaking soliton equation, sine-Gordon equation and double sine-Gordon equation, are studied by means of the mapping method proposed by the author recently. Many new results are presented. A simple review of the method is finally given.
Entropy and convexity for nonlinear partial differential equations.
Ball, John M; Chen, Gui-Qiang G
2013-12-28
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.
Logarithmic singularities of solutions to nonlinear partial differential equations
Tahara, Hidetoshi
2007-01-01
We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.
Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility
Institute of Scientific and Technical Information of China (English)
Mostafa F. El-Sabbagh; Ahmad T. Ali
2011-01-01
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
Approximate solution of a nonlinear partial differential equation
Vajta, M.
2007-01-01
Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for t
Residual models for nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Garry Pantelis
2005-11-01
Full Text Available Residual terms that appear in nonlinear PDEs that are constructed to generate filtered representations of the variables of the fully resolved system are examined by way of a consistency condition. It is shown that certain commonly used empirical gradient models for the residuals fail the test of consistency and therefore cannot be validated as approximations in any reliable sense. An alternate method is presented for computing the residuals. These residual models are independent of free or artificial parameters and there direct link with the functional form of the system of PDEs which describe the fully resolved system are established.
Modified Homotopy Analysis Method for Nonlinear Fractional Partial Differential Equations
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D. Ziane
2017-05-01
Full Text Available In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM. The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations
Sachdev, PL
2010-01-01
A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/boundary conditions. This title presents the constructive mathematical techniques. It deals with the asymptotic methods which include self-similarity, balancing argument, and matched asymptotic expansions
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
Exact travelling wave solutions of nonlinear partial differential equations
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Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt)]. E-mail: asoliman_99@yahoo.com; Abdou, M.A. [Theoretical Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-04-15
An extended Fan-sub equation method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The key idea of this method is to take full advantage of the general elliptic equation, involving five parameters, which has more new solutions and whose degeneracies can lead to special sub equation involving three parameters. As an illustration of the extended Fan method, more new solutions are obtained for three models namely, generalized KdV, Drinfeld-Sokolov system and RLW equation.
Symposium on Nonlinear Semigroups, Partial Differential Equations and Attractors
Zachary, Woodford
1987-01-01
The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.
Painlevé analysis for nonlinear partial differential equations
Musette, M
1998-01-01
The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and syste...
DIFFERENCE METHODS FOR A NON-LINEAR ELLIPTIC SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS,
DIFFERENCE EQUATIONS, ITERATIONS), (*ITERATIONS, DIFFERENCE EQUATIONS), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), EQUATIONS, FUNCTIONS(MATHEMATICS), SEQUENCES(MATHEMATICS), NONLINEAR DIFFERENTIAL EQUATIONS
Scalable nonlinear iterative methods for partial differential equations
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Cai, X-C
2000-10-29
We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.
Nonlinear partial differential equations for scientists and engineers
Debnath, Lokenath
1997-01-01
"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership. Topics and key features: * Thorough coverage of derivation and methods of soluti...
Nonlinear evolution operators and semigroups applications to partial differential equations
Pavel, Nicolae H
1987-01-01
This research monograph deals with nonlinear evolution operators and semigroups generated by dissipative (accretive), possibly multivalued operators, as well as with the application of this theory to partial differential equations. It shows that a large class of PDE's can be studied via the semigroup approach. This theory is not available otherwise in the self-contained form provided by these Notes and moreover a considerable part of the results, proofs and methods are not to be found in other books. The exponential formula of Crandall and Liggett, some simple estimates due to Kobayashi and others, the characterization of compact semigroups due to Brézis, the proof of a fundamental property due to Ursescu and the author and some applications to PDE are of particular interest. Assuming only basic knowledge of functional analysis, the book will be of interest to researchers and graduate students in nonlinear analysis and PDE, and to mathematical physicists.
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
Calculus of variations and nonlinear partial differential equations
Marcellini, Paolo
2008-01-01
This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro (Italy) in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. The topics discussed are transport equations for nonsmooth vector fields, homogenization, viscosity methods for the infinite Laplacian, weak KAM theory and geometrical aspects of symmetrization. A historical overview of all CIME courses on the calculus of variations and partial differential equations is contributed by Elvira Mascolo.
Differential geometry techniques for sets of nonlinear partial differential equations
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
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Zhao Xiqiang [Department of Mathematics, Ocean University of China, Qingdao Shandong 266071 (China)] e-mail: zhaodss@yahoo.com.cn; Wang Limin [Shandong University of Technology, Zibo Shandong 255049 (China); Sun Weijun [Shandong University of Technology, Zibo Shandong 255049 (China)
2006-04-01
In this letter, a new method, called the repeated homogeneous balance method, is proposed for seeking the traveling wave solutions of nonlinear partial differential equations. The Burgers-KdV equation is chosen to illustrate our method. It has been confirmed that more traveling wave solutions of nonlinear partial differential equations can be effectively obtained by using the repeated homogeneous balance method.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2016-07-01
Full Text Available In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2017-02-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2016-08-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
Renormalizing Partial Differential Equations
Bricmont, J.; Kupiainen, A.
1994-01-01
In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Oscillation criteria for a class of nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Robert Marik
2002-03-01
Full Text Available This paper presents sufficient conditions on the function $c(x$ to ensure that every solution of partial differential equation $$ sum_{i=1}^{n}{partial over partial x_i} Phi_{p}({partial u over partial x_i}+B(x,u=0, quad Phi_p(u:=|u|^{p-1}mathop{ m sgn} u. quad p>1 $$ is weakly oscillatory, i.e. has zero outside of every ball in $mathbb{R}^n$. The main tool is modified Riccati technique developed for Schrodinger operator by Noussair and Swanson [11].
Simple equation method for nonlinear partial differential equations and its applications
Directory of Open Access Journals (Sweden)
Taher A. Nofal
2016-04-01
Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
YeCaier; PanZuliang
2003-01-01
Nonlinear partial differetial equation(NLPDE)is converted into ordinary differential equation(ODE)via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
Energy Technology Data Exchange (ETDEWEB)
Zhang, Sheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China)]. E-mail: zhshaeng@yahoo.com.cn; Xia, Tiecheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China); Department of Mathematics, Shanghai University, Shanghai 200444 (China)
2007-04-09
In this Letter, a generalized new auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the combined KdV-mKdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
Analytic solutions of a class of nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
ZHANG Hong-qing; DING Qi
2008-01-01
An approach is presented for computing the adjoint operator vector of a class of nonlinear (that is,partial-nonlinear) operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author.A unified theory is then given to solve a class of nonlinear (partial-nonlinear and including all linear)and non-homogeneous differential equations with a mathematical mechanization method.In other words,a transformation is constructed by homogenization and triangulation,which reduces the original system to a simpler diagonal system.Applications are given to solve some elasticity equations.
On Degenerate Partial Differential Equations
Chen, Gui-Qiang G.
2010-01-01
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, fundamental problems in fluid mechanics and differential geometry. The solution to these fundamental problems greatly requires a deep understanding of nonlinear degenerate parti...
A higher order lattice BGK model for simulating some nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
LAI HuiLin; MA ChangFeng
2009-01-01
In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux+βunux-γuxx+δuxxx= F(U). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.
A higher order lattice BGK model for simulating some nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.
New Exact Solutions for New Model Nonlinear Partial Differential Equation
Directory of Open Access Journals (Sweden)
A. Maher
2013-01-01
Full Text Available In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
The numerical dynamic for highly nonlinear partial differential equations
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
An effective analytic approach for solving nonlinear fractional partial differential equations
Ma, Junchi; Zhang, Xiaolong; Liang, Songxin
2016-08-01
Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.
Analytic continuation of solutions of some nonlinear convolution partial differential equations
Directory of Open Access Journals (Sweden)
Hidetoshi Tahara
2015-01-01
Full Text Available The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.
A unified lattice Boltzmann model for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Chai Zhenhua [State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074 (China); Shi Baochang [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)], E-mail: sbchust@126.com; Zheng Lin [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2008-05-15
In this paper, a unified and novel lattice Boltzmann model is proposed for solving nonlinear partial differential equation that has the form DU{sub t} + {alpha}UU{sub x} + {beta}U{sup n}U{sub x} - {gamma}U{sub xx} + {delta} U{sub xxx} = F(x,t). Numerical results agree well with the analytical solutions and results derived by existing literature, which indicates the present model is satisfactory and efficient on solving nonlinear partial differential equations.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAICheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Soliton solution for nonlinear partial differential equations by cosine-function method
Energy Technology Data Exchange (ETDEWEB)
Ali, A.H.A. [Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom (Egypt); Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt)], E-mail: asoliman_99@yahoo.com; Raslan, K.R. [Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo (Egypt)
2007-08-20
In this Letter, we established a traveling wave solution by using Cosine-function algorithm for nonlinear partial differential equations. The method is used to obtain the exact solutions for five different types of nonlinear partial differential equations such as, general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKdV), general improved Korteweg-de Vries equation (GIKdV), and Coupled equal width wave equations (CEWE), which are the important soliton equations.
Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Shingareva, Inna K
2011-01-01
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an
Directory of Open Access Journals (Sweden)
M. R.Odekunle
2014-08-01
Full Text Available Tau method which is an economized polynomial technique for solving ordinary and partial differential equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution of linearizedPartial differential Equation without first rewriting them in terms of other known functions as commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique. The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial differential equations.
Directory of Open Access Journals (Sweden)
Elsayed M.E. Zayed
2016-02-01
Full Text Available In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.
Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2+1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.
Energy Technology Data Exchange (ETDEWEB)
Zhou Yubin; Wang Mingliang; Miao Tiande
2004-03-15
The periodic wave solutions for a class of nonlinear partial differential equations, including the Davey-Stewartson equations and the generalized Zakharov equations, are obtained by using the F-expansion method, which can be regarded as an overall generalization of the Jacobi elliptic function expansion method recently proposed. In the limit cases the solitary wave solutions of the equations are also obtained.
Directory of Open Access Journals (Sweden)
Yusuf Pandir
2012-01-01
Full Text Available We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
Energy Technology Data Exchange (ETDEWEB)
Jerome L.V. Lewandowski
2005-01-25
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
Nonlinear perturbations of systems of partial differential equations with constant coefficients
Directory of Open Access Journals (Sweden)
Carmen J. Vanegas
2000-01-01
Full Text Available In this article, we show the existence of solutions to boundary-value problems, consisting of nonlinear systems of partial differential equations with constant coefficients. For this purpose, we use the right inverse of an associated operator and a fix point argument. As illustrations, we apply this method to Helmholtz equations and to second order systems of elliptic equations.
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
Institute of Scientific and Technical Information of China (English)
2008-01-01
Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Institute of Scientific and Technical Information of China (English)
WANG Shundin; ZHANG Hua
2008-01-01
Using functional derivative technique In quantum field theory,the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations.The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by Introducing the time translation operator.The functional partial differential evolution equations were solved by algebraic dynam-ics.The algebraic dynamics solutions are analytical In Taylor series In terms of both initial functions and time.Based on the exact analytical solutions,a new nu-merical algorithm-algebraic dynamics algorithm was proposed for partial differ-ential evolution equations.The difficulty of and the way out for the algorithm were discussed.The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Recent topics in non-linear partial differential equations 4
Mimura, M
1989-01-01
This fourth volume concerns the theory and applications of nonlinear PDEs in mathematical physics, reaction-diffusion theory, biomathematics, and in other applied sciences. Twelve papers present recent work in analysis, computational analysis of nonlinear PDEs and their applications.
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Developmental Partial Differential Equations
Duteil, Nastassia Pouradier; Rossi, Francesco; Boscain, Ugo; Piccoli, Benedetto
2015-01-01
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold's geometry. DPDE is used to study a diffusion equation with source on a growing surface whose gro...
Modified extended tanh-function method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-03-15
Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers-Fisher equation and Drinfeld-Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.
Operator splitting for partial differential equations with Burgers nonlinearity
Holden, Helge; Risebro, Nils Henrik
2011-01-01
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers' equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $H^r$ for initial data in $H^{r+5}$ with arbitrary $r\\ge 1$.
Application of homotopy-perturbation to non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Cveticanin, L. [Faculty of Technical Sciences, 21000 Novi Sad, Trg D. Obradovica 6 (Serbia)], E-mail: cveticanin@uns.ns.ac.yu
2009-04-15
In this paper He's homotopy perturbation method has been adopted for solving non-linear partial differential equations. An approximate solution of the differential equation which describes the longitudinal vibration of a beam is obtained. The solution is compared with that found using the variational iteration method introduced by He. The difference between the two solutions is negligible.
Techniques in Linear and Nonlinear Partial Differential Equations.
1987-09-14
Vibration proolems. Flame propagation, symmetry and antisymmetry of solutions. Singular solutions of Euler equations. u3t *onstants in Sobolev...Majda on weak, singualr, solutions of the Euler equations in 2-dimensions by showing that certain kinds of singular solutions were simply not possible. % % Al c ooop
Fokas, A. S.; De Lillo, S.
2014-03-01
So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.
LAGUERRE PSEUDOSPECTRAL METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Cheng-long Xu; Ben-yu Guo
2002-01-01
The Laguerre Gauss-Radau interpolation is investigated. Some approximation results are obtained. As an example, the Laguerre pseudospectral scheme is constructed for the BBM equation. The stability and the convergence of proposed scheme are proved. The numerical results show the high accuracy of this approch.
Banks, H. T.; Kunisch, K.
1982-01-01
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
Picone's identity for a system of first-order nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Jaroslav Jaros
2013-06-01
Full Text Available We established a Picone identity for systems of nonlinear partial differential equations of first-order. With the help of this formula, we obtain qualitative results such as an integral inequality of Wirtinger type and the existence of zeros for the first components of solutions in a given bounded domain.
Directory of Open Access Journals (Sweden)
Abaker. A. Hassaballa.
2015-10-01
Full Text Available - In recent years, many more of the numerical methods were used to solve a wide range of mathematical, physical, and engineering problems linear and nonlinear. This paper applies the homotopy perturbation method (HPM to find exact solution of partial differential equation with the Dirichlet and Neumann boundary conditions.
Directory of Open Access Journals (Sweden)
Jun Shuai
2013-11-01
Full Text Available A new approach using optimization technique for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs is presented. After the spatial basis functions of the nonlinear PDEs are chosen, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation to obtain a high-dimensional dynamical system. Secondly, modes of lower-dimensional dynamical systems are obtained by linear combination from the modes of the high-dimensional dynamical systems (ordinary differential equations of nonlinear PDEs. An error function for matrix of the linear combination coefficients is derived, and a simple algorithm to determine the optimal combination matrix is also introduced. A numerical example shows that the optimal dynamical system can use much smaller number of modes to capture the dynamics of nonlinear partial differential equations.
Nonlinear grid error effects on numerical solution of partial differential equations
Dey, S. K.
1980-01-01
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
Directory of Open Access Journals (Sweden)
Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
A new mapping method and its applications to nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Zeng Xin [Department of Mathematics, Zhengzhou University, Zhengzhou 450052 (China)], E-mail: zeng79723@163.com; Yong Xuelin [Department of Mathematics and Physics, North China Electric Power University, Beijing 102206 (China)
2008-10-27
In this Letter, a new mapping method is proposed for constructing more exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional Konopelchenko-Dubrovsky equation and the (2+1)-dimensional KdV equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
Exact solutions of some nonlinear partial differential equations using functional variable method
Indian Academy of Sciences (India)
A Nazarzadeh; M Eslami; M Mirzazadeh
2013-08-01
The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrödinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.
Lu, Bin
2012-06-01
In this Letter, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.
Variational iteration method for solving non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Hemeda, A.A. [Department of Mathematics, Faculty of Science, University of Tanta, Tanta (Egypt)], E-mail: aahemeda@yahoo.com
2009-02-15
In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV-MKdV equation and Camassa-Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
A Maple Package for the Painlevé Test of Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
徐桂琼; 李志斌
2003-01-01
A Maple package, named PLtest, is presented to study whether or not nonlinear partial differential equations the standard WTC algorithm and the Kruskal simplification algorithm. Therefore, we not only study whether the given PDEs pass the test or not, but also obtain its truncated expansion form related to some integrability properties. Several well-known nonlinear models with physical interests illustrate the effectiveness of this package.
Sharma, Dinkar; Singh, Prince; Chauhan, Shubha
2016-01-01
In this paper, a combined form of the Laplace transform method with the homotopy perturbation method (HPTM) is applied to solve nonlinear systems of partial differential equations viz. the system of third order KdV Equations and the systems of coupled Burgers' equations in one- and two- dimensions. The nonlinear terms can be easily handled by the use of He's polynomials. The results shows that the HPTM is very efficient, simple and avoids the round-off errors. Four test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM) which shows that this method is a suitable method for solving systems of partial differential equations.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.
An ansatz for solving nonlinear partial differential equations in mathematical physics
Akbar, M. Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin–Bona–Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general sol...
Calatroni, Luca
2013-08-01
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H -1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.
Homotopy perturbation method for nonlinear partial differential equations of fractional order
Energy Technology Data Exchange (ETDEWEB)
Momani, Shaher [Department of Mathematics and Physics, Qatar University (Qatar)]. E-mail: shahermm@yahoo.com; Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa' Applied University, Salt (Jordan)]. E-mail: odibat@bau.edu.jo
2007-06-11
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.
Directory of Open Access Journals (Sweden)
Yusuf Pandir
2013-01-01
Full Text Available We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE, we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.
Huang, Qing; Wang, Li-Zhen; Zuo, Su-Li
2016-02-01
In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann-Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada-Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed. Supported by the National Natural Science Foundation of China under Grant Nos. 11101332, 11201371, 11371293 and the Natural Science Foundation of Shaanxi Province under Grant No. 2015JM1037
Arithmetic partial differential equations
Buium, Alexandru; Simanca, Santiago R.
2006-01-01
We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``flows'' on them that are the arithmetic analogues of the heat and wave...
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, an extended method is proposed for constructing new forms ofexact travelling wave solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to the asymmetric Nizhnik-Novikov-Vesselov equation and the coupled Drinfel'd-Sokolov-Wilson equation and successfully cover the previously known travelling wave solutions found by Chen's method [Y. Chen, et al. Chaos, Solitons and Fractals 22 (2004) 675; Y. Chen, et al. Int. J. Mod. Phys. C 4 (2004) 595].
Filimonov, M. Yu.
2016-12-01
An analytical method for representation of solutions of nonlinear partial differential equations in the form of special series with recurrently computed coefficients is presented. The coefficients recurrent obtaining from linear differential equations is achieved by specificity of the considered equations. It turns out that due to the functional arbitrariness which possibly is contained in special series, one can prove global convergence of the constructed series to solution of considered nonlinear partial differential equations.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
He-Laplace Method for Linear and Nonlinear Partial Differential Equations
Directory of Open Access Journals (Sweden)
Hradyesh Kumar Mishra
2012-01-01
Full Text Available A new treatment for homotopy perturbation method is introduced. The new treatment is called He-Laplace method which is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The method is implemented on linear and nonlinear partial differential equations. It is found that the proposed scheme provides the solution without any discretization or restrictive assumptions and avoids the round-off errors.
Energy Technology Data Exchange (ETDEWEB)
Ravi Kanth, A.S.V. [Applied Mathematics Division, School of Science and Humanities, V.I.T. University, Vellore-632 014, Tamil Nadu (India)], E-mail: asvravikanth@yahoo.com; Aruna, K. [Applied Mathematics Division, School of Science and Humanities, V.I.T. University, Vellore-632 014, Tamil Nadu (India)
2008-11-17
In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.
Directory of Open Access Journals (Sweden)
Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Partial differential equations
Levine, Harold
1997-01-01
The subject matter, partial differential equations (PDEs), has a long history (dating from the 18th century) and an active contemporary phase. An early phase (with a separate focus on taut string vibrations and heat flow through solid bodies) stimulated developments of great importance for mathematical analysis, such as a wider concept of functions and integration and the existence of trigonometric or Fourier series representations. The direct relevance of PDEs to all manner of mathematical, physical and technical problems continues. This book presents a reasonably broad introductory account of the subject, with due regard for analytical detail, applications and historical matters.
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE, (2 + 1-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.
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A. H. Bhrawy
2014-01-01
Full Text Available One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.
An ansatz for solving nonlinear partial differential equations in mathematical physics.
Akbar, M Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Partial differential equations possessing Frobenius integrable decompositions
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Ma, Wen-Xiu [Department of Mathematics, University of South Florida, Tampa, FL 33620-5700 (United States)]. E-mail: mawx@cas.usf.edu; Wu, Hongyou [Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888 (United States)]. E-mail: wu@math.niu.edu; He, Jingsong [Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026 (China)]. E-mail: jshe@ustc.edu.cn
2007-04-16
Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Baecklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
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S. S. Motsa
2014-01-01
Full Text Available This paper presents a new application of the homotopy analysis method (HAM for solving evolution equations described in terms of nonlinear partial differential equations (PDEs. The new approach, termed bivariate spectral homotopy analysis method (BISHAM, is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.
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Fukang Yin
2013-01-01
Full Text Available This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs. The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.
Institute of Scientific and Technical Information of China (English)
LI Hua-Mei
2003-01-01
In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.
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Elsayed Mohamed Elsayed ZAYED
2014-07-01
Full Text Available In this article, many new exact solutions of the (2+1-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.doi:10.14456/WJST.2014.14
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Yang Zhang
2013-01-01
Full Text Available We introduce a continuum modeling method to approximate a class of large wireless networks by nonlinear partial differential equations (PDEs. This method is based on the convergence of a sequence of underlying Markov chains of the network indexed by N, the number of nodes in the network. As N goes to infinity, the sequence converges to a continuum limit, which is the solution of a certain nonlinear PDE. We first describe PDE models for networks with uniformly located nodes and then generalize to networks with nonuniformly located, and possibly mobile, nodes. Based on the PDE models, we develop a method to control the transmissions in nonuniform networks so that the continuum limit is invariant under perturbations in node locations. This enables the networks to maintain stable global characteristics in the presence of varying node locations.
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Brajesh Kumar Singh
2016-01-01
Full Text Available This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.
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Lu, Bin, E-mail: lubinhb@163.com [School of Mathematical Sciences, Anhui University, Hefei 230601 (China)
2012-06-04
In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Exact travelling wave solutions for a class of nonlinear partial differential equations
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Xie Fuding E-mail: xiefd@sohu.com; Gao Xiaoshan
2004-03-01
In this paper, the tanh-method is improved by means of a proper rational transformation based upon a coupled projective Riccati equations. The ansatz can be applied to find more and new exact solutions of the partial differential equations with the aid of symbolic computation system, Maple. We choose an example, which includes phi{sup 4} equation, Klein-Gordon equation, Duffing equation, Landau-Ginburg-Higgs equation and Sine-Gordon equation, to illustrate the method.
Oscillations of the Solutions of Nonlinear Delay Hyperbolic Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
LIU An-ping; GUO Yan-feng; YANG Xiang-hui
2004-01-01
In this paper, oscillatory properties of solutions of certain hyperbolic partial differential equations with multi-delays are investigated and a series of sufficient conditions for oscillations of the equations are established. Theresults fully indicate that the oscillations are caused by delays.
Stochastic partial differential equations
Chow, Pao-Liu
2014-01-01
Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Itô Type Lévy Processes and Stochastic IntegralsStochastic Differential Equations of Lévy Type Comments Scalar Equations of First Order Introduction Generalized Itô's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat EquationLinear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Grad
Algebraic Approaches to Partial Differential Equations
Xu, Xiaoping
2012-01-01
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations...
Institute of Scientific and Technical Information of China (English)
LIU Chun-Ping; LING Zhi
2005-01-01
By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.
Elements of partial differential equations
Sneddon, Ian N
2006-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Symmetries of partial differential equations
Gaussier, Hervé; Merker, Joël
2004-01-01
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.
3D early embryogenesis image filtering by nonlinear partial differential equations.
Krivá, Z; Mikula, K; Peyriéras, N; Rizzi, B; Sarti, A; Stasová, O
2010-08-01
We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which
Katzav, E; Nechaev, S; Vasilyev, O
2007-06-01
We report some observations concerning the statistics of longest increasing subsequences (LIS). We argue that the expectation of LIS, its variance, and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity.
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Zenchuk, A I, E-mail: zenchuk@itp.ac.r [Institute of Problems of Chemical Physics, RAS Acad. Semenov av., 1 Chernogolovka, Moscow region 142432 (Russian Federation)
2010-06-18
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical C- and S-integrable partial differential equations (PDEs). Generalizations of (1+1)-dimensional C-integrable and (2+1)-dimensional S-integrable N-wave equations are derived among examples. Examples of multidimensional second-order PDEs are represented as well.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher
2015-07-01
Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
Discrete Surface Modelling Using Partial Differential Equations.
Xu, Guoliang; Pan, Qing; Bajaj, Chandrajit L
2006-02-01
We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
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Hasibun Naher
2012-01-01
Full Text Available We construct new analytical solutions of the (3+1-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.
Hyperbolic partial differential equations
Lax, Peter D
2006-01-01
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of soluti
Stochastic partial differential equations
Lototsky, Sergey V
2017-01-01
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected ...
Basic linear partial differential equations
Treves, Francois
2006-01-01
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
Introduction to partial differential equations
Borthwick, David
2016-01-01
This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.Within each section the author creates a narrative that answers the five questions: (1) What is the scientific problem we are trying to understand? (2) How do we model that with PDE? (3) What techniques can we use to analyze the PDE? (4) How do those techniques apply to this equation? (5) What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.
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Jose Ernie C. Lope
2013-12-01
Full Text Available In their 2012 work, Lope, Roque, and Tahara considered singular nonlinear partial differential equations of the form tut = F(t; x; u; ux, where the function F is assumed to be continuous in t and holomorphic in the other variables. They have shown that under some growth conditions on the coefficients of the partial Taylor expansion of F as t 0, the equation has a unique solution u(t; x with the same growth order as that of F(t; x; 0; 0. Koike considered systems of partial differential equations using the Banach fixed point theorem and the iterative method of Nishida and Nirenberg. In this paper, we prove the result obtained by Lope and others using the method of Koike, thereby avoiding the repetitive step of differentiating a recursive equation with respect to x as was done by the aforementioned authors.
Partial Differential Equations of Physics
Geroch, Robert
1996-01-01
Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of differential equations so formulated. Examples of such features include hyperbolicity of the equations, constraints and their roles (e.g., in connection with the initial-value formulation), how diffeomorphism freedom is manifest, and how interactions betwee...
Kramer, Sean; Bollt, Erik M
2013-09-01
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a partial differential equation (PDE) model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modeling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing are discussed.
Zia, Haider
2017-06-01
This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.
A note on the auxiliary equation method for solving nonlinear partial differential equations
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Liu, Chunping [Institute of Mathematics, Yangzhou University, Yangzhou 225002 (China)]. E-mail: yzslcp@pub.yz.jsinfo.net; Liu, Xiaoping [Gaoyou Branch, Yangzhou Education College, Gaoyou 225600 (China)
2006-01-02
First, we pick up some solutions of an auxiliary ordinary differential equation, which were neglected by Sirendaoreji and Sun Jiong in the auxiliary equation method. Then, we give the classification of the solutions for the auxiliary ordinary differential equation depending on its three parameters. Finally, we consider the (2+1)-dimensional dispersive long wave equations and get its more exact solitary wave solutions and reveal the relation of the exact solitary wave solutions obtained by Sirendaoreji and Sun Jiong in their paper.
Partial Differential Equations An Introduction
Choudary, A. D. R.; Parveen, Saima; Varsan, Constantin
2010-01-01
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating mathe...
Partial Differential Equations An Introduction
Choudary, A D R; Varsan, Constantin
2010-01-01
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating mathematical methods.
Directory of Open Access Journals (Sweden)
Shaheed N. Huseen
2013-01-01
Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
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Taha Aziz
2012-01-01
Full Text Available The unsteady unidirectional flow of an incompressible fourth grade fluid bounded by a suddenly moved rigid plate is studied. The governing nonlinear higher order partial differential equation for this flow in a semiinfinite domain is modelled. Translational symmetries in variables and are employed to construct two different classes of closed-form travelling wave solutions of the model equation. A conditional symmetry solution of the model equation is also obtained. The physical behavior and the properties of various interesting flow parameters on the structure of the velocity are presented and discussed. In particular, the significance of the rheological effects are mentioned.
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Jagdev Singh
2014-01-01
Full Text Available The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.
On averaging methods for partial differential equations
Verhulst, F.
2001-01-01
The analysis of weakly nonlinear partial differential equations both qualitatively and quantitatively is emerging as an exciting eld of investigation In this report we consider specic results related to averaging but we do not aim at completeness The sections and contain important material which
Interpolation and partial differential equations
MALIGRANDA, Lech; Persson, Lars-Erik; Wyller, John
1994-01-01
One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations (PDEs). Nowadays interpolation theory has been developed in an almost unbelievable way {see the bibliography of Maligranda [Interpolation of Operators and Applications (1926-1990), 2nd ed. (Luleå University, Luleå, 1993), p. 154]}. In this article some model examples are presented which display how powerful this theory is when dealing with PDEs. One main aim i...
Abstract methods in partial differential equations
Carroll, Robert W
2012-01-01
Detailed, self-contained treatment examines modern abstract methods in partial differential equations, especially abstract evolution equations. Suitable for graduate students with some previous exposure to classical partial differential equations. 1969 edition.
Variational principles for some nonlinear partial differential equations with variable coefficients
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He Jihuan E-mail: jhhe@dhu.edu.cn
2004-03-01
Variational principles for generalized Korteweg-de Vries equation and nonlinear Schroedinger's equation are obtained by the semi-inverse method. The most interesting features of the proposed method are its extreme simplicity and concise forms of variational functionals for a wide range of nonlinear problems. Comparison with the results obtained by the Noether's theorem is made, revealing the present theorem is a straightforward and attracting mathematical tool.
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T. D. Frank
2016-12-01
Full Text Available In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker–Planck equation approach is used to describe stochastic aspects of active Nambu systems. Different thermostatistic settings are considered that are characterized by appropriately-defined entropy measures, such as the Boltzmann–Gibbs–Shannon entropy and the Tsallis entropy. In general, the free energy Fokker–Planck equations associated with these generalized entropy measures correspond to nonlinear partial differential equations. Irrespective of the entropy-related nonlinearities occurring in these nonlinear partial differential equations, it is shown that semi-analytical solutions for the stationary probability densities of the active Nambu systems can be obtained provided that the pumping mechanisms of the active systems assume the so-called canonical-dissipative form and depend explicitly only on Nambu invariants. Applications are presented both for purely-dissipative and for active systems illustrating that the proposed framework includes as a special case stochastic equilibrium systems.
Partial differential equations an introduction
Colton, David
2004-01-01
Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
Ishii, H.; Lions, P. L.
We investigate comparison and existence results for viscosity solutions of fully nonlinear, second-order, elliptic, possibly degenerate equations. These results complement those recently obtained by R. Jensen and H. Ishii. We consider various boundary conditions like for instance Dirichlet and Neumann conditions. We also apply these methods and results to quasilinear Monge-Ampère equations. Finally, we also address regularity questions.
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Mittal, R. C.; Jain, R. K.
2012-12-01
In this paper, a numerical method is proposed to approximate the solution of the nonlinear parabolic partial differential equation with Neumann's boundary conditions. The method is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and its derivatives, which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK3 scheme. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization. Four illustrative examples are included to demonstrate the validity and applicability of the technique. In numerical test problems, the performance of this method is shown by computing L∞andL2error norms for different time levels. Results shown by this method are found to be in good agreement with the known exact solutions.
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Wang Qi [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China) and Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080 (China)]. E-mail: wangqi_dlut@yahoo.com.cn; Chen Yong [Nonlinear Science Center, Department of Mathematics, Ningbo University, Ningbo 315211 (China)
2007-01-15
With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.
Cortes, Adriano Mauricio
2014-01-01
In this paper we present PetIGA, a high-performance implementation of Isogeometric Analysis built on top of PETSc. We show its use in solving nonlinear and time-dependent problems, such as phase-field models, by taking advantage of the high-continuity of the basis functions granted by the isogeometric framework. In this work, we focus on the Cahn-Hilliard equation and the phase-field crystal equation.
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available A novel approach is proposed to deal with a class of nonlinear partial equations including integer and noninteger order derivative. This class of equations cannot be handled with any other commonly used analytical technique. The proposed method is based on the multi-Laplace transform. We solved as an example some complicated equations. Three illustrative examples are presented to confirm the applicability of the proposed method. We have presented in detail the stability, the convergence and the uniqueness analysis of some examples.
3-D zebrafish embryo image filtering by nonlinear partial differential equations.
Rizzi, Barbara; Campana, Matteo; Zanella, Cecilia; Melani, Camilo; Cunderlik, Robert; Krivá, Zuzana; Bourgine, Paul; Mikula, Karol; Peyriéras, Nadine; Sarti, Alessandro
2007-01-01
We discuss application of nonlinear PDE based methods to filtering of 3-D confocal images of embryogenesis. We focus on the mean curvature driven and the regularized Perona-Malik equations, where standard as well as newly suggested edge detectors are used. After presenting the related mathematical models, the practical results are given and discussed by visual inspection and quantitatively using the mean Hausdorff distance.
Superdiffusions and positive solutions of non-linear partial differential equations
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Dynkin, E B [Cornell University, New York (United States)
2004-02-28
By using super-Brownian motion, all positive solutions of the non-linear differential equation {delta}u=u{sup {alpha}} with 1<{alpha}{<=}2 in a bounded smooth domain E are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case {alpha}=2 was treated by B. Mselati in 2002.
Beck, Christian; E, Weinan; Jentzen, Arnulf
2017-01-01
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, ...
Institute of Scientific and Technical Information of China (English)
FAN En-Gui
2001-01-01
Two new applications of homogeneous balance (HB) method are presented.It is shown that HB methodcan be extended to search for the Backlund transformations and similarity reductions of nonlinear partial differentialequations.The close relations among the HB method,Weiss-Tabor-Carnevale method and Clarkson-Kruskal directreduction method are also found.KdV-MKdV equation is considered as an illustrative example,and its one kind of Backlund transformation,three kinds of similarity reductions and several kinds of travelling wave solutions are obtained by using extended HB method.
Caplan, R M
2011-01-01
An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-square value of the solution at the boundaries. Application of the MSD boundary condition to the nonlinear Schr\\"odinger equation is shown, and numerical simulations are performed to demonstrate its usefulness and advantages over other simple boundary conditions.
Hamiltonian partial differential equations and applications
Nicholls, David; Sulem, Catherine
2015-01-01
This book is a unique selection of work by world-class experts exploring the latest developments in Hamiltonian partial differential equations and their applications. Topics covered within are representative of the field’s wide scope, including KAM and normal form theories, perturbation and variational methods, integrable systems, stability of nonlinear solutions as well as applications to cosmology, fluid mechanics and water waves. The volume contains both surveys and original research papers and gives a concise overview of the above topics, with results ranging from mathematical modeling to rigorous analysis and numerical simulation. It will be of particular interest to graduate students as well as researchers in mathematics and physics, who wish to learn more about the powerful and elegant analytical techniques for Hamiltonian partial differential equations.
PARTIAL DIFFERENTIAL EQUATIONS FOR DENSITIES OF RANDOM PROCESSES,
PARTIAL DIFFERENTIAL EQUATIONS , STOCHASTIC PROCESSES), (*STOCHASTIC PROCESSES, PARTIAL DIFFERENTIAL EQUATIONS ), EQUATIONS, STATISTICAL FUNCTIONS, STATISTICAL PROCESSES, PROBABILITY, NUMERICAL METHODS AND PROCEDURES
Partial differential equations of mathematical physics
Sobolev, S L
1964-01-01
Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied math
Directory of Open Access Journals (Sweden)
Jacqueline Fleckinger
2001-12-01
Full Text Available We study the asymptotic behavior of positive solutions $u$ of $$ -Delta_p u(x = V(x u(x^{p-1}, quad p>1; x in Omega,$$ and related partial differential inequalities, as well as conditions for existence of such solutions. Here, $Omega$ contains the exterior of a ball in $mathbb{R}^N$ $1
ERC Workshop on Geometric Partial Differential Equations
Novaga, Matteo; Valdinoci, Enrico
2013-01-01
This book is the outcome of a conference held at the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. The aim of the conference was to discuss recent results on nonlinear partial differential equations, and more specifically geometric evolutions and reaction-diffusion equations. Particular attention was paid to self-similar solutions, such as solitons and travelling waves, asymptotic behaviour, formation of singularities and qualitative properties of solutions. These problems arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years.
Owolabi, Kolade M.
2017-03-01
In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.
Partial Differential Equations and Solitary Waves Theory
Wazwaz, Abdul-Majid
2009-01-01
"Partial Differential Equations and Solitary Waves Theory" is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirota’s bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons. While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II w...
Solution techniques for elementary partial differential equations
Constanda, Christian
2012-01-01
Incorporating a number of enhancements, Solution Techniques for Elementary Partial Differential Equations, Second Edition presents some of the most important and widely used methods for solving partial differential equations (PDEs). The techniques covered include separation of variables, method of characteristics, eigenfunction expansion, Fourier and Laplace transformations, Green’s functions, perturbation methods, and asymptotic analysis.
Partial differential equations for scientists and engineers
Farlow, Stanley J
1993-01-01
Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing th
Lin, Ji; Ren, Bo; Li, Hua-mei; Li, Yi-Shen
2008-03-01
Two Darboux transformations of the (1+1) -dimensional Wu-Zhang (WZ) equation and the two-component Camassa-Holm (2CH) system with the reciprocal transformation are obtained. One-loop and two-loop soliton solutions and multisoliton(like) solutions of the 2CH system are obtained by using the Darboux transformations and selecting different seed solutions of the corresponding equations. The bidirectional soliton solutions of the (1+1) -dimensional WZ equation are also obtained. The interactions of two-soliton head-on and overtaking collisions for the WZ equation and the evolution of the two-soliton(-like) solutions for the 2CH system are studied.
Partial differential equations mathematical techniques for engineers
Epstein, Marcelo
2017-01-01
This monograph presents a graduate-level treatment of partial differential equations (PDEs) for engineers. The book begins with a review of the geometrical interpretation of systems of ODEs, the appearance of PDEs in engineering is motivated by the general form of balance laws in continuum physics. Four chapters are devoted to a detailed treatment of the single first-order PDE, including shock waves and genuinely non-linear models, with applications to traffic design and gas dynamics. The rest of the book deals with second-order equations. In the treatment of hyperbolic equations, geometric arguments are used whenever possible and the analogy with discrete vibrating systems is emphasized. The diffusion and potential equations afford the opportunity of dealing with questions of uniqueness and continuous dependence on the data, the Fourier integral, generalized functions (distributions), Duhamel's principle, Green's functions and Dirichlet and Neumann problems. The target audience primarily comprises graduate s...
Nonlinear Second-Order Partial Differential Equation-Based Image Smoothing Technique
Directory of Open Access Journals (Sweden)
Tudor Barbu
2016-09-01
Full Text Available A second-order nonlinear parabolic PDE-based restoration model is provided in this article. The proposed anisotropic diffusion-based denoising approach is based on some robust versions of the edge-stopping function and of the conductance parameter. Two stable and consistent approximation schemes are then developed for this differential model. Our PDE-based filtering technique achieves an efficient noise removal while preserving the edges and other image features. It outperforms both the conventional filters and also many PDE-based denoising approaches, as it results from the successful experiments and method comparison applied.
Boundary value problems for partial differential equations with exponential dichotomies
Laederich, Stephane
We are extending the notion of exponential dichotomies to partial differential evolution equations on the n-torus. This allows us to give some simple geometric criteria for the existence of solutions to certain nonlinear Dirichlet boundary value problems.
Introduction to partial differential equations with applications
Zachmanoglou, E C
1988-01-01
This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers.
Developments in the Theory of Nonlinear First-Order Partial Differential Equations.
1983-12-01
weakenings of the classical notion of solution lead to nonuniqueness . However, in view of the way these problems arise in applications - in particular...S(ii) If u and v are uniformly continuous and (H3) holds, then u v. (iii) If u and v are Lipschitz continuous, then u = v. This result in fact...special case of viscosity solutions which are Lipschitz continuous (and hence satisfy the equation almost everywhere). Other uniqueness results
Numerical Analysis of Partial Differential Equations
Lui, S H
2011-01-01
A balanced guide to the essential techniques for solving elliptic partial differential equations Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis
Partial Differential Equations Modeling and Numerical Simulation
Glowinski, Roland
2008-01-01
This book is dedicated to Olivier Pironneau. For more than 250 years partial differential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at first and then those originating from human activity and technological development. Mechanics, physics and their engineering applications were the first to benefit from the impact of partial differential equations on modeling and design, but a little less than a century ago the Schrödinger equation was the key opening the door to the application of partial differential equations to quantum chemistry, for small atomic and molecular systems at first, but then for systems of fast growing complexity. Mathematical modeling methods based on partial differential equations form an important part of contemporary science and are widely used in engineering and scientific applications. In this book several experts in this field present their latest results and discuss trends in the numerical analy...
Combat modeling with partial differential equations
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Protopopescu, V.; Santoro, R.T.; Dockery, J.; Cox, R.L.; Barnes, J.M.
1987-11-01
A new analytic model based on coupled nonlinear partial differential equations is proposed to describe the temporal and spatial evolution of opposing forces in combat. Analytic descriptions of combat have been developed previously using relatively simpler models based on ordinary differential equations (.e.g, Lanchester's equations of combat) that capture only the global temporal variation of the forces, but not their spatial movement (advance, retreat, flanking maneuver, etc.). The rationale for analytic models and, particularly, the motivation for the present model are reviewed. A detailed description of this model in terms of the mathematical equations together with the possible and plausible military interpretation are presented. Numerical solutions of the nonlinear differential equation model for a large variety of parameters (battlefield length, initial force ratios, initial spatial distribution of forces, boundary conditions, type of interaction, etc.) are implemented. The computational methods and computer programs are described and the results are given in tabular and graphic form. Where possible, the results are compared with the predictions given by the traditional Lanchester equations. Finally, a PC program is described that uses data downloaded from the mainframe computer for rapid analysis of the various combat scenarios. 11 refs., 10 figs., 5 tabs.
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Mathematical physics with partial differential equations
Kirkwood, James
2011-01-01
Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform
Contact Structures of Partial Differential Equations
Eendebak, P.T.
2007-01-01
We study the geometry of contact structures of partial differential equations. The main classes we study are first order systems of two equations in two independent and two dependent variables and the second order scalar equations in two independent variables. The contact distribution in these two c
A New Approach for Solving Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
Fanwei Meng
2013-01-01
Full Text Available We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011.
Particle Systems and Partial Differential Equations I
Gonçalves, Patricia
2014-01-01
This book presents the proceedings of the international conference Particle Systems and Partial Differential Equations I, which took place at the Centre of Mathematics of the University of Minho, Braga, Portugal, from the 5th to the 7th of December, 2012. The purpose of the conference was to bring together world leaders to discuss their topics of expertise and to present some of their latest research developments in those fields. Among the participants were researchers in probability, partial differential equations and kinetics theory. The aim of the meeting was to present to a varied public the subject of interacting particle systems, its motivation from the viewpoint of physics and its relation with partial differential equations or kinetics theory, and to stimulate discussions and possibly new collaborations among researchers with different backgrounds. The book contains lecture notes written by François Golse on the derivation of hydrodynamic equations (compressible and incompressible Euler and Navie...
Superprocesses and Partial Differential Equations
E. B. Dynkin
1993-01-01
The subject of this article is a class of measure-valued Markov processes. A typical example is super-Brownian motion. The Laplacian $\\Delta$ plays a fundamental role in the theory of Brownian motion. For super-Brownian motion, an analogous role is played by the operator $\\Delta u - \\psi(u)$, where a nonlinear function $\\psi$ describes the branching mechanism. The class of admissible functions $\\psi$ includes the family $\\psi(u) = u^\\alpha, 1 < \\alpha \\leq 2$. Super-Brownian motion belongs to...
Transform methods for solving partial differential equations
Duffy, Dean G
2004-01-01
Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach. Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements: New in the Second Edition: ·...
Numerical Methods for Stochastic Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Sharp, D.H.; Habib, S.; Mineev, M.B.
1999-07-08
This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
A minicourse on stochastic partial differential equations
Rassoul-Agha, Firas
2009-01-01
In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time keeping a realistic pace for the student of the subject.
Diffusions, superdiffusions and partial differential equations
Dynkin, E B
2002-01-01
Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Exp-function method for solving fractional partial differential equations.
Zheng, Bin
2013-01-01
We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.
Li, Biao; Chen, Yong; Zhang, Hongqing
2004-02-01
In this paper, by introducing some proper transformations, the applied range of the homogenous balance (HB) method is extended. With the help of Mathematica, we obtain three auto-Bäcklund transformations (BT) for the generalized Fithugh-Nagumo equation, the generalized Burgers-Fisher equation, the generalized Burgers-Huxley equation, respectively, by use of the extended HB method. From these BTs, some exact solutions for these equations are derived.
Stochastic partial differential equations an introduction
Liu, Wei
2015-01-01
This book provides an introduction to the theory of stochastic partial differential equations (SPDEs) of evolutionary type. SPDEs are one of the main research directions in probability theory with several wide ranging applications. Many types of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. The theory of SPDEs is based both on the theory of deterministic partial differential equations, as well as on modern stochastic analysis. Whilst this volume mainly follows the ‘variational approach’, it also contains a short account on the ‘semigroup (or mild solution) approach’. In particular, the volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. Various types of generalized coercivity conditions are shown to guarantee non-explosion, but also a systematic approach to treat SPDEs with explosion in finite time is developed. It is, so far, the only book where the latter and t...
Numerical approximation of partial differential equations
Bartels, Sören
2016-01-01
Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular ...
Numerical Methods for Partial Differential Equations
Guo, Ben-yu
1987-01-01
These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering. Numerical methods both for boundary value problems of elliptic equations and for initial-boundary value problems of evolution equations, such as hyperbolic systems and parabolic equations, are involved. The 16 papers of this volume present recent or new unpublished results and provide a good overview of current research being done in this field in China.
Underdetermined systems of partial differential equations
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Bender, Carl M. [Department of Physics, Washington University, St. Louis, Missouri 63130 (United States); Dunne, Gerald V. [Department of Physics, University of Connecticut, Storrs, Connecticut 06269 (United States); Mead, Lawrence R. [Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, Mississippi 39406-5046 (United States)
2000-09-01
This paper examines underdetermined systems of partial differential equations in which the independent variables may be classical c-numbers or even quantum operators. One can view an underdetermined system as expressing the kinematic constraints on a set of dynamical variables that generate a Lie algebra. The arbitrariness in the general solution reflects the freedom to specify the dynamics of such a system. (c) 2000 American Institute of Physics.
Observability of discretized partial differential equations
Cohn, Stephen E.; Dee, Dick P.
1988-01-01
It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.
Adaptive grid methods for partial differential equations
Anderson, D. A.
1983-01-01
A number of techniques for constructing adaptive mesh generators for use in solving partial differential equations are reviewed in this paper. Techniques reviewed include methods based on steady grid generation schemes and those which are explicitly designed to determine grid speeds in a time-dependent or space-marching problem. Results for candidate methods are included and suggestions for areas of future research are suggested.
Ambit processes and stochastic partial differential equations
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole; Benth, Fred Espen; Veraart, Almut
Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between...... ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis....
Involutive characteristic sets of algebraic partial differential equation systems
Institute of Scientific and Technical Information of China (English)
陈玉福; 高小山
2003-01-01
This paper presents an algorithm to reduce a nonlinear algebraic partial differential equation system into the involutive characteristic set with respect to an abstract involutive prolongation direction, which covers the existing algorithms based on Riquier method, Thomas method, and Pommaret method. It also provides new algorithms for computing involutive characteristic sets due to the existence of new involutive directions. Experiments show that these new algorithms may be used to significantly reduce the computational steps in Wu-Ritt's characteristic set method for algebraic partial differential equations.
Handbook of differential equations stationary partial differential equations
Chipot, Michel
2006-01-01
This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Ke
Stability of the second order partial differential equations
Ghaemi MB; Cho YJ; Alizadeh B; Gordji M Eshaghi
2011-01-01
Abstract We say that a functional equation (ξ) is stable if any function g satisfying the functional equation (ξ) approximately is near to a true solution of (ξ). In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms: y x ( x , t ) = f ( x , t , y ( x , t ) ) , a y x ( x , t ) + b y t ( x , t ) = f ( x , t , y ( x , t ) ) , p (...
Darboux transformations and linear parabolic partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Arrigo, Daniel J.; Hickling, Fred [Department of Mathematics, University of Central Arkansas, Conway, AR (United States)
2002-07-19
Solutions for a class of linear parabolic partial differential equation are provided. These solutions are obtained by first solving a system of (n+1) nonlinear partial differential equations. This system arises as the coefficients of a Darboux transformation and is equivalent to a matrix Burgers' equation. This matrix equation is solved using a generalized Hopf-Cole transformation. The solutions for the original equation are given in terms of solutions of the heat equation. These results are applied to the (1+1)-dimensional Schroedinger equation where all bound state solutions are obtained for a 2n-parameter family of potentials. As a special case, the solutions for integral members of the regular and modified Poeschl-Teller potentials are recovered. (author). Letter-to-the-editor.
Coherence and Chaos in Integrable PDEs (Partial Differential Equations)
1991-03-01
01 Aug 88 to 30 Sep 9n 4. AMSUB"=S. PUNOUUS NU"蕁 COHERENCE AND CHAOS IN INTEGRABLE PDEs ( PARTIAL DIFFERENTIAL EQUATIONS ) AFOSR-83-0195 _61102F... Differential Equations , Parts 1 and 2; Lectures in Appl. Math. 23, edited by Basil Nicolaenko, Darrel Holm, and and J. Mac Hyman (American Mathematical...Coherent Structures, edited by David Campbell, Alan C. Newell, R. Schrieffer, and Harvey Segur, Physica 18D (1986). 4. Nonlinear Systems of Partial
Algebraic and geometric structures of analytic partial differential equations
Kaptsov, O. V.
2016-11-01
We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.
Partial differential equation models in macroeconomics.
Achdou, Yves; Buera, Francisco J; Lasry, Jean-Michel; Lions, Pierre-Louis; Moll, Benjamin
2014-11-13
The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
Boundary value problems and partial differential equations
Powers, David L
2005-01-01
Boundary Value Problems is the leading text on boundary value problems and Fourier series. The author, David Powers, (Clarkson) has written a thorough, theoretical overview of solving boundary value problems involving partial differential equations by the methods of separation of variables. Professors and students agree that the author is a master at creating linear problems that adroitly illustrate the techniques of separation of variables used to solve science and engineering.* CD with animations and graphics of solutions, additional exercises and chapter review questions* Nearly 900 exercises ranging in difficulty* Many fully worked examples
Modern methods in partial differential equations
Schechter, Martin
2013-01-01
Upon its initial 1977 publication, this volume made recent accomplishments in its field available to advanced undergraduates and beginning graduate students of mathematics. Requiring only some familiarity with advanced calculus and rudimentary complex function theory, it covered discoveries of the previous three decades, a particularly fruitful era. Now it remains a permanent, much-cited contribution to the ever-expanding literature on partial differential equations. Author Martin Schechter chose subjects that will motivate students and introduce them to techniques with wide applicability to p
Partial Differential Equations and Spectral Theory
Demuth, Michael; Witt, Ingo
2011-01-01
This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory, written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research, with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress, provide additional motivation, and develop the necessary tools. Moreover, the authors share their views on futur
Generalized functions and partial differential equations
Friedman, Avner
2005-01-01
This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research.Geared toward upper-level undergraduates and
Lie group analysis method for two classes of fractional partial differential equations
Chen, Cheng; Jiang, Yao-Lin
2015-09-01
In this paper we deal with two classes of fractional partial differential equation: n order linear fractional partial differential equation and nonlinear fractional reaction diffusion convection equation, by using the Lie group analysis method. The infinitesimal generators general formula of n order linear fractional partial differential equation is obtained. For nonlinear fractional reaction diffusion convection equation, the properties of their infinitesimal generators are considered. The four special cases are exhaustively investigated respectively. At the same time some examples of the corresponding case are also given. So it is very convenient to solve the infinitesimal generator of some fractional partial differential equation.
Abstract Operators and Higher-order Linear Partial Differential Equation
Institute of Scientific and Technical Information of China (English)
BI Guang-qing; BI Yue-kai
2011-01-01
We summarize several relevant principles for the application of abstract operators in partial differential equations,and combine abstract operators with the Laplace transform.Thus we have developed the theory of partial differential equations of abstract operators and obtained the explicit solutions of initial value problems for a class of higher-order linear partial differential equations.
Solving Partial Differential Equations on Overlapping Grids
Energy Technology Data Exchange (ETDEWEB)
Henshaw, W D
2008-09-22
We discuss the solution of partial differential equations (PDEs) on overlapping grids. This is a powerful technique for efficiently solving problems in complex, possibly moving, geometry. An overlapping grid consists of a set of structured grids that overlap and cover the computational domain. By allowing the grids to overlap, grids for complex geometries can be more easily constructed. The overlapping grid approach can also be used to remove coordinate singularities by, for example, covering a sphere with two or more patches. We describe the application of the overlapping grid approach to a variety of different problems. These include the solution of incompressible fluid flows with moving and deforming geometry, the solution of high-speed compressible reactive flow with rigid bodies using adaptive mesh refinement (AMR), and the solution of the time-domain Maxwell's equations of electromagnetism.
Algorithm refinement for stochastic partial differential equations.
Energy Technology Data Exchange (ETDEWEB)
Alexander, F. J. (Francis J.); Garcia, Alejandro L.,; Tartakovsky, D. M. (Daniel M.)
2001-01-01
A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. A variety of numerical experiments were performed for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a non-stochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except within the particle region, far from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
Asymptotic problems for stochastic partial differential equations
Salins, Michael
Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an Lp sense. This strengthens previous results where convergence was proved in probability.
Elliptic partial differential equations existence and regularity of distributional solutions
Boccardo, Lucio
2013-01-01
Elliptic partial differential equations is one of the main and most active areas in mathematics. In our book we study linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason the book is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.
Spectral methods for time dependent partial differential equations
Gottlieb, D.; Turkel, E.
1983-01-01
The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.
Nonlocal diffusion second order partial differential equations
Benedetti, I.; Loi, N. V.; Malaguti, L.; Taddei, V.
2017-02-01
The paper deals with a second order integro-partial differential equation in Rn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.
Extrapolation methods for dynamic partial differential equations
Turkel, E.
1978-01-01
Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. Computational results are presented confirming the analytic discussions.
Hilbert space methods for partial differential equations
Directory of Open Access Journals (Sweden)
Ralph E. Showalter
1994-09-01
Full Text Available This book is an outgrowth of a course which we have given almost periodically over the last eight years. It is addressed to beginning graduate students of mathematics, engineering, and the physical sciences. Thus, we have attempted to present it while presupposing a minimal background: the reader is assumed to have some prior acquaintance with the concepts of ``linear'' and ``continuous'' and also to believe $L^2$ is complete. An undergraduate mathematics training through Lebesgue integration is an ideal background but we dare not assume it without turning away many of our best students. The formal prerequisite consists of a good advanced calculus course and a motivation to study partial differential equations.
Inverse problems for partial differential equations
Isakov, Victor
2017-01-01
This third edition expands upon the earlier edition by adding nearly 40 pages of new material reflecting the analytical and numerical progress in inverse problems in last 10 years. As in the second edition, the emphasis is on new ideas and methods rather than technical improvements. These new ideas include use of the stationary phase method in the two-dimensional elliptic problems and of multi frequencies\\temporal data to improve stability and numerical resolution. There are also numerous corrections and improvements of the exposition throughout. This book is intended for mathematicians working with partial differential equations and their applications, physicists, geophysicists, and financial, electrical, and mechanical engineers involved with nondestructive evaluation, seismic exploration, remote sensing, and various kinds of tomography. Review of the second edition: "The first edition of this excellent book appeared in 1998 and became a standard reference for everyone interested in analysis and numerics of...
Partial differential equations in several complex variables
Chen, So-Chin
2001-01-01
This book is intended both as an introductory text and as a reference book for those interested in studying several complex variables in the context of partial differential equations. In the last few decades, significant progress has been made in the fields of Cauchy-Riemann and tangential Cauchy-Riemann operators. This book gives an up-to-date account of the theories for these equations and their applications. The background material in several complex variables is developed in the first three chapters, leading to the Levi problem. The next three chapters are devoted to the solvability and regularity of the Cauchy-Riemann equations using Hilbert space techniques. The authors provide a systematic study of the Cauchy-Riemann equations and the \\bar\\partial-Neumann problem, including L^2 existence theorems on pseudoconvex domains, \\frac 12-subelliptic estimates for the \\bar\\partial-Neumann problems on strongly pseudoconvex domains, global regularity of \\bar\\partial on more general pseudoconvex domains, boundary ...
A Method for Image Decontamination Based on Partial Differential Equation
Directory of Open Access Journals (Sweden)
Hou Junping
2015-01-01
Full Text Available This paper will introduce the method to apply partial differential equations for the decontamination processing of images. It will establish continuous partial differential mathematical models for image information and use specific solving methods to conduct decontamination processing to images during the process of solving partial differential equations, such as image noise reduction, image denoising and image segmentation. This paper will study the uniqueness of solution for the partial differential equations and the monotonicity that functional constrain has on multipliers by making analysis of the ROF model in the partial differential mathematical model.
From ordinary to partial differential equations
Esposito, Giampiero
2017-01-01
This book is addressed to mathematics and physics students who want to develop an interdisciplinary view of mathematics, from the age of Riemann, Poincaré and Darboux to basic tools of modern mathematics. It enables them to acquire the sensibility necessary for the formulation and solution of difficult problems, with an emphasis on concepts, rigour and creativity. It consists of eight self-contained parts: ordinary differential equations; linear elliptic equations; calculus of variations; linear and non-linear hyperbolic equations; parabolic equations; Fuchsian functions and non-linear equations; the functional equations of number theory; pseudo-differential operators and pseudo-differential equations. The author leads readers through the original papers and introduces new concepts, with a selection of topics and examples that are of high pedagogical value.
Partial Differential Equations in General Relativity
Energy Technology Data Exchange (ETDEWEB)
Choquet-Bruhat, Yvonne
2008-09-07
General relativity is a physical theory basic in the modeling of the universe at the large and small scales. Its mathematical formulation, the Einstein partial differential equations, are geometrically simple, but intricate for the analyst, involving both hyperbolic and elliptic PDE, with local and global problems. Many problems remain open though remarkable progress has been made recently towards their solutions. Alan Rendall's book states, in a down-to-earth form, fundamental results used to solve different types of equations. In each case he gives applications to special models as well as to general properties of Einsteinian spacetimes. A chapter on ODE contains, in particular, a detailed discussion of Bianchi spacetimes. A chapter entitled 'Elliptic systems' treats the Einstein constraints. A chapter entitled 'Hyperbolic systems' is followed by a chapter on the Cauchy problem and a chapter 'Global results' which contains recently proved theorems. A chapter is dedicated to the Einstein-Vlasov system, of which the author is a specialist. On the whole, the book surveys, in a concise though precise way, many essential results of recent interest in mathematical general relativity, and it is very clearly written. Each chapter is followed by an up to date bibliography. In conclusion, this book will be a valuable asset to relativists who wish to learn clearly-stated mathematical results and to mathematicians who want to penetrate into the subtleties of general relativity, as a mathematical and physical theory. (book review)
Compatible Spatial Discretizations for Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Arnold, Douglas, N, ed.
2004-11-25
From May 11--15, 2004, the Institute for Mathematics and its Applications held a hot topics workshop on Compatible Spatial Discretizations for Partial Differential Equations. The numerical solution of partial differential equations (PDE) is a fundamental task in science and engineering. The goal of the workshop was to bring together a spectrum of scientists at the forefront of the research in the numerical solution of PDEs to discuss compatible spatial discretizations. We define compatible spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. A wide variety of discretization methods applied across a wide range of scientific and engineering applications have been designed to or found to inherit or mimic intrinsic spatial structure and reproduce fundamental properties of the solution of the continuous PDE model at the finite dimensional level. A profusion of such methods and concepts relevant to understanding them have been developed and explored: mixed finite element methods, mimetic finite differences, support operator methods, control volume methods, discrete differential forms, Whitney forms, conservative differencing, discrete Hodge operators, discrete Helmholtz decomposition, finite integration techniques, staggered grid and dual grid methods, etc. This workshop seeks to foster communication among the diverse groups of researchers designing, applying, and studying such methods as well as researchers involved in practical solution of large scale problems that may benefit from advancements in such discretizations; to help elucidate the relations between the different methods and concepts; and to generally advance our understanding in the area of compatible spatial discretization methods for PDE. Particular points of emphasis included: + Identification of intrinsic properties of PDE models that are critical for the fidelity of numerical
Fast methods for static Hamilton-Jacobi Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Vladimirsky, Alexander Boris
2001-05-01
The authors develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems. In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. The techniques use partial information about the characteristic directions to de-couple the system. Previously known fast methods, available for isotropic problems, are discussed in detail. They introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on the analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology.
Fast methods for static Hamilton-Jacobi Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Vladimirsky, Alexander Boris [Univ. of California, Berkeley, CA (United States)
2001-01-01
The authors develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems. In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. The techniques use partial information about the characteristic directions to de-couple the system. Previously known fast methods, available for isotropic problems, are discussed in detail. They introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on the analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology.
Teaching Modeling with Partial Differential Equations: Several Successful Approaches
Myers, Joseph; Trubatch, David; Winkel, Brian
2008-01-01
We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…
Teaching Modeling with Partial Differential Equations: Several Successful Approaches
Myers, Joseph; Trubatch, David; Winkel, Brian
2008-01-01
We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…
A Novel Partial Differential Algebraic Equation (PDAE) Solver
DEFF Research Database (Denmark)
Lim, Young-il; Chang, Sin-Chung; Jørgensen, Sten Bay
2004-01-01
accuracy and stability. The space-time CE/SE method is successfully implemented to solve PDAE systems through combining an iteration procedure for nonlinear algebraic equations. For illustration, chromatographic adsorption problems including convection, diffusion and reaction terms with a linear......For solving partial differential algebraic equations (PDAEs), the space-time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution...
Local structure-preserving algorithms for partial differential equations
Institute of Scientific and Technical Information of China (English)
2008-01-01
In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural generalization of the corresponding global SPAs. Local SPAs for the problems with proper boundary conditions are global SPAs, but the inverse is not necessarily valid. The concept of the local SPAs can explain the difference between different SPAs and provide a basic theory for analyzing and constructing high performance SPAs. Furthermore, it enlarges the applicable scopes of SPAs. We also discuss the application and the construction of local SPAs and derive several new SPAs for the nonlinear Klein-Gordon equation.
The Finite Element Method An Introduction with Partial Differential Equations
Davies, A J
2011-01-01
The finite element method is a technique for solving problems in applied science and engineering. The essence of this book is the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. The method is developed for the solution of Poisson's equation, in a weighted-residual context, and then proceeds to time-dependent and nonlinear problems. The relationship with the variational approach is alsoexplained. This book is written at an introductory level, developing all the necessary concepts where required. Co
Partial differential equations theory and completely solved problems
Hillen, Thomas; van Roessel, Henry
2014-01-01
Uniquely provides fully solved problems for linear partial differential equations and boundary value problems Partial Differential Equations: Theory and Completely Solved Problems utilizes real-world physical models alongside essential theoretical concepts. With extensive examples, the book guides readers through the use of Partial Differential Equations (PDEs) for successfully solving and modeling phenomena in engineering, biology, and the applied sciences. The book focuses exclusively on linear PDEs and how they can be solved using the separation of variables technique. The authors begin
A note on the Lie symmetries of complex partial differential equations and their split real systems
Indian Academy of Sciences (India)
F M Mahomed; Rehana Naz
2011-09-01
Folklore suggests that the split Lie-like operators of a complex partial differential equation are symmetries of the split system of real partial differential equations. However, this is not the case generally. We illustrate this by using the complex heat equation, wave equation with dissipation, the nonlinear Burgers equation and nonlinear KdV equations. We split the Lie symmetries of a complex partial differential equation in the real domain and obtain real Lie-like operators. Further, the complex partial differential equation is split into two coupled or uncoupled real partial differential equations which constitute a system of two equations for two real functions of two real variables. The Lie symmetries of this system are constructed by the classical Lie approach. We compare these Lie symmetries with the split Lie-like operators of the given complex partial differential equation for the examples considered. We conclude that the split Lie-like operators of complex partial differential equations are not in general symmetries of the split system of real partial differential equations. We prove a proposition that gives the criteria when the Lie-like operators are symmetries of the split system.
Calculation of similarity solutions of partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1980-08-01
When a partial differential equation in two independent variables is invariant to a group G of stretching transformations, it has similarity solutions that can be found by solving an ordinary differential equation. Under broad conditions, this ordinary differential equation is also invariant to another stretching group G', related to G. The invariance of the ordinary differential equation to G' can be used to simplify its solution, particularly if it is of second order. Then a method of Lie's can be used to reduce it to a first-order equation, the study of which is greatly facilitated by analysis of its direction field. The method developed here is applied to three examples: Blasius's equation for boundary layer flow over a flat plate and two nonlinear diffusion equations, cc/sub t/ = c/sub zz/ and c/sub t/ = (cc/sub z/)/sub z/.
Directory of Open Access Journals (Sweden)
Hossein Jafari
2016-04-01
Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.
Solution of partial differential equations using a gridless method
Energy Technology Data Exchange (ETDEWEB)
Syms, G.F. [National Research Council of Canada, Inst. for Aerospace Research, Ottawa, Ontario (Canada)]. E-mail: Jerry.Syms@nrc-cnrc.gc.ca
2004-07-01
A set of algorithms to solve linear and nonlinear partial differential evolution equations in one dimension using a gridless method was developed. The potential flexibility of the method is connected to the fact that the points in the grid are not. In the gridless method, the spatial derivatives are computed from the analytic differentiation of a local approximation to the function while the temporal integration is carried out using standard ordinary differential equation techniques. Clouds of points were used to determine the local function approximation. Two sets of basis functions were implemented: ordinary polynomials, x{sup j}, and focus-centred polynomials, (x - x{sup (i)}){sup j}. Overdetermined matrix systems defining the polynomial coefficients were solved through a linear least-squares procedure using either the normal equations or orthogonal triangulation. It was found that the choice of the basis functions and solution procedure could greatly affect the matrix condition number and thus the accuracy of the function reconstruction. The ability of the gridless method to solve partial differential equations was demonstrated by applying the method to the linear convection-diffusion equation and the nonlinear Burger's equation. The stability of the method was found to be negatively affected when reconstructions from over-determined systems were used. (author)
Witten, Matthew
1983-01-01
Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq
Bollt, Erik
2012-01-01
Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a PDE model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modelling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing is discussed.
A complex Noether approach for variational partial differential equations
Naz, R.; Mahomed, F. M.
2015-10-01
Scalar complex partial differential equations which admit variational formulations are studied. Such a complex partial differential equation, via a complex dependent variable, splits into a system of two real partial differential equations. The decomposition of the Lagrangian of the complex partial differential equation in the real domain is shown to yield two real Lagrangians for the split system. The complex Maxwellian distribution, transonic gas flow, Maxwellian tails, dissipative wave and Klein-Gordon equations are considered. The Noether symmetries and gauge terms of the split system that correspond to both the Lagrangians are constructed by the Noether approach. In the case of coupled split systems, the same Noether symmetries are obtained. The Noether symmetries for the uncoupled split systems are different. The conserved vectors of the split system which correspond to both the Lagrangians are compared to the split conserved vectors of the complex partial differential equation for the examples. The split conserved vectors of the complex partial differential equation are the same as the conserved vectors of the split system of real partial differential equations in the case of coupled systems. Moreover a Noether-like theorem for the split system is proved which provides the Noether-like conserved quantities of the split system from knowledge of the Noether-like operators. An interesting result on the split characteristics and the conservation laws is shown as well. The Noether symmetries and gauge terms of the Lagrangian of the split system with the split Noether-like operators and gauge terms of the Lagrangian of the given complex partial differential equation are compared. Folklore suggests that the split Noether-like operators of a Lagrangian of a complex Euler-Lagrange partial differential equation are symmetries of the Lagrangian of the split system of real partial differential equations. This is not the case. They are proved to be the same if the
New Fractional Complex Transform for Conformable Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
Çenesiz Y.
2016-12-01
Full Text Available Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.
Parameter Estimation of Partial Differential Equation Models
Xun, Xiaolei
2013-09-01
Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from long-range infrared light detection and ranging data. Supplementary materials for this article are available online. © 2013 American Statistical Association.
ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
MEI Shu-li; LU Qi-shao; ZHANG Sen-wen; JIN Li
2005-01-01
The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations (ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.
Institute of Scientific and Technical Information of China (English)
苏道毕力格; 王晓民; 乌云莫日根
2014-01-01
In this paper, we study the application of the symmetry classification to the boundary value problem of nonli-near partial differential equations. Firstly, by using differential characteristic set algorithm for the complete symmetry classification of partial differential equations, the complete symmetry classification of a given boundary value problem of nonlinear partial differential equations is proposed. Secondly, by using an extended symmetry, the boundary value problem of nonlinear partial differential equations is reduced to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.%研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用。首先，利用偏微分方程(组)完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类；其次，利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题；最后，利用龙格-库塔法求解了常微分方程组初值问题的数值解。
Constrained Optimization and Optimal Control for Partial Differential Equations
Leugering, Günter; Griewank, Andreas
2012-01-01
This special volume focuses on optimization and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since 2006. The book is organized in sections which cover almost the entire spectrum of modern research in this emerging field. Indeed, even though the field of optimal control and optimization for PDE-constrained problems has undergone a dramatic increase of interest during the last four decades, a full theory for nonlinear problems is still lacking. The cont
Fourier Series in Several Variables with Applications to Partial Differential Equations
Shapiro, Victor L
2011-01-01
Fourier Series in Several Variables with Applications to Partial Differential Equations illustrates the value of Fourier series methods in solving difficult nonlinear partial differential equations (PDEs). Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory. The book first presents four summability methods used in studying multiple Fourier series: iterated Fejer, Bochner-Riesz, Abel, a
Topics in numerical partial differential equations and scientific computing
2016-01-01
Numerical partial differential equations (PDEs) are an important part of numerical simulation, the third component of the modern methodology for science and engineering, besides the traditional theory and experiment. This volume contains papers that originated with the collaborative research of the teams that participated in the IMA Workshop for Women in Applied Mathematics: Numerical Partial Differential Equations and Scientific Computing in August 2014.
Lagrangian vector field and Lagrangian formulation of partial differential equations
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M.Chen
2005-01-01
Full Text Available In this paper we consider the Lagrangian formulation of a system of second order quasilinear partial differential equations. Specifically we construct a Lagrangian vector field such that the flows of the vector field satisfy the original system of partial differential equations.
Solving Fractional Partial Differential Equations with Corrected Fourier Series Method
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Nor Hafizah Zainal
2014-01-01
Full Text Available The corrected Fourier series (CFS is proposed for solving partial differential equations (PDEs with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.
Canonical coordinates for partial differential equations
Hunt, L. R.; Villarreal, Ramiro
1988-01-01
Necessary and sufficient conditions are found under which operators of the form Sigma (m, j=1) x (2) sub j + X sub O can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.
Fem Formulation of Coupled Partial Differential Equations for Heat Transfer
Ameer Ahamad, N.; Soudagar, Manzoor Elahi M.; Kamangar, Sarfaraz; Anjum Badruddin, Irfan
2017-08-01
Heat Transfer in any field plays an important role for transfer of energy from one region to another region. The heat transfer in porous medium can be simulated with the help of two partial differential equations. These equations need an alternate and relatively easy method due to complexity of the phenomenon involved. This article is dedicated to discuss the finite element formulation of heat transfer in porous medium in Cartesian coordinates. A triangular element is considered to discretize the governing partial differential equations and matrix equations are developed for 3 nodes of element. Iterative approach is used for the two sets of matrix equations involved representing two partial differential equations.
The Painlevé property for partial differential equations
Weiss, John; Tabor, M.; Carnevale, George
1983-03-01
In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equations (Burgers' equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.
Effective action for stochastic partial differential equations
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Hochberg, David [Laboratorio de Astrofisica Espacial y Fisica Fundamental, Apartado 50727, 28080 Madrid, (Spain); Centro de Astrobiologia, INTA, Carratera Ajalvir, Km. 4, 28850 Torrejon, Madrid, (Spain); Molina-Paris, Carmen [Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Perez-Mercader, Juan [Laboratorio de Astrofisica Espacial y Fisica Fundamental, Apartado 50727, 28080 Madrid, (Spain); Visser, Matt [Physics Department, Washington University, Saint Louis, Missouri 63130-4899 (United States)
1999-12-01
Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern formation, and the structural development of the universe itself. It is reasonably well known that certain SPDEs can be manipulated to be equivalent to (nonquantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these nonquantum field theories are fully interacting. The limitation to one loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the Becchi-Rouet-Stora-Tyutin formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this ''direct approach'' is the SPDE analog of canonical quantization using physical fields). After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from
Effective action for stochastic partial differential equations.
Hochberg, D; Molina-París, C; Pérez-Mercader, J; Visser, M
1999-12-01
Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern formation, and the structural development of the universe itself. It is reasonably well known that certain SPDEs can be manipulated to be equivalent to (nonquantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these nonquantum field theories are fully interacting. The limitation to one loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the Becchi-Rouet-Stora-Tyutin formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this "direct approach" is the SPDE analog of canonical quantization using physical fields). After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from QFT to the case of
Learning partial differential equations via data discovery and sparse optimization
Schaeffer, Hayden
2017-01-01
We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.
Existence of solutions for a system of elliptic partial differential equations
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Robert Dalmasso
2011-05-01
Full Text Available In this article, we establish the existence of radial solutions for a system of nonlinear elliptic partial differential equations with Dirichlet boundary conditions. Also we discuss the question of uniqueness, and illustrate our results with examples.
Partial differential equations with variable exponents variational methods and qualitative analysis
Radulescu, Vicentiu D
2015-01-01
Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive meth
Stability of the second order partial differential equations
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Ghaemi MB
2011-01-01
Full Text Available Abstract We say that a functional equation (ξ is stable if any function g satisfying the functional equation (ξ approximately is near to a true solution of (ξ. In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms: y x ( x , t = f ( x , t , y ( x , t , a y x ( x , t + b y t ( x , t = f ( x , t , y ( x , t , p ( x , t y x t ( x , t + q ( x , t y t ( x , t + p t ( x , t y x ( x , t - p x ( x , t y t ( x , t = f ( x , t , y ( x , t , p ( x , t y x x ( x , t + q ( x , t y x ( x , t = f ( x , t , y ( x , t . 2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99.
Impacts of noise on a class of partial differential equations
Lv, Guangying; Duan, Jinqiao
2015-03-01
This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen-Cahn equation does not have finite time singularities and the unique solution exists globally.
Partial differential equations & boundary value problems with Maple
Articolo, George A
2009-01-01
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple''s animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website. Ancillary list: Maple files- http://www.elsevierdirect.com/companion.jsp?ISBN=9780123747327 Provides a quick overview of the software w/simple commands needed to get startedIncludes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equationsIncorporates an early introduction to Sturm-L...
FORMAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS AND THE PROJECTIVE LIMIT
Institute of Scientific and Technical Information of China (English)
施惟慧; 沈臻
2003-01-01
Based on stratification theory, the existence theorems of formal solutions of partial differential equation (PDE) are given. And the relationship between formal solutions and projective limit of Ehresmann chain is presented.
Lie symmetry analysis of some time fractional partial differential equations
El Kinani, E. H.; Ouhadan, A.
2015-04-01
This paper uses Lie symmetry analysis to reduce the number of independent variables of time fractional partial differential equations. Then symmetry properties have been employed to construct some exact solutions.
Numerical identifications of parameters in partial differential equations
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Li Jingzhi; Zou Jun [Department of Mathematics, Chinese University of Hong Kong, Shatin, N. T., Hong Kong (China)
2005-01-01
In this paper, we will review some recent theoretical and algorithmic developments in parameter identifications in partial differential equations by our research group, focusing on such aspects as variational formulations, convergence analysis, choice strategies of regularization parameters and algorithmic implementations.
Introduction to partial differential equations and Hilbert space methods
Gustafson, Karl E
1997-01-01
Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.
International Conference on Multiscale Methods and Partial Differential Equations.
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Thomas Hou
2006-12-12
The International Conference on Multiscale Methods and Partial Differential Equations (ICMMPDE for short) was held at IPAM, UCLA on August 26-27, 2005. The conference brought together researchers, students and practitioners with interest in the theoretical, computational and practical aspects of multiscale problems and related partial differential equations. The conference provided a forum to exchange and stimulate new ideas from different disciplines, and to formulate new challenging multiscale problems that will have impact in applications.
BOUNDARY VALUE PROBLEMS, PARTIAL DIFFERENTIAL EQUATIONS ), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), (*NUMERICAL ANALYSIS, BOUNDARY VALUE PROBLEMS), FUNCTIONS(MATHEMATICS), DIFFERENCE EQUATIONS
First-order partial differential equations in classical dynamics
Smith, B. R.
2009-12-01
Carathèodory's classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton-Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Sparse dynamics for partial differential equations.
Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D; Osher, Stanley
2013-04-23
We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.
Data-driven discovery of partial differential equations
Rudy, Samuel; Brunton, Steven; Proctor, Joshua; Kutz, J. Nathan
2016-11-01
Fluid dynamics is inherently governed by spatial-temporal interactions which can be characterized by partial differential equations (PDEs). Emerging sensor and measurement technologies allowing for rich, time-series data collection motivate new data-driven methods for discovering governing equations. We present a novel computational technique for discovering governing PDEs from time series measurements. A library of candidate terms for the PDE including nonlinearities and partial derivatives is computed and sparse regression is then used to identify a subset which accurately reflects the measured dynamics. Measurements may be taken either in a Eulerian framework to discover field equations or in a Lagrangian framework to study a single stochastic trajectory. The method is shown to be robust, efficient, and to work on a variety of canonical equations. Data collected from a simulation of a flow field around a cylinder is used to accurately identify the Navier-Stokes vorticity equation and the Reynolds number to within 1%. A single trace of Brownian motion is also used to identify the diffusion equation. Our method provides a novel approach towards data enabled science where spatial-temporal information bolsters classical machine learning techniques to identify physical laws.
Partial differential equations of parabolic type
Friedman, Avner
2008-01-01
This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman - Director of the Mathematical Biosciences Institute at The Ohio State University - offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations. Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamenta
Relations between Stochastic and Partial Differential Equations in Hilbert Spaces
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I. V. Melnikova
2012-01-01
Full Text Available The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation and solutions to the deterministic partial differential (with derivatives in Hilbert spaces equation for the probability characteristic is proved. Interpretation of objects in the equations is given.
Spectral methods for partial differential equations
Hussaini, M. Y.; Streett, C. L.; Zang, T. A.
1984-01-01
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid-dynamical applications are emphasized.
Wahle, Chris W; Ross, David S; Thurston, George M
2012-07-21
We mathematically design sets of static light scattering experiments to provide for model-independent measurements of ternary liquid mixing free energies to a desired level of accuracy. A parabolic partial differential equation (PDE), linearized from the full nonlinear PDE [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)], describes how data noise affects the free energies to be inferred. The linearized PDE creates a net of spacelike characteristic curves and orthogonal, timelike curves in the composition triangle, and this net governs diffusion of information coming from light scattering measurements to the free energy. Free energy perturbations induced by a light scattering perturbation diffuse along the characteristic curves and towards their concave sides, with a diffusivity that is proportional to the local characteristic curvature radius. Consequently, static light scattering can determine mixing free energies in regions with convex characteristic curve boundaries, given suitable boundary data. The dielectric coefficient is a Lyapunov function for the dynamical system whose trajectories are PDE characteristics. Information diffusion is heterogeneous and system-dependent in the composition triangle, since the characteristics depend on molecular interactions and are tangent to liquid-liquid phase separation coexistence loci at critical points. We find scaling relations that link free energy accuracy, total measurement time, the number of samples, and the interpolation method, and identify the key quantitative tradeoffs between devoting time to measuring more samples, or fewer samples more accurately. For each total measurement time there are optimal sample numbers beyond which more will not improve free energy accuracy. We estimate the degree to which many-point interpolation and optimized measurement concentrations can improve accuracy and save time. For a modest light scattering setup, a sample calculation shows that less than two
Hilbert space methods in partial differential equations
Showalter, Ralph E
1994-01-01
This graduate-level text opens with an elementary presentation of Hilbert space theory sufficient for understanding the rest of the book. Additional topics include boundary value problems, evolution equations, optimization, and approximation.1979 edition.
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
Second volume of a highly regarded two-volume set, fully usable on its own, examines physical systems that can usefully be modeled by equations of the first order. Examples are drawn from a wide range of scientific and engineering disciplines. The book begins with a consideration of pairs of quasilinear hyperbolic equations of the first order and goes on to explore multicomponent chromatography, complications of counter-current moving-bed adsorbers, the adiabatic adsorption column, and chemical reaction in countercurrent reactors. Exercises appear at the end of most sections. Accessible to any
Variational integrators for nonvariational partial differential equations
Kraus, Michael; Maj, Omar
2015-08-01
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.
Partial differential equations and calculus of variations
Leis, Rolf
1988-01-01
This volume contains 18 invited papers by members and guests of the former Sonderforschungsbereich in Bonn (SFB 72) who, over the years, collaborated on the research group "Solution of PDE's and Calculus of Variations". The emphasis is on existence and regularity results, on special equations of mathematical physics and on scattering theory.
Optimal moving grids for time-dependent partial differential equations
Wathen, A. J.
1992-01-01
Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.
Partial differential equations with numerical methods
Larsson, Stig
2003-01-01
The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.
An Implementation Solution for Fractional Partial Differential Equations
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Nicolas Bertrand
2013-01-01
Full Text Available The link between fractional differentiation and diffusion equation is used in this paper to propose a solution for the implementation of fractional diffusion equations. These equations permit us to take into account species anomalous diffusion at electrochemical interfaces, thus permitting an accurate modeling of batteries, ultracapacitors, and fuel cells. However, fractional diffusion equations are not addressed in most commercial software dedicated to partial differential equations simulation. The proposed solution is evaluated in an example.
Numerical Analysis of Partial Differential Equations
Lions, Jacques-Louis
2011-01-01
S. Albertoni: Alcuni metodi di calcolo nella teoria della diffusione dei neutroni.- I. Babuska: Optimization and numerical stability in computations.- J.H. Bramble: Error estimates in elliptic boundary value problems.- G. Capriz: The numerical approach to hydrodynamic problems.- A. Dou: Energy inequalities in an elastic cylinder.- T. Doupont: On the existence of an iterative method for the solution of elliptic difference equation with an improved work estimate.- J. Douglas, J.R. Cannon: The approximation of harmonic and parabolic functions of half-spaces from interior data.- B.E. Hubbard: Erro
Directory of Open Access Journals (Sweden)
Shehu Maitama
2016-01-01
Full Text Available A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM and a well-known method, Homotopy Perturbation Method (HPM. In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.
Generating functionals and Lagrangian partial differential equations
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Vankerschaver, Joris; Liao, Cuicui; Leok, Melvin [Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept. 0112, La Jolla, California 92093-0112 (United States)
2013-08-15
The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.
Automated computational modelling for complicated partial differential equations
Ølgaard, K.B.
2013-01-01
In engineering, physical phenomena are often described mathematically by partial differential equations (PDEs), and a commonly used method to solve these equations is the finite element method (FEM). Implementing a solver based on this method for a given PDE in a computer program written in source c
Partial differential equations modeling, analysis and numerical approximation
Le Dret, Hervé
2016-01-01
This book is devoted to the study of partial differential equation problems both from the theoretical and numerical points of view. After presenting modeling aspects, it develops the theoretical analysis of partial differential equation problems for the three main classes of partial differential equations: elliptic, parabolic and hyperbolic. Several numerical approximation methods adapted to each of these examples are analyzed: finite difference, finite element and finite volumes methods, and they are illustrated using numerical simulation results. Although parts of the book are accessible to Bachelor students in mathematics or engineering, it is primarily aimed at Masters students in applied mathematics or computational engineering. The emphasis is on mathematical detail and rigor for the analysis of both continuous and discrete problems. .
Entropy methods for diffusive partial differential equations
Jüngel, Ansgar
2016-01-01
This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.
Elliptic partial differential equations of second order
Gilbarg, David
2001-01-01
From the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student. Although the material has been developed from lectures at Stanford, it has developed into an almost systematic coverage that is much longer than could be covered in a year's lectures". Newsletter, New Zealand Mathematical Society, 1985 "Primarily addressed to graduate students this elegant book is accessible and useful to a broad spectrum of applied mathematicians". Revue Roumaine de Mathématiques Pures et Appliquées,1985.
Partial Differential Equations A unified Hilbert Space Approach
Picard, Rainer
2011-01-01
This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev Lattice structure, a simple extension of the well established notion of a chain (or scale) of Hilbert spaces. Thefocus on a Hilbert space setting is a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations.This global point of view is takenby focussing on the issues involved in determining the appropriate func
Feng, Qing-Hua
2014-08-01
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.
CIME course on Control of Partial Differential Equations
Alabau-Boussouira, Fatiha; Glass, Olivier; Le Rousseau, Jérôme; Zuazua, Enrique
2012-01-01
The term “control theory” refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010. Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a fri...
On System of Time-fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
Abid KAMRAN
2013-01-01
Full Text Available In this paper, we apply Homotopy Perturbation (HPM using Laplace transform to tackle time- fractional system of Partial Differential equations. The proposed technique is fully compatible with the complexity of these problems and obtained results are highly encouraging. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm.
Stability test for a parabolic partial differential equation
Vajta, Miklos
2001-01-01
The paper describes a stability test applied to coupled parabolic partial differential equations. The PDE's describe the temperature distribution of composite structures with linear inner heat sources. The distributed transfer functions are developed based on the transmission matrix of each layer.
Nyquist stability test for a parabolic partial differential equation
Vajta, Miklos; Hamza, M.H.
2000-01-01
The paper describes a Nyquist stability test applied to a parabolic partial differential equation. The PDE describes the temperature distribution of composite structures with linear inner heat source. The distributed transfer functions have been developed by the transmission matrix method. To
Riccati equations for second order spatially invariant partial differential systems
Curtain, Ruth F.
Recently, the class of spatially invariant systems was introduced with motivating examples of partial differential equations on an infinite domain. For these it was shown that by taking Fourier transforms, one obtains infinitely many finite-dimensional systems with a scalar parameter. The idea is
Function spaces and partial differential equations volume 2 : contemporary analysis
Taheri, Ali
2015-01-01
This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour.
Function spaces and partial differential equations 2 volume set
Taheri, Ali
2015-01-01
This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour.
On a perturbation method for partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Fernandez, Francisco M. [CEQUINOR (Conicet), Departamento de Quimica, Facultad de Ciencias Exactas, Universidad Nacional de la Plata, La Plata (Argentina)]. E-mail: framfer@isis.unlp.edu.ar
2001-06-08
We show that a recently developed perturbation method for partial differential equations can be rewritten in the form of an interaction picture. In this way it is possible to compare this approach with others such as the standard perturbation theory and a straightforward temporal expansion of the evolution operator. We choose a simple, exactly solvable model as an illustrative example. (author)
Conservation laws, differential identities, and constraints of partial differential equations
Zharinov, V. V.
2015-11-01
We consider specific cohomological properties such as low-dimensional conservation laws and differential identities of systems of partial differential equations (PDEs). We show that such properties are inherent to complex systems such as evolution systems with constraints. The mathematical tools used here are the algebraic analysis of PDEs and cohomologies over differential algebras and modules.
非线性偏微分方程的约化和精确解%REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
叶彩儿; 潘祖梁
2003-01-01
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differentialequation (ODE) via a new ansatzUsing undetermined function method ,the ODE obtained aboveis replaced by a set of algebraic equations which are solved out with the aid of MathematicaTheexact solutions and solitary solutions of NLPDE are obtained.
Stochastic partial differential equations in turbulence related problems
Chow, P.-L.
1978-01-01
The theory of stochastic partial differential equations (PDEs) and problems relating to turbulence are discussed by employing the theories of Brownian motion and diffusion in infinite dimensions, functional differential equations, and functional integration. Relevant results in probablistic analysis, especially Gaussian measures in function spaces and the theory of stochastic PDEs of Ito type, are taken into account. Linear stochastic PDEs are analyzed through linearized Navier-Stokes equations with a random forcing. Stochastic equations for waves in random media as well as model equations in turbulent transport theory are considered. Markovian models in fully developed turbulence are discussed from a stochastic equation viewpoint.
An introduction to partial differential equations with Matlab
Coleman, Matthew P
2013-01-01
Introduction What are Partial Differential Equations? PDEs We Can Already Solve Initial and Boundary Conditions Linear PDEs-Definitions Linear PDEs-The Principle of Superposition Separation of Variables for Linear, Homogeneous PDEs Eigenvalue Problems The Big Three PDEsSecond-Order, Linear, Homogeneous PDEs with Constant CoefficientsThe Heat Equation and Diffusion The Wave Equation and the Vibrating String Initial and Boundary Conditions for the Heat and Wave EquationsLaplace's Equation-The Potential Equation Using Separation of Variables to Solve the Big Three PDEs Fourier Series Introduction
Stability of solutions to stochastic partial differential equations
Gess, Benjamin; Tölle, Jonas M.
2016-03-01
We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models.
Simulation of Stochastic Partial Differential Equations and Stochastic Active Contours
Lang, Annika
2007-01-01
This thesis discusses several aspects of the simulation of stochastic partial differential equations. First, two fast algorithms for the approximation of infinite dimensional Gaussian random fields with given covariance are introduced. Later Hilbert space-valued Wiener processes are constructed out of these random fields. A short introduction to infinite-dimensional stochastic analysis and stochastic differential equations is given. Furthermore different definitions of numerical stability for...
Partial differential equations and boundary-value problems with applications
Pinsky, Mark A
2011-01-01
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems-rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate th
Discretization of partial differential equations preserving their physical symmetries
Energy Technology Data Exchange (ETDEWEB)
Valiquette, F; Winternitz, P [Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128, succ. Centre-ville, Montreal, QC, H3C 3J7 (Canada)
2005-11-11
A procedure for obtaining a 'minimal' discretization of a partial differential equation, preserving all of its Lie point symmetries, is presented. 'Minimal' in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variables. We restrict ourselves to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Korteweg-de Vries equations are presented. Some exact solutions of the discrete schemes are obtained.
On the stability of nonautonomous binary dynamical systems of partial differential equations
Directory of Open Access Journals (Sweden)
Salvatore Rionero
2013-01-01
Full Text Available Nonlinear nonautonomoua binary reaction-diffusion dynamical systems of partial differential equations (PDE are considered. Stability criteria - via a nonautonomous L²-energy - are obtained. Applications to nonautonomous Lotka-volterra systems of PDEs and to “preys” struggle for the life, are furnished.
Directory of Open Access Journals (Sweden)
І.С. Клюс
2007-01-01
Full Text Available The correctness of a problem with multi–point conditions on temporary variable of linear partial differential equations not solved as to the highest derivative with respect to time, perturbed by the nonlinear integro-differential operator is investigated
A Partial Differential Equation for the Rank One Convex Envelope
Oberman, Adam M.; Ruan, Yuanlong
2017-02-01
A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn-Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates.
Similarity Solutions of Partial Differential Equations in Probability
Directory of Open Access Journals (Sweden)
Mario Lefebvre
2011-01-01
Full Text Available Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.
Cellular non-deterministic automata and partial differential equations
Kohler, D.; Müller, J.; Wever, U.
2015-09-01
We define cellular non-deterministic automata (CNDA) in the spirit of non-deterministic automata theory. They are different from the well-known stochastic automata. We propose the concept of deterministic superautomata to analyze the dynamical behavior of a CNDA and show especially that a CNDA can be embedded in a deterministic cellular automaton. As an application we discuss a connection between certain partial differential equations and CNDA.
Multigrid methods for space fractional partial differential equations
Jiang, Yingjun; Xu, Xuejun
2015-12-01
We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level. Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings.
Certified reduced basis methods for parametrized partial differential equations
Hesthaven, Jan S; Stamm, Benjamin
2016-01-01
This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.
A stability analysis for a semilinear parabolic partial differential equation
Chafee, N.
1973-01-01
The parabolic partial differential equation considered is u sub t = u sub xx + f(u), where minus infinity x plus infinity and o t plus infinity. Under suitable hypotheses pertaining to f, a class of initial data is exhibited: phi(x), minus infinity x plus infinity, for which the corresponding solutions u(x,t) appraoch zero as t approaches the limit of plus infinity. This convergence is uniform with respect to x on any compact subinterval of the real axis.
Unified algorithm for partial differential equations and examples of numerical computation
Energy Technology Data Exchange (ETDEWEB)
Watanabe, Tsuguhiro [National Inst. for Fusion Science, Toki, Gifu (Japan)
1999-04-01
A new unified algorithm is proposed to solve partial differential equations which describe nonlinear boundary value problems, eigenvalue problems and time developing boundary value problems. The algorithm is composed of implicit difference scheme and multiple shooting scheme and is named as HIDM (Higher order Implicit Difference Method). A new prototype computer programs for 2-dimensional partial differential equations is constructed and tested successfully to several problems. Extension of the computer programs to 3 or more higher order dimension problems will be easy due to the direct product type difference scheme. (author)
Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach
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Toutounji, Mohamad, E-mail: Mtoutounji@uaeu.ac.ae
2015-02-15
This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.
Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach
Toutounji, Mohamad
2015-02-01
This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron-phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.
Solving Partial Differential Equations Using a New Differential Evolution Algorithm
Directory of Open Access Journals (Sweden)
Natee Panagant
2014-01-01
Full Text Available This paper proposes an alternative meshless approach to solve partial differential equations (PDEs. With a global approximate function being defined, a partial differential equation problem is converted into an optimisation problem with equality constraints from PDE boundary conditions. An evolutionary algorithm (EA is employed to search for the optimum solution. For this approach, the most difficult task is the low convergence rate of EA which consequently results in poor PDE solution approximation. However, its attractiveness remains due to the nature of a soft computing technique in EA. The algorithm can be used to tackle almost any kind of optimisation problem with simple evolutionary operation, which means it is mathematically simpler to use. A new efficient differential evolution (DE is presented and used to solve a number of the partial differential equations. The results obtained are illustrated and compared with exact solutions. It is shown that the proposed method has a potential to be a future meshless tool provided that the search performance of EA is greatly enhanced.
Plane waves and spherical means applied to partial differential equations
John, Fritz
2004-01-01
Elementary and self-contained, this heterogeneous collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. The first chapter deals with the decomposition of arbitrary functions into functions of the type of plane waves. Succeeding chapters introduce the first application of the Radon transformation and examine the solution of the initial value problem for homogeneous hyperbolic equations with con
On fractional partial differential equations related to quantum mechanics
Purohit, S. D.; Kalla, S. L.
2011-01-01
In this paper, we investigate the solutions of generalized fractional partial differential equations involving the Caputo time-fractional derivative and the Liouville space-fractional derivatives. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms. Several special cases as solutions of one-dimensional non-homogeneous fractional equations occurring in quantum mechanics are presented in the concluding section. The results given earlier by Debnath (2003 Fract. Calc. Appl. Anal. 6 119-55), Saxena et al (2010 Appl. Math. Comput. 216 1412-7) and Pagnini and Mainardi (2010 J. Comput. Appl. Math. 233 1590-5) follow as special cases of our findings.
On fractional partial differential equations related to quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Purohit, S D [Department of Basic-Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur-313001 (India); Kalla, S L, E-mail: sunil_a_purohit@yahoo.com, E-mail: shyamkalla@gmail.com [Institute of Mathematics, VIHE, 15 B, Pal-Link Road, Jodhpur-342008 (India)
2011-01-28
In this paper, we investigate the solutions of generalized fractional partial differential equations involving the Caputo time-fractional derivative and the Liouville space-fractional derivatives. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms. Several special cases as solutions of one-dimensional non-homogeneous fractional equations occurring in quantum mechanics are presented in the concluding section. The results given earlier by Debnath (2003 Fract. Calc. Appl. Anal. 6 119-55), Saxena et al (2010 Appl. Math. Comput. 216 1412-7) and Pagnini and Mainardi (2010 J. Comput. Appl. Math. 233 1590-5) follow as special cases of our findings.
The application of partial differential equation in interferogram denoising
Liu, Jingfeng; Li, Yanqiu; Liu, Ke
2008-03-01
The presence of noise in interferograms is unavoidable, it may be introduced in acquisition and transmission. These random distortions make it difficult to perform any required processing. Removing noise is often the first step in interferograms analysis. In recent yeas, partial differential equations(PDEs) method in image processing have received extensive concern. compared with traditional approaches such as median filter, average filter, low pass filter etc, PDEs method can not only remove noise but also keep much more details without blurring or changing the location of the edges. In this paper, a fourth-order partial differential equation was applied to optimize the trade-off between noise removal and edges preservation. The time evolution of these PDEs seeks to minimize a cost function which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood. these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image .piecewise planar images look more nature than step images which anisotropic diffusion (second order PDEs)uses to approximate an observed image .The simulation results make it clear that the fourth-order partial differential equatoin can effectively remove noise and preserve interferogram edges.
Partial differential equation transform - Variational formulation and Fourier analysis.
Wang, Yang; Wei, Guo-Wei; Yang, Siyang
2011-12-01
Nonlinear partial differential equation (PDE) models are established approaches for image/signal processing, data analysis and surface construction. Most previous geometric PDEs are utilized as low-pass filters which give rise to image trend information. In an earlier work, we introduced mode decomposition evolution equations (MoDEEs), which behave like high-pass filters and are able to systematically provide intrinsic mode functions (IMFs) of signals and images. Due to their tunable time-frequency localization and perfect reconstruction, the operation of MoDEEs is called a PDE transform. By appropriate selection of PDE transform parameters, we can tune IMFs into trends, edges, textures, noise etc., which can be further utilized in the secondary processing for various purposes. This work introduces the variational formulation, performs the Fourier analysis, and conducts biomedical and biological applications of the proposed PDE transform. The variational formulation offers an algorithm to incorporate two image functions and two sets of low-pass PDE operators in the total energy functional. Two low-pass PDE operators have different signs, leading to energy disparity, while a coupling term, acting as a relative fidelity of two image functions, is introduced to reduce the disparity of two energy components. We construct variational PDE transforms by using Euler-Lagrange equation and artificial time propagation. Fourier analysis of a simplified PDE transform is presented to shed light on the filter properties of high order PDE transforms. Such an analysis also offers insight on the parameter selection of the PDE transform. The proposed PDE transform algorithm is validated by numerous benchmark tests. In one selected challenging example, we illustrate the ability of PDE transform to separate two adjacent frequencies of sin(x) and sin(1.1x). Such an ability is due to PDE transform's controllable frequency localization obtained by adjusting the order of PDEs. The
Institute of Scientific and Technical Information of China (English)
陆海霞
2013-01-01
首先研究通过椭圆型偏微分方程歧点的连通分支的性质,然后得到椭圆型偏微分方程边值问题至少有一个正解存在结果.主要研究方法是全局分歧理论.%In this paper,the properties of the connected component containing the bifurcation points of a nonlinear elliptic partial differential equation is firstly studied.Then the existence of at least one positive solution for the boundary value problems of the equation is proved.The method to show our main results is the global bifurcation theol.
Adaptive interpolation wavelet and homotopy perturbation method for partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ma, Q; Mei, S [College of Information and Electrical Engineering, China Agricultural University, 17 Qinghua Donglu Road, Beijing 100083 (China)], E-mail: meishuli@163.com
2008-02-15
The homotopy perturbation method proposed by Ji-Huan He has been developed to solve nonlinear matrix differential equations. This paper constructs an adaptive multilevel quasi-wavelet operator according to the interpolation wavelet theory, with which the nonlinear partial differential equations can be discretized adaptively in physical spaces as a matrix differential equation, its numerical solution can be obtained by using the homotopy perturbation method. Numerical results show that the homotopy perturbation method is not sensitive to the time step, so the arithmetic error mainly arises in the space step. Burgers equation is taken as examples to illustrate its effectiveness and convenience.
Numerical methods for stochastic partial differential equations with white noise
Zhang, Zhongqiang
2017-01-01
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical compa...
Partial differential equation models in the socio-economic sciences
Burger, Martin
2014-10-06
Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges. In this introductory article of the Theme Issue, we will provide an overview of the field and its recent boosting topics. Moreover, we will put the contributions to the Theme Issue in an appropriate perspective.
Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues
Canada, Antonio
2011-01-01
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \\ 1 \\leq p \\leq +\\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Polynomial solutions of quasi_homogeneous partial differential equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
By means of a method of analytic number theory the following theorem is proved. Let p be a quasi_homogeneous linear partial differential operator with degree m, m>0, w.r.t a dilation {δ-τ}-{τ<0} given by (a-1,…, a-n). Assume that either a1,…, an are positive rational numbers or m=∑nj=1α-jα-j for some α=(α-1, …, αn)∈I{}+n-+. Then the dimension of the space of polynomial solutions of the equation p[u]=0 on R+n must be infinite.
Fourth-order partial differential equations for effective image denoising
Directory of Open Access Journals (Sweden)
Seongjai Kim
2009-04-01
Full Text Available This article concerns mathematical image denoising methods incorporating fourth-order partial differential equations (PDEs. We introduce and analyze piecewise planarity conditions (PPCs with which unconstrained fourth-order variational models in continuum converge to a piecewise planar image. It has been observed that fourth-order variational models holding PPCs can restore better images than models without PPCs and second-order models. Numerical schemes are presented in detail and various examples in image denoising are provided to verify the claim.
Neuron Segmentation in Electron Microscopy Images Using Partial Differential Equations.
Jones, Cory; Sayedhosseini, Mojtaba; Ellisman, Mark; Tasdizen, Tolga
2013-01-01
In connectomics, neuroscientists seek to identify the synaptic connections between neurons. Segmentation of cell membranes using supervised learning algorithms on electron microscopy images of brain tissue is often done to assist in this effort. Here we present a partial differential equation with a novel growth term to improve the results of a supervised learning algorithm. We also introduce a new method for representing the resulting image that allows for a more dynamic thresholding to further improve the result. Using these two processes we are able to close small to medium sized gaps in the cell membrane detection and improve the Rand error by as much as 9% over the initial supervised segmentation.
Optimal Control Problems for Partial Differential Equations on Reticulated Domains
Kogut, Peter I
2011-01-01
In the development of optimal control, the complexity of the systems to which it is applied has increased significantly, becoming an issue in scientific computing. In order to carry out model-reduction on these systems, the authors of this work have developed a method based on asymptotic analysis. Moving from abstract explanations to examples and applications with a focus on structural network problems, they aim at combining techniques of homogenization and approximation. Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference tool for gradu
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
Lagaris, I E; Fotiadis, D I
1997-01-01
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.
Beyer, Horst Reinhard
2007-01-01
The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.
Structure analysis of growing network based on partial differential equations
Directory of Open Access Journals (Sweden)
Junbo JIA
2016-04-01
Full Text Available The topological structure is one of the most important contents in the complex network research. Therein the node degree and the degree distribution are the most basic characteristic quantities to describe topological structure. In order to calculate the degree distribution, first of all, the node degree is considered as a continuous variable. Then, according to the Markov Property of growing network, the cumulative distribution function's evolution equation with time can be obtained. Finally, the partial differential equation (PDE model can be established through distortion processing. Taking the growing network with preferential and random attachment mechanism as an example, the PDE model is obtained. The analytic expression of degree distribution is obtained when this model is solved. Besides, the degree function over time is the same as the characteristic line of PDE. At last, the model is simulated. This PDE method of changing the degree distribution calculation into problem of solving PDE makes the structure analysis more accurate.
Modeling tree crown dynamics with 3D partial differential equations.
Beyer, Robert; Letort, Véronique; Cournède, Paul-Henry
2014-01-01
We characterize a tree's spatial foliage distribution by the local leaf area density. Considering this spatially continuous variable allows to describe the spatiotemporal evolution of the tree crown by means of 3D partial differential equations. These offer a framework to rigorously take locally and adaptively acting effects into account, notably the growth toward light. Biomass production through photosynthesis and the allocation to foliage and wood are readily included in this model framework. The system of equations stands out due to its inherent dynamic property of self-organization and spontaneous adaptation, generating complex behavior from even only a few parameters. The density-based approach yields spatially structured tree crowns without relying on detailed geometry. We present the methodological fundamentals of such a modeling approach and discuss further prospects and applications.
Bringing partial differential equations to life for students
José Cano, María; Chacón-Vera, Eliseo; Esquembre, Francisco
2015-05-01
Teaching partial differential equations (PDEs) carries inherent difficulties that an interactive visualization might help overcome in an active learning process. However, the generation of this kind of teaching material implies serious difficulties, mainly in terms of coding efforts. This work describes how to use an authoring tool, Easy Java Simulations, to build interactive simulations using FreeFem++ (Hecht F 2012 J. Numer. Math. 20 251) as a PDE solver engine. It makes possible to build simulations where students can change parameters, the geometry and the equations themselves getting an immediate feedback. But it is also possible for them to edit the simulations to set deeper changes. The process is ilustrated with some basic examples. These simulations show PDEs in a pedagogic manner and can be tuned by no experts in the field, teachers or students. Finally, we report a classroom experience and a survey from the third year students in the Degree of Mathematics at the University of Murcia.
Modeling Tree Crown Dynamics with 3D Partial Differential Equations
Directory of Open Access Journals (Sweden)
Robert eBeyer
2014-07-01
Full Text Available We characterize a tree's spatial foliage distribution by the local leaf area density. Considering this spatially continuous variable allows to describe the spatiotemporal evolution of the tree crown by means of 3D partial differential equations. These offer a framework to rigorously take locally and adaptively acting effects into account, notably the growth towards light. Biomass production through photosynthesis and the allocation to foliage and wood are readily included in this model framework. The system of equations stands out due to its inherent dynamic property of self-organization and spontaneous adaptation, generating complex behavior from even only a few parameters. The density-based approach yields spatially structured tree crowns without relying on detailed geometry. We present the methodological fundamentals of such a modeling approach and discuss further prospects and applications.
Algorithm Refinement for Stochastic Partial Differential Equations. I. Linear Diffusion
Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.
2002-10-01
A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
Algorithm refinement for stochastic partial differential equations I. linear diffusion
Alexander, F J; Tartakovsky, D M
2002-01-01
A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
Controllability and Stabilization of Bilinear and Semilinear Partial Differential Equations
DEFF Research Database (Denmark)
Krishnaswamy, Vijayaraghavan
The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method.......The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method....
Bove, Antonio; Murthy, MK Venkatesha
2009-01-01
This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as "sums of squares" of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.
Solutions of multidimensional partial differential equations representable as a one-dimensional flow
Zenchuk, A. I.
2014-03-01
We propose an algorithm for reducing an (M+ 1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow ut + (u, ux uxx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+ 1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.
Function Substitution in Partial Differential Equations: Nonhomogeneous Boundary Conditions
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T. V. Oblakova
2017-01-01
Full Text Available In this paper we consider a mixed initial-boundary value problem for a parabolic equation with nonhomogeneous boundary conditions. The classical methods of searching for an analytical solution of such problems in the first stage involve variable substitution , leading to a problem with homogeneous boundary conditions. In the reference literature ([1], as a rule, the simplest types of variable substitutions are given, under which the new and old unknown functions differ by a term linear in the spatial variable. The form of this additional term depends on the type of the boundary conditions, but is in no way connected with the equation under consideration. Moreover, in the case of the second boundary-value problem, it is necessary to use quadratic additives, since a linear replacement for this type of conditions may not exist. In the educational literature ([2] - [4], it is usually limited to considering only the first boundary-value problem in the general formulation.In this paper, we consider a substitution that takes into account in principle the form of a linear differential operator. Namely, as an additive term, it is proposed to use the parametrically time-dependent solution of the boundary value problem for an ordinary differential equation obtained from the original partial differential equation by the method of separation of the Fourier variables.The existence of the proposed replacement for boundary conditions of any type is proved on the example of a nonstationary heat equation in the presence of heat exchange with the surrounding medium. In this case, the additional term is a linear combination of hyperbolic functions. It is shown that in addition to the "insensitivity" to the type of boundary conditions, the advantages of a new replacement in comparison with the traditional linear (or quadratic substitution include a much simpler structure of the resulting solution. Namely, the described approach allows one to obtain a solution
Pullback permanence for non-autonomous partial differential equations
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Jose A. Langa
2002-08-01
Full Text Available A system of differential equations is permanent if there exists a fixed bounded set of positive states strictly bounded away from zero to which, from a time on, any positive initial data enter and remain. However, this fact does not happen for a differential equation with general non-autonomous terms. In this work we introduce the concept of pullback permanence, defined as the existence of a time dependent set of positive states to which all solutions enter and remain for suitable initial time. We show this behaviour in the non-autonomous logistic equation $u_{t}-Delta u=lambda u-b(tu^{3}$, with $b(t>0$ for all $tin mathbb{R}$, $lim_{to infty }b(t=0$. Moreover, a bifurcation scenario for the asymptotic behaviour of the equation is described in a neighbourhood of the first eigenvalue of the Laplacian. We claim that pullback permanence can be a suitable tool for the study of the asymptotic dynamics for general non-autonomous partial differential equations.
Partial differential equations in action from modelling to theory
Salsa, Sandro
2016-01-01
The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear bo...
Partial differential equations in action from modelling to theory
Salsa, Sandro
2015-01-01
The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear bo...
A partial differential equation model of metastasized prostatic cancer.
Friedman, Avner; Jain, Harsh Vardhan
2013-06-01
Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.
A data storage model for novel partial differential equation descretizations.
Energy Technology Data Exchange (ETDEWEB)
Doyle, Wendy S.K.; Thompson, David C.; Pebay, Philippe Pierre
2007-04-01
The purpose of this report is to define a standard interface for storing and retrieving novel, non-traditional partial differential equation (PDE) discretizations. Although it focuses specifically on finite elements where state is associated with edges and faces of volumetric elements rather than nodes and the elements themselves (as implemented in ALEGRA), the proposed interface should be general enough to accommodate most discretizations, including hp-adaptive finite elements and even mimetic techniques that define fields over arbitrary polyhedra. This report reviews the representation of edge and face elements as implemented by ALEGRA. It then specifies a convention for storing these elements in EXODUS files by extending the EXODUS API to include edge and face blocks in addition to element blocks. Finally, it presents several techniques for rendering edge and face elements using VTK and ParaView, including the use of VTK's generic dataset interface for interpolating values interior to edges and faces.
Variational and potential formulation for stochastic partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Munoz S, A G [Laboratorio de AstronomIa y Fisica Teorica (LAFT), Departamento de Fisica, Facultad de Ciencias, La Universidad del Zulia, Maracaibo, 4004 (Venezuela); Ojeda, J [Laboratorio de AstronomIa y Fisica Teorica (LAFT), Departamento de Fisica, Facultad de Ciencias, La Universidad del Zulia, Maracaibo, 4004 (Venezuela); Sierra D, P [Laboratorio de AstronomIa y Fisica Teorica (LAFT), Departamento de Fisica, Facultad de Ciencias, La Universidad del Zulia, Maracaibo, 4004 (Venezuela); Centro de Estudios Matematicos y FIsicos (CeMaFi), Departamento de Matematica y Fisica, La Universidad del Zulia, Maracaibo, 4004 (Venezuela); Soldovieri, T [Laboratorio de AstronomIa y Fisica Teorica (LAFT), Departamento de Fisica, Facultad de Ciencias, La Universidad del Zulia, Maracaibo, 4004 (Venezuela)
2006-01-27
Recently there has been interest in finding a potential formulation for stochastic partial differential equations (SPDEs). The rationale behind this idea lies in obtaining all the dynamical information of the system under study from one single expression. In this letter we formally provide a general Lagrangian formalism for SPDEs using the Hojman et al method. We show that it is possible to write the corresponding effective potential starting from an s-equivalent Lagrangian, and that this potential is able to reproduce all the dynamics of the system once a special differential operator has been applied. This procedure can be used to study the complete time evolution and spatial inhomogeneities of the system under consideration, and is also suitable for the statistical mechanics description of the problem. (letter to the editor)
Optimized difference schemes for multidimensional hyperbolic partial differential equations
Directory of Open Access Journals (Sweden)
Adrian Sescu
2009-04-01
Full Text Available In numerical solutions to hyperbolic partial differential equations in multidimensions, in addition to dispersion and dissipation errors, there is a grid-related error (referred to as isotropy error or numerical anisotropy that affects the directional dependence of the wave propagation. Difference schemes are mostly analyzed and optimized in one dimension, wherein the anisotropy correction may not be effective enough. In this work, optimized multidimensional difference schemes with arbitrary order of accuracy are designed to have improved isotropy compared to conventional schemes. The derivation is performed based on Taylor series expansion and Fourier analysis. The schemes are restricted to equally-spaced Cartesian grids, so the generalized curvilinear transformation method and Cartesian grid methods are good candidates.
Initial condition strategies for multiple-time partial differential equations.
Energy Technology Data Exchange (ETDEWEB)
Coffey, Todd Stirling
2005-02-01
Many highly oscillatory circuits have a wide separation of time scales between the underlying oscillation and the behavior of interest. This is particularly true of communication circuits. Multiple-time Partial Differential Equation (MPDE) methods offer substantial speed-up for these circuits by introducing a periodic artificial time variable that represents the highly oscillatory behavior. This leaves just the slowly changing behavior of interest, which can be integrated with much larger steps. One problem of particular interest is the larger initial condition that must be specified for this periodic artificial time variable. One possible solution is to formulate an optimization problem in the hopes of increasing the step sizes taken in the slow time direction. This talk will discuss one possible unconstrained optimization problem for determining this initial condition. Numerical results and comparisons to several other initial condition strategies will be presented in addition to MPDE background research and implementation issues.
Cubature Methods For Stochastic (Partial) Differential Equations In Weighted Spaces
Doersek, Philipp; Veluscek, Dejan
2012-01-01
The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded payoffs. We first introduce a recently described flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da Prato-Zabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced assumption on weak symmetry of the cubature formula. In finite dimensions, we use the UFG condition to obtain optimal rates of convergence on non-uniform meshes for nonsmooth payoffs with exponential growth.
Reduced basis methods for partial differential equations an introduction
Quarteroni, Alfio; Negri, Federico
2016-01-01
This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examp...
Essential partial differential equations analytical and computational aspects
Griffiths, David F; Silvester, David J
2015-01-01
This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems. The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test unde...
Directory of Open Access Journals (Sweden)
A. A. Hemeda
2013-01-01
Full Text Available An extension of the so-called new iterative method (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iteration method (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
PARTIAL DIFFERENTIAL EQUATIONS , THEORY), (*COMPLEX VARIABLES, PARTIAL DIFFERENTIAL EQUATIONS ), FUNCTIONS(MATHEMATICS), BOUNDARY VALUE PROBLEMS, INEQUALITIES, TRANSFORMATIONS (MATHEMATICS), TOPOLOGY, SET THEORY
Institute of Scientific and Technical Information of China (English)
ZHENG Bin
2012-01-01
In this paper, the ( G'/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.
BOOK REVIEW: Partial Differential Equations in General Relativity
Halburd, Rodney G.
2008-11-01
Although many books on general relativity contain an overview of the relevant background material from differential geometry, very little attention is usually paid to background material from the theory of differential equations. This is understandable in a first course on relativity but it often limits the kinds of problems that can be studied rigorously. Einstein's field equations lie at the heart of general relativity. They are a system of partial differential equations (PDEs) relating the curvature of spacetime to properties of matter. A central part of most problems in general relativity is to extract information about solutions of these equations. Most standard texts achieve this by studying exact solutions or numerical and analytical approximations. In the book under review, Alan Rendall emphasises the role of rigorous qualitative methods in general relativity. There has long been a need for such a book, giving a broad overview of the relevant background from the theory of partial differential equations, and not just from differential geometry. It should be noted that the book also covers the basic theory of ordinary differential equations. Although there are many good books on the rigorous theory of PDEs, methods related to the Einstein equations deserve special attention, not only because of the complexity and importance of these equations, but because these equations do not fit into any of the standard classes of equations (elliptic, parabolic, hyperbolic) that one typically encounters in a course on PDEs. Even specifying exactly what ones means by a Cauchy problem in general relativity requires considerable care. The main problem here is that the manifold on which the solution is defined is determined by the solution itself. This means that one does not simply define data on a submanifold. Rendall's book gives a good overview of applications and results from the qualitative theory of PDEs to general relativity. It would be impossible to give detailed
A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications
Dai, Wanyang
2011-01-01
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the dr...
Furihata, Daisuke
2010-01-01
Nonlinear Partial Differential Equations (PDEs) have become increasingly important in the description of physical phenomena. Unlike Ordinary Differential Equations, PDEs can be used to effectively model multidimensional systems. The methods put forward in Discrete Variational Derivative Method concentrate on a new class of ""structure-preserving numerical equations"" which improves the qualitative behaviour of the PDE solutions and allows for stable computing. The authors have also taken care to present their methods in an accessible manner, which means that the book will be useful to engineer
Institute of Scientific and Technical Information of China (English)
Chaolu Temuer; Yu-shan BAI
2009-01-01
In this paper,we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations (PDEs)with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example,the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm,i.e.,Wu's method,in differential equations.
Fast solution of phase unwrapping partial differential equation using wavelets
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Maryam Rahnemoonfar
2016-03-01
Full Text Available Phase unwrapping is the most critical step in the processing of synthetic aperture radar interferometry. The phase obtained by SAR interferometry is wrapped over a range from $-\\pi$ to $\\pi$. Phase unwrapping must be performed to obtain the true phase. The least square approach attains the unwrapped phase by minimizing the difference between the discrete partial derivatives of the wrapped phase and the discrete partial derivatives of the unwrapped solution. The least square solution will result in discrete version of the Poisson's partial differential equation. Solving the discretized Poisson's equation with the classical method of Gauss-Seidel relaxation has extremely slow convergence. In this paper we have used Wavelet techniques which overcome this limitation by transforming low-frequency components of error into high frequency components which consequently can be removed quickly by using the Gauss-Seidel relaxation method. In Discrete Wavelet Transform (DWT two operators, decomposition (analysis and reconstruction (synthesis, are used. In the decomposition stage an image is separated into one low-frequency component (approximation and three high-frequency components (details. In the reconstruction stage, the image is reconstructed by synthesizing the approximated and detail components. We tested our algorithm on both simulated and real data and on both unweighted and weighted forms of discretized Poisson's equation. The experimental results show the effectiveness of the proposed method.
Application of Stochastic Partial Differential Equations to Reservoir Property Modelling
Potsepaev, R.
2010-09-06
Existing algorithms of geostatistics for stochastic modelling of reservoir parameters require a mapping (the \\'uvt-transform\\') into the parametric space and reconstruction of a stratigraphic co-ordinate system. The parametric space can be considered to represent a pre-deformed and pre-faulted depositional environment. Existing approximations of this mapping in many cases cause significant distortions to the correlation distances. In this work we propose a coordinate free approach for modelling stochastic textures through the application of stochastic partial differential equations. By avoiding the construction of a uvt-transform and stratigraphic coordinates, one can generate realizations directly in the physical space in the presence of deformations and faults. In particular the solution of the modified Helmholtz equation driven by Gaussian white noise is a zero mean Gaussian stationary random field with exponential correlation function (in 3-D). This equation can be used to generate realizations in parametric space. In order to sample in physical space we introduce a stochastic elliptic PDE with tensor coefficients, where the tensor is related to correlation anisotropy and its variation is physical space.
Mathematical analysis of partial differential equations modeling electrostatic MEMS
Esposito, Pierpaolo; Guo, Yujin
2010-01-01
Micro- and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniature-size mechanical devices, are essential components of modern technology. It is the mathematical model describing "electrostatically actuated" MEMS that is addressed in this monograph. Even the simplified models that the authors deal with still lead to very interesting second- and fourth-order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). While nonlinear eigenvalue problems-where the stationary MEMS models fit-are a well-developed
Isostable reduction with applications to time-dependent partial differential equations.
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.
Isostable reduction with applications to time-dependent partial differential equations
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.
The Stampacchia maximum principle for stochastic partial differential equations and applications
Chekroun, Mickaël D.; Park, Eunhee; Temam, Roger
2016-02-01
Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for SPDEs are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction-diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee-Infante equation perturbed by a nonlinear noise are also derived.
Nonlocal symmetry generators and explicit solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Qin Maochang [School of Science, Chongqing Technology and Business University, Chongqing 400067 (China)
2007-04-27
The nonlocal symmetry of a partial differential equation is studied in this paper. The partial differential equation written as a conservation law can be transformed into an equivalent system by introducing a suitable potential. The nonlocal symmetry group generators of original partial differential equations can be obtained through their equivalent system. Further, new explicit solutions can be constructed from the newly obtained symmetry generators. The Burgers equation is chosen as an example; many new valuable explicit solutions and nonlocal symmetry generators are presented.
Stochastic partial differential equations a modeling, white noise functional approach
Holden, Helge; Ubøe, Jan; Zhang, Tusheng
1996-01-01
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in r...
Composition of processes and related partial differential equations
D'Ovidio, Mirko
2010-01-01
In this paper different types of compositions involving independent fractional Brownian motions $B^j_{H_j}(t)$, $t>0$, $j=1,2$ are examined. The partial differential equations governing the distributions of $I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|)$, $t>0$ and $J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1})$, $t>0$ are derived by different methods and compared with those existing in the literature and with those related to $B^1(|B^2_{H_2}(t)|)$, $t>0$. The process of iterated Brownian motion $I^n_F(t)$, $t>0$ is examined in detail and its moments are calculated. Furthermore for $J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H})$, $t>0$ the following factorization is proved $J^{n-1}_F(t)=\\prod_{j=1}^{n} B^j_{\\frac{H}{n}}(t)$, $t>0$. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.
Wavelets, turbulence, and boundary value problems for partial differential equations
Weiss, John E.
1995-04-01
In this paper the qualitative properties of an inviscid, incompressible two-dimensional fluid are examined by numerical methods based on the compactly supported wavelets (the wavelet- Galerkin method). In particular, we examine the behavior of the spatial gradients of the vorticity. The growth of these gradients is related to the transfer of enstrophy (integral of squared vorticity) to the small-scales of the fluid motion. Implicit time differencing and wavelet-Galerkin space discretization allow a direct investigation of the long time behavior of the inviscid fluid. The effects of hyperviscosity on the long time limit are examined. To solve boundary problems we developed a new numerical method for the solution of partial differential equations in nonseparable domains. The method uses a wavelet-Galerkin solver with a nontrivial adaptation of the standard capacitance matrix method. The numerical solutions exhibit spectral convergence with regard to the order of the compactly supported, Daubechies wavelet basis. Furthermore, the rate of convergence is found to be independent of the geometry. We solve the Helmholtz equation since, for the indefinite case, the solutions have qualitative properties that well illustrate the applications of our method.
BOOK REVIEW: Partial Differential Equations in General Relativity
Choquet-Bruhat, Yvonne
2008-09-01
General relativity is a physical theory basic in the modeling of the universe at the large and small scales. Its mathematical formulation, the Einstein partial differential equations, are geometrically simple, but intricate for the analyst, involving both hyperbolic and elliptic PDE, with local and global problems. Many problems remain open though remarkable progress has been made recently towards their solutions. Alan Rendall's book states, in a down-to-earth form, fundamental results used to solve different types of equations. In each case he gives applications to special models as well as to general properties of Einsteinian spacetimes. A chapter on ODE contains, in particular, a detailed discussion of Bianchi spacetimes. A chapter entitled 'Elliptic systems' treats the Einstein constraints. A chapter entitled 'Hyperbolic systems' is followed by a chapter on the Cauchy problem and a chapter 'Global results' which contains recently proved theorems. A chapter is dedicated to the Einstein Vlasov system, of which the author is a specialist. On the whole, the book surveys, in a concise though precise way, many essential results of recent interest in mathematical general relativity, and it is very clearly written. Each chapter is followed by an up to date bibliography. In conclusion, this book will be a valuable asset to relativists who wish to learn clearly-stated mathematical results and to mathematicians who want to penetrate into the subtleties of general relativity, as a mathematical and physical theory.
A constrained backpropagation approach for the adaptive solution of partial differential equations.
Rudd, Keith; Di Muro, Gianluca; Ferrari, Silvia
2014-03-01
This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multipliers, CPROP solves the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination. As a result, CPROP reduces the dimensionality of the optimization problem, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic and parabolic PDEs with changing parameters and nonhomogeneous terms.
Multiscale functions, scale dynamics, and applications to partial differential equations
Cresson, Jacky; Pierret, Frédéric
2016-05-01
Modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism, randomness, and differentiability. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following: "With a finite set of data concerning a phenomenon, can we recover its underlying nature? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus, and scale dynamics based on the time scale calculus [see Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, 2001)] which is used to introduce the notion of scale equations. These definitions will be illustrated on the multi-scale Okamoto's functions. Scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime. The introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant. Typical examples of such equations are given by the scale Euler-Lagrange equation. We illustrate our results using the scale Newton's equation which gives rise to a non-linear diffusion equation or a non-linear Schrödinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered.
Grothaus, Martin
2012-01-01
In this paper a length-conserving numerical scheme for a nonlinear fourth order system of partial differential algebraic equations arising in technical textile industry is studied. Applying a semidiscretization in time, the resulting sequence of nonlinear elliptic systems with algebraic constraint is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is performed by projected gradient methods on finite element spaces which determine the computational effort and approximation quality of the algorithm. Simulation results are presented and discussed in view of the application of an elastic inextensible fiber motion.
Institute of Scientific and Technical Information of China (English)
侯祥林; 刘铁林; 翟中海
2011-01-01
针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法.%For elliptic nonlinear partial differential equations with boundary value problem, based on difference method and dynamic design variable optimization method, by taking unknown function value on discrete net point as design variables, difference equation of all the discrete net points is constructed as an objective function. A kind of optimization algorithm about solving unknown function value on discrete net point is proposed. Universal computing program is designed. Practical example is analyzed. By comparing the computing result with the analytical solution, effectiveness and feasibility are verified. Thus complicated nonlinear mathematical physics equations can be solved by the numerical calculation method.
Partial differential equations of mathematical physics and integral equations
Guenther, Ronald B
1996-01-01
This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t
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Verhoeven, C; Musette, M [Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels (Belgium)
2003-02-28
In this letter, we analyse two bidirectional sixth-order partial differential equations, which are reductions in (1 + 1) dimensions of equations belonging to the KP hierarchy. They have fourth-order and fifth-order Lax pairs, respectively. We derive their Baecklund transformations and, from the nonlinear superposition formula, we can build their soliton solutions like a Grammian. The interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head-on collision of solitary waves of different type and shape. (letter to the editor)
Fourth-order partial differential equations for noise removal.
You, Y L; Kaveh, M
2000-01-01
A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.
Grid generation for the solution of partial differential equations
Eiseman, Peter R.; Erlebacher, Gordon
1989-01-01
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.
Final Report: Symposium on Adaptive Methods for Partial Differential Equations
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Pernice, M.; Johnson, C.R.; Smith, P.J.; Fogelson, A.
1998-12-10
OAK-B135 Final Report: Symposium on Adaptive Methods for Partial Differential Equations. Complex physical phenomena often include features that span a wide range of spatial and temporal scales. Accurate simulation of such phenomena can be difficult to obtain, and computations that are under-resolved can even exhibit spurious features. While it is possible to resolve small scale features by increasing the number of grid points, global grid refinement can quickly lead to problems that are intractable, even on the largest available computing facilities. These constraints are particularly severe for three dimensional problems that involve complex physics. One way to achieve the needed resolution is to refine the computational mesh locally, in only those regions where enhanced resolution is required. Adaptive solution methods concentrate computational effort in regions where it is most needed. These methods have been successfully applied to a wide variety of problems in computational science and engineering. Adaptive methods can be difficult to implement, prompting the development of tools and environments to facilitate their use. To ensure that the results of their efforts are useful, algorithm and tool developers must maintain close communication with application specialists. Conversely it remains difficult for application specialists who are unfamiliar with the methods to evaluate the trade-offs between the benefits of enhanced local resolution and the effort needed to implement an adaptive solution method.
Shape Morphing of Complex Geometries Using Partial Differential Equations
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Gabriela González Castro
2007-11-01
Full Text Available An alternative technique for shape morphing using a surface generating method using partial differential equations is outlined throughout this work. The boundaryvalue nature that is inherent to this surface generation technique together with its mathematical properties are hereby exploited for creating intermediate shapes between an initial shape and a final one. Four alternative shape morphing techniques are proposed here. The first one is based on the use of a linear combination of the boundary conditions associated with the initial and final surfaces, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. The fourth of these alternatives is based on the manipulation of the spine of the surfaces, which is computed as a by-product of the solution. Results of morphing sequences between two topologically nonequivalent surfaces are presented. Thus, it is shown that the PDE based approach for morphing is capable of obtaining smooth intermediate surfaces automatically in most of the methodologies presented in this work and the spine has been revealed as a powerful tool for morphing surfaces arising from the method proposed here.
Partial differential equations an accessible route through theory and applications
Vasy, András
2015-01-01
This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses. The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the fea...
Preconditioning for partial differential equation constrained optimization with control constraints
Stoll, Martin
2011-10-18
Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal-dual active set method. We also consider the Moreau-Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. © 2011 John Wiley & Sons, Ltd.
The Application of Partial Differential Equations in Medical Image Processing
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Mohammad Madadpour Inallou
2013-10-01
Full Text Available Mathematical models are the foundation of biomedical computing. Partial Differential Equations (PDEs in Medical Imaging is concerned with acquiring images of the body for research, diagnosis and treatment. Biomedical Image Processing and its influence has undergoing a revolution in the past decade. Image processing has become an important component in contemporary science and technology and has been an interdisciplinary research field attracting expertise from applied mathematics, biology, computer sciences, engineering, statistics, microscopy, radiologic sciences, physics, medicine and etc. Medical imaging equipment is taking on an increasingly critical role in healthcare as the industry strives to lower patient costs and achieve earlier disease prediction using noninvasive means. The subsections of medical imaging are categorized to two: Conventional (X-Ray and Ultrasound and Computed (CT, MRI, fMRI, SPECT, PET and etc. This paper is organized as fallow: First section describes some kind of image processing. Second section is about techniques and requirements, and in the next sections the proceeding of Analyzing, Smoothing, Segmentation, De-noising and Registration in Medical Image Processing Equipment by PDEs Framework will be regarded
Fan sub-equation method for Wick-type stochastic partial differential equations
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Zhang Sheng, E-mail: zhshaeng@yahoo.com.c [Department of Mathematics, Bohai University, Jinzhou 121013 (China); School of Mathematical Sciences, Dalian University of Technology, Dalian 116024 (China); Zhang Hongqing [School of Mathematical Sciences, Dalian University of Technology, Dalian 116024 (China)
2010-09-13
An improved algorithm is devised for using Fan sub-equation method to solve Wick-type stochastic partial differential equations. Applying the improved algorithm to the Wick-type generalized stochastic KdV equation, we obtain more general Jacobi and Weierstrass elliptic function solutions, hyperbolic and trigonometric function solutions, exponential function solutions and rational solutions.
On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations
Dubrovin, Boris; Grava, Tamara; Klein, Christian; Moro, Antonio
2015-06-01
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P) equation or its fourth-order analogue P. As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
control theory to systems described by partial differential equations. The intent is not to advance the theory of partial differential equations per se. Thus all considerations will be restricted to the more familiar equations of the type which often occur in mathematical physics. Specifically, the distributed parameter systems under consideration are represented by a set of field
Schiesser, William E
2014-01-01
Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the com
Asymptotics for dissipative nonlinear equations
Hayashi, Nakao; Kaikina, Elena I; Shishmarev, Ilya A
2006-01-01
Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We give an equivalent construction of the infinitesimal time translation operator for partial differential evolution equation in the algebraic dynamics algorithm proposed by Shun-Jin Wang and his students. Our construction involves only simple partial differentials and avoids the derivative terms of δ function which appear in the course of computation by means of Wang-Zhang operator. We prove Wang’s equivalent theorem which says that our construction and Wang-Zhang’s are equivalent. We use our construction to deal with several typical equations such as nonlinear advection equation, Burgers equation, nonlinear Schrodinger equation, KdV equation and sine-Gordon equation, and obtain at least second order approximate solutions to them. These equations include the cases of real and complex field variables and the cases of the first and the second order time derivatives.
Liu, Chengshi
2010-08-01
We give an equivalent construction of the infinitesimal time translation operator for partial differential evolution equation in the algebraic dynamics algorithm proposed by Shun-Jin Wang and his students. Our construction involves only simple partial differentials and avoids the derivative terms of δ function which appear in the course of computation by means of Wang-Zhang operator. We prove Wang’s equivalent theorem which says that our construction and Wang-Zhang’s are equivalent. We use our construction to deal with several typical equations such as nonlinear advection equation, Burgers equation, nonlinear Schrodinger equation, KdV equation and sine-Gordon equation, and obtain at least second order approximate solutions to them. These equations include the cases of real and complex field variables and the cases of the first and the second order time derivatives.
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Petzold, L; Cao, Y; Li, S; Serban, R
2005-08-09
Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.
SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN IRn
Institute of Scientific and Technical Information of China (English)
TANG SHANJIAN
2005-01-01
This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.
Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form
Directory of Open Access Journals (Sweden)
Phil Broadbridge
2008-09-01
Full Text Available The entropy evolution behaviour of a partial differential equation (PDE in conservation form, may be readily discerned from the sign of the local source term of Shannon information density. This can be easily used as a diagnostic tool to predict smoothing and non-smoothing properties, as well as positivity of solutions with conserved mass. The familiar fourth order diffusion equations arising in applications do not have increasing Shannon entropy. However, we obtain a new class of nonlinear fourth order diffusion equations that do indeed have this property. These equations also exhibit smoothing properties and they maintain positivity. The counter-intuitive behaviour of fourth order diffusion, observed to occur or not occur on an apparently ad hoc basis, can be predicted from an easily calculated entropy production rate. This is uniquely defined only after a technical definition of the irreducible source term of a reaction diffusion equation.
Modeling boyciana-fish-human interaction with partial differential algebraic equations.
Jiang, Yushan; Zhang, Qingling; Wang, Haiyan
2016-07-01
Under the influence of human population distribution, the boyciana-fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system's persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population.
Tang, Chen; Wang, Linlin; Yan, Haiqing
2012-07-10
In this paper, we first present the general description for partial differential equations (PDEs) based image processing methods, including the basic idea, the main advantages and disadvantages, a few representative PDE models, and the derivation of PDE models. Then we review our contributions on PDE-based anisotropic filtering methods for electronic speckle pattern interferometry, including the second-order, fourth-order, and coupled nonoriented PDE filtering models and the second-order and coupled nonlinear oriented PDE filtering models. We have summarized the features of each model.
Griffiths, Graham
2010-01-01
Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by p
A NUMERICAL SOLUTION OF A SYSTEM OF LINEAR INTEGRO-PARTIAL DIFFERENTIAL EQUATIONS,
The numerical solution to a system of linear integro- partial differential equations is treated. A numerical solution to the system was obtained by...using difference approximations to the partial differential equations . To assure convergence, a stability condition derived from the related plate
A note on the Dirichlet problem for model complex partial differential equations
Ashyralyev, Allaberen; Karaca, Bahriye
2016-08-01
Complex model partial differential equations of arbitrary order are considered. The uniqueness of the Dirichlet problem is studied. It is proved that the Dirichlet problem for higher order of complex partial differential equations with one complex variable has infinitely many solutions.
Directory of Open Access Journals (Sweden)
Magdy A. El-Tawil
2012-10-01
Full Text Available In this paper, the random finite difference method with three points is used in solving random partial differential equations problems mainly: random parabolic, elliptic and hyperbolic partial differential equations. The conditions of the mean square convergence of the numerical solutions are studied. The numerical solutions are computed through some numerical case studies.
Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations
Bhowmik, S.K.; Stolk, C.C.
2011-01-01
We investigate the application of windowed Fourier frames to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given
In-out intermittency in partial differential equation and ordinary differential equation models.
Covas, Eurico; Tavakol, Reza; Ashwin, Peter; Tworkowski, Andrew; Brooke, John M.
2001-06-01
We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities. (c) 2001 American Institute of Physics.
Institute of Scientific and Technical Information of China (English)
Yang Jun; Wang Chunyan; Li Jing; Meng Zhijuan
2006-01-01
This paper is concerned with the oscillations of neutral hyperbolic partial differential equations with delays. Necessary and sufficient conditions are obtained for the oscillations of all solutions of the equations, and these results are illustrated by some examples.
On Complex Singularity Analysis for Some Linear Partial Differential Equations in
Directory of Open Access Journals (Sweden)
A. Lastra
2013-01-01
Full Text Available We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011. The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.
Partial Differential Equations of an Epidemic Model with Spatial Diffusion
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El Mehdi Lotfi
2014-01-01
Full Text Available The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.
Partial differential equations II elements of the modern theory equations with constant coefficients
Shubin, M
1994-01-01
This book, the first printing of which was published as Volume 31 of the Encyclopaedia of Mathematical Sciences, contains a survey of the modern theory of general linear partial differential equations and a detailed review of equations with constant coefficients. Readers will be interested in an introduction to microlocal analysis and its applications including singular integral operators, pseudodifferential operators, Fourier integral operators and wavefronts, a survey of the most important results about the mixed problem for hyperbolic equations, a review of asymptotic methods including short wave asymptotics, the Maslov canonical operator and spectral asymptotics, a detailed description of the applications of distribution theory to partial differential equations with constant coefficients including numerous interesting special topics.
Toomarian, N.; Fijany, A.; Barhen, J.
1993-01-01
Evolutionary partial differential equations are usually solved by decretization in time and space, and by applying a marching in time procedure to data and algorithms potentially parallelized in the spatial domain.
On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations
ElHassan ElJaoui; Said Melliani
2016-01-01
We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.
On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations
Directory of Open Access Journals (Sweden)
ElHassan ElJaoui
2016-01-01
Full Text Available We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.
A Distributed Problem Solving Environment (PSE) for Partial Differential Equation Based Problems
National Research Council Canada - National Science Library
TERAMOTO, Takayuki; NAKAMURA, Takashi; KAWATA, Shigeo; MATIDE, Syunsuke; HAYASAKA, Koji; NONAKA, Hidetaka; SASAKI, Eiji; SANADA, Yasuhiro
2001-01-01
...) for partial differential equation (PDE) based problems. The system inputs a problem information including a discretization and computation scheme, and outputs a program flow and also a C-language source code for the problem...
3rd International Conference on Particle Systems and Partial Differential Equations
Soares, Ana
2016-01-01
The main focus of this book is on different topics in probability theory, partial differential equations and kinetic theory, presenting some of the latest developments in these fields. It addresses mathematical problems concerning applications in physics, engineering, chemistry and biology that were presented at the Third International Conference on Particle Systems and Partial Differential Equations, held at the University of Minho, Braga, Portugal in December 2014. The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, providing a venue for them to present their latest findings and discuss their areas of expertise. Further, it was intended to introduce a vast and varied public, including young researchers, to the subject of interacting particle systems, its underlying motivation, and its relation to partial differential equations. This book will appeal to probabilists, analysts and those mathematicians whose wor...
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Khovanskii, A G [Institute for System Analysis , Russian Academy of Sciences, Moscow (Russian Federation); Chulkov, S P [Independent University of Moscow, Moscow (Russian Federation)
2006-02-28
We consider systems of linear partial differential equations with analytic coefficients and discuss existence and uniqueness theorems for their formal and analytic solutions. Using elementary methods, we define and describe an analogue of the Hilbert polynomial for such systems.
Quasi-sure Limit Theorem of Parabolic Stochastic Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
Xi Cheng ZHANG
2004-01-01
In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions, by the way, we obtain the quasi-sure continuity of the solution.
Zhang, Xu
2010-01-01
The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations.
Institute of Scientific and Technical Information of China (English)
WANG Yu-lan; CHAO Lu
2008-01-01
How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.
Application of Variational Iteration Method to Fractional Hyperbolic Partial Differential Equations
Directory of Open Access Journals (Sweden)
Fadime Dal
2009-01-01
Full Text Available The solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iteration method. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equations. Application of variational iteration technique to this problem has shown the rapid convergence of the sequence constructed by this method to the exact solution.
Oaku, Toshinori
1986-01-01
We give a general formulation of boundary value problems in the framework of hyperfunctions both for systems of linear partial differential equations with non-characteristic boundary and for Fuchsian systems of partial differential equations in a unified manner. We also give a microlocal formulation, which enables us to prove new results on propagation of micro-analyticity up to the boundary for solutions of systems micro-hyperbolic in a weak sense.
Analytic method for solitary solutions of some partial differential equations
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Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya@firat.edu.tr
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
Separation of variables for partial differential equations an eigenfunction approach
Cain, George
2005-01-01
Potential, Heat, and Wave Equations. Basic Approximation Theory. Sturm-Liouville Problems. Fourier Series. Eigenfunction Expansions for Equations in Two Independent Variables. One-Dimensional Diffusion Equation. One-Dimensional Wave Equation. Potential Problems in the Plane. Multidimensional Problems. Bibliography. Index.
Intuitive Understanding of Solutions of Partially Differential Equations
Kobayashi, Y.
2008-01-01
This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…
Intuitive Understanding of Solutions of Partially Differential Equations
Kobayashi, Y.
2008-01-01
This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…
Arshad, Muhammad; Lu, Dianchen; Wang, Jun
2017-07-01
In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order partial differential equations (PDEs) can be resolved effectively. The most distinct aspect of this method is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. We utilize the proposed scheme on some initial value problems and approximate numerical solutions of linear and nonlinear time fractional PDEs are obtained, which shows that the method is highly accurate and simple to apply. The proposed technique is thus an influential technique for solving the fractional PDEs and fractional order problems occurring in the field of engineering, physics etc. Numerical results are obtained for verification and demonstration purpose by using Mathematica software.
Energy Technology Data Exchange (ETDEWEB)
Gauthier-Villars [ed.
1998-12-31
This book is dedicated to the French mathematician J.L.Lions. It represents a compilation of articles from about 80 authors. The topics treated are diverse but the more or less commune matter is the study of the characteristics of some partial differential equations. Stability, optimal approximation, numerical resolution, particular applications are among the subjects reviewed. An article deals with the MHD stability of fusion plasmas in tokamaks, another presents the scientific and technical challenges of nuclear energy in France. The latter that contains no equations can be considered as an enjoyable break in a sea of about 40 mathematical articles. (A.C.)
Institute of Scientific and Technical Information of China (English)
Qingfeng ZHU; Yufeng SHI
2012-01-01
Backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP) with non-Lipschitz coefficients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.
Finite element wavelets for solving partial differential equations
Nguyen, Hoang
2005-01-01
This thesis deals with the application of wavelet bases for the numerical solution of operator equations, as boundary value problems and boundary integral equations. The use of suitable wavelet bases has the advantage that the arising stiffness matrices are well-conditioned uniformly in their sizes,
2010-01-01
The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global un...
Relaxation approximations and bounded variation estimates for some partial differential equations
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Francisco Caicedo
2002-02-01
Full Text Available In this paper, we introduce a new technique for studying solutions of bounded variation for some conservation laws of first order partial differential equations and for some degenerate parabolic equations in multi-dimensional space. The connection between these two types of equations is the vanishing relaxation method.
A first course in partial differential equations with complex variables and transform methods
Weinberger, H F
1995-01-01
Suitable for advanced undergraduate and graduate students, this text presents the general properties of partial differential equations, including the elementary theory of complex variables. Topics include one-dimensional wave equation, properties of elliptic and parabolic equations, separation of variables and Fourier series, nonhomogeneous problems, and analytic functions of a complex variable. Solutions. 1965 edition.
An Effective Method for Seeking Conservation Laws of Partial Differential Equations
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QIN Mao-Chang; MEI Feng-Xiang; FAN Gui-Hong
2006-01-01
This paper introduces an effective method for seeking localconservation laws of general partial differential equations (PDEs). The well-known variational principle does not involve in this method. Alternatively, the conservation laws can be derived from symmetries, which include the symmetries of the associated linearized equation of the PDEs,and the adjoint symmetries of the adjoint equation of the PDEs.
OSCILLATION OF IMPULSIVE HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH DELAY
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, oscillation properties of the solutions of impulsive hyperbolic equation with delay are investigated via the method of differential inequalities. Sufficient conditions for oscillations of the solutions are established.
Partial differential equations in action complements and exercises
Salsa, Sandro
2015-01-01
This textbook presents problems and exercises at various levels of difficulty in the following areas: Classical Methods in PDEs (diffusion, waves, transport, potential equations); Basic Functional Analysis and Distribution Theory; Variational Formulation of Elliptic Problems; and Weak Formulation for Parabolic Problems and for the Wave Equation. Thanks to the broad variety of exercises with complete solutions, it can be used in all basic and advanced PDE courses.
Multiscale functions, Scale dynamics and Applications to partial differential equations
Cresson, Jacky
2015-01-01
Modeling phenomena from experimental data, always begin with a \\emph{choice of hypothesis} on the observed dynamics such as \\emph{determinism}, \\emph{randomness}, \\emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \\emph{"With a finite set of data concerning a phenomenon, can we recover its underlying nature ?} From this problem, we introduce in this paper the definition of \\emph{multi-scale functions}, \\emph{scale calculus} and \\emph{scale dynamics} based on the \\emph{time-scale calculus} (see \\cite{bohn}). These definitions will be illustrated on the \\emph{multi-scale Okamoto's functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \\emph{scale regimes} whereas the equation is \\emph{scale invariant}. A typical example of such an equation, is the \\emph{Euler-Lagrange equation} and particularly the \\emph{Newton's equation} ...
Pettersson, Mass Per; Nordström, Jan
2015-01-01
This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The approach described in the text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dime...
Lectures on Cauchy's problem in linear partial differential equations
Hadamard, Jacques
2003-01-01
Would well repay study by most theoretical physicists."" - Physics Today""An overwhelming influence on subsequent work on the wave equation."" - Science Progress""One of the classical treatises on hyperbolic equations."" - Royal Naval Scientific ServiceDelivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbol
"Pulling out" as a procedural resource when solving partial differential equations
Modir, Bahar
2016-01-01
We investigate how students solve partial differential equations and partial derivatives in the context of quantum mechanics. We use the resources framework to investigate students' discussion in a group problem-solving environment to investigate the fine-grain elements of their problem solving. We analyze an example of students' use of separation of variables to solve a partial differential equation for a free particle problem. We identify a mathematical action called "pulling out" as a procedural resource to help students with separating the time part from the space part of the wave function in the course of solving the time-dependent Schrodinger equation. We discuss how students use "pulling out" as a procedural step in solving partial differential equations and sense-making.
Structure analysis of growing network based on partial differential equations
Junbo JIA; Jin, Zhen
2016-01-01
The topological structure is one of the most important contents in the complex network research. Therein the node degree and the degree distribution are the most basic characteristic quantities to describe topological structure. In order to calculate the degree distribution, first of all, the node degree is considered as a continuous variable. Then, according to the Markov Property of growing network, the cumulative distribution function's evolution equation with time can be obtained. Finally...
Calculus for cognitive scientists partial differential equation models
Peterson, James K
2016-01-01
This book shows cognitive scientists in training how mathematics, computer science and science can be usefully and seamlessly intertwined. It is a follow-up to the first two volumes on mathematics for cognitive scientists, and includes the mathematics and computational tools needed to understand how to compute the terms in the Fourier series expansions that solve the cable equation. The latter is derived from first principles by going back to cellular biology and the relevant biophysics. A detailed discussion of ion movement through cellular membranes, and an explanation of how the equations that govern such ion movement leading to the standard transient cable equation are included. There are also solutions for the cable model using separation of variables, as well an explanation of why Fourier series converge and a description of the implementation of MatLab tools to compute the solutions. Finally, the standard Hodgkin - Huxley model is developed for an excitable neuron and is solved using MatLab.
Ohmori, Shousuke; Yamazaki, Yoshihiro
2016-01-01
Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
REGULARITY THEORY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.
$C_{0}$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain
Jacob, Birgit; Morris, Kirsten; Zwart, Hans
2015-01-01
Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result of this paper is a simple test for $C_{0}$-semigrou
Multigrid for high dimensional elliptic partial differential equations on non-equidistant grids
bin Zubair, H.; Oosterlee, C.E.; Wienands, R.
2006-01-01
This work presents techniques, theory and numbers for multigrid in a general d-dimensional setting. The main focus is the multigrid convergence for high-dimensional partial differential equations (PDEs). As a model problem we have chosen the anisotropic diffusion equation, on a unit hypercube. We
Multigrid for high dimensional elliptic partial differential equations on non-equidistant grids
bin Zubair, H.; Oosterlee, C.E.; Wienands, R.
2006-01-01
This work presents techniques, theory and numbers for multigrid in a general d-dimensional setting. The main focus is the multigrid convergence for high-dimensional partial differential equations (PDEs). As a model problem we have chosen the anisotropic diffusion equation, on a unit hypercube. We pr
RBSDE's with jumps and the related obstacle problems for integral-partial differential equations
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FAN; Yulian
2006-01-01
The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.
The Ergodicity of Stochastic Partial Differential Equations with Lévy Jump
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Guo Li ZHOU; Zhen Ting HOU
2011-01-01
In this article,the authors prove the uniqueness in law of a class of stochastic equations in infinite dimension,then we apply it to establish the existence and uniqueness of invariant measure of the generalized stochastic partial differential equation perturbed by Lévy process.
Subich, Christopher J.
2015-08-01
This work extends the machinery of the moving mesh partial differential equation (MMPDE) method to the spectral collocation discretization of time-dependent partial differential equations. Unlike previous approaches which bootstrap the moving grid from a lower-order, finite-difference discretization, this work uses a consistent spectral collocation discretization for both the grid movement problem and the underlying, physical partial differential equation. Additionally, this work develops an error monitor function based on filtering in the spectral domain, which concentrates grid points in areas of locally poor resolution without relying on an assumption of locally steep gradients. This makes the MMPDE method more robust in the presence of rarefaction waves which feature rapid change in higher-order derivatives.
Mobile point sensors and actuators in the controllability theory of partial differential equations
Khapalov, Alexander Y
2017-01-01
This book presents a concise study of controllability theory of partial differential equations when they are equipped with actuators and/or sensors that are finite dimensional at every moment of time. Based on the author’s extensive research in the area of controllability theory, this monograph specifically focuses on the issues of controllability, observability, and stabilizability for parabolic and hyperbolic partial differential equations. The topics in this book also cover related applied questions such as the problem of localization of unknown pollution sources based on information obtained from point sensors that arise in environmental monitoring. Researchers and graduate students interested in controllability theory of partial differential equations and its applications will find this book to be an invaluable resource to their studies.
Wavelet transforms as solutions of partial differential equations
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Zweig, G.
1997-10-01
This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). Wavelet transforms are useful in representing transients whose time and frequency structure reflect the dynamics of an underlying physical system. Speech sound, pressure in turbulent fluid flow, or engine sound in automobiles are excellent candidates for wavelet analysis. This project focused on (1) methods for choosing the parent wavelet for a continuous wavelet transform in pattern recognition applications and (2) the more efficient computation of continuous wavelet transforms by understanding the relationship between discrete wavelet transforms and discretized continuous wavelet transforms. The most interesting result of this research is the finding that the generalized wave equation, on which the continuous wavelet transform is based, can be used to understand phenomena that relate to the process of hearing.
On positivity of certain systems of partial differential equations.
Parmeggiani, Alberto
2007-01-16
We extend the Fefferman-Phong inequality to N x N systems of PDEs with symbol p(x, xi) = A(x)e(xi) + B(x, xi) + C(x) = p(x, xi)* > or = -cl, (x, xi)[abstract: see text] where e is a positive homogeneous quadratic form, and B(x, xi) = Sigma(l=1)(n) B(l)(x)xi(l). We thus generalize a result by L.-Y. Sung that was obtained for systems of ordinary differential equations. Our proof exploits a Calderón-Zygmund decomposition of the phase-space )[abstract: see text] of the kind introduced by C. Fefferman and D. H. Phong for studying subelliptic differential operators and goes by induction on the size N of the system.
Bui-Thanh, Tan
2015-08-01
By revisiting the basic Godunov approach for system of linear hyperbolic Partial Differential Equations (PDEs) we show that it is hybridizable. As such, it is a natural recipe for us to constructively and systematically establish a unified hybridized discontinuous Galerkin (HDG) framework for a large class of PDEs including those of Friedrichs' type. The unification is fourfold. First, it provides a single constructive procedure to devise HDG schemes for elliptic, parabolic, hyperbolic, and mixed-type PDEs. The key that we exploit is the fact that, for many PDEs, irrespective of their type, the first order form is a hyperbolic system. Second, it reveals the nature of the trace unknowns as the upwind states. Third, it provides a parameter-free HDG framework, and hence eliminating the "usual complaint" that HDG is a parameter-dependent method. Fourth, it allows us to rediscover most existing HDG methods and furthermore discover new ones. We apply the proposed unified framework to three different PDEs: the convection-diffusion-reaction equation, the Maxwell equation in frequency domain, and the Stokes equation. The purpose is to present a step-by-step construction of various HDG methods, including the most economic ones with least trace unknowns, by exploiting the particular structure of the underlying PDEs. The well-posedness of the resulting HDG schemes, i.e. the existence and uniqueness of the HDG solutions, is proved. The well-posedness result is also extended and proved for abstract Friedrichs' systems. We also discuss variants of the proposed unified framework and extend them to the popular Lax-Friedrichs flux and to nonlinear PDEs. Numerical results for transport equation, convection-diffusion equation, compressible Euler equation, and shallow water equation are presented to support the unification framework.
Partial differential equations-based segmentation for radiotherapy treatment planning.
Gibou, Frederic; Levy, Doron; Cardenas, Carlos; Liu, Pingyu; Boyer, Arthur
2005-04-01
The purpose of this study is to develop automatic algorithms for the segmentation phase of radiotherapy treatment planning. We develop new image processing techniques that are based on solving a partial diferential equation for the evolution of the curve that identifies the segmented organ. The velocity function is based on the piecewise Mumford-Shah functional. Our method incorporates information about the target organ into classical segmentation algorithms. This information, which is given in terms of a three- dimensional wireframe representation of the organ, serves as an initial guess for the segmentation algorithm. We check the performance of the new algorithm on eight data sets of three diferent organs: rectum, bladder, and kidney. The results of the automatic segmentation were compared with a manual seg- mentation of each data set by radiation oncology faculty and residents. The quality of the automatic segmentation was measured with the k-statistics", and with a count of over- and undersegmented frames, and was shown in most cases to be very close to the manual segmentation of the same data. A typical segmentation of an organ with sixty slices takes less than ten seconds on a Pentium IV laptop.
Early-warning signs for pattern-formation in stochastic partial differential equations
Gowda, Karna; Kuehn, Christian
2015-05-01
There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. In particular, we study stochastic versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a scaling law of the covariance operator in a regime where linearization is expected to be a good approximation for the local fluctuations around deterministic steady states. We compare these results to direct numerical simulation, and study the influence of noise level, noise color, distance to bifurcation and domain size on early-warning signs.
Lamb, George L
1995-01-01
INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. With Emphasis on Wave Propagation and Diffusion. This is the ideal text for students and professionals who have some familiarity with partial differential equations, and who now wish to consolidate and expand their knowledge. Unlike most other texts on this topic, it interweaves prior knowledge of mathematics and physics, especially heat conduction and wave motion, into a presentation that demonstrates their interdependence. The result is a superb teaching text that reinforces the reader's understanding of both mathematics and physic
Tang, Chen; Zhang, Fang; Yan, Haiqing; Chen, Zhanqing
2006-04-01
Denoising in electronic speckle pattern interferometry fringes is the key problem in electronic speckle pattern interferometry. We present the new filtering method based on partial differential equations (called PDE filtering method) to electronic speckle pattern interferometry fringes. The PDE filtering method transforms the image processing to solving the partial differential equations. We test the proposed method on experimentally obtained electronic speckle pattern interferometry fringes, and compare with traditional mean filtering and low-pass Fourier filtering methods. The experimental results show that the technique is capable of effectively removing noise. The PDE filtering method is flexible and has fast computational speed and stable results.
Kudryashov, Nikolay A.; Volkov, Alexandr K.
2017-01-01
We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.
Directory of Open Access Journals (Sweden)
Yi-Fei Pu
2013-01-01
Full Text Available The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.
Institute of Scientific and Technical Information of China (English)
Hongmei Zhang; Fawang Liu
2007-01-01
In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.
Farshid Mirzaee; Mohammad Komak Yari
2016-01-01
In this paper, we introduce three-dimensional fuzzy differential transform method and we utilize it to solve fuzzy partial differential equations. This technique is a successful method because of reducing such problems to solve a system of algebraic equations; so, the problem can be solved directly. A considerable advantage of this method is to obtain the analytical solutions if the equation has an exact solution that is a polynomial function. Numerical examples are included to demonstrate th...
On Exact Solutions to Partial Differential Equations by the Modified Homotopy Perturbation Method
Institute of Scientific and Technical Information of China (English)
Gang YANG; Ru-yun CHEN; Luo-gen YAO
2012-01-01
Based on the modified homotopy perturbation method (MHPM),exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions.Under suitable initial conditions,the PDE is transformed into an ODE.Some illustrative examples reveal the efficiency of the proposed method.
Lagrange's early contributions to the theory of first-order partial differential equations
Engelsman, S.B.
1980-01-01
In 1776, J. L. Lagrange gave a definition of the concept of a “complete solution” of a first-order partial differential equation. This definition was entirely different from the one given earlier by Euler. One of the sources for Lagrange's reformulation of this concept can be found in his attempt to
Houwen, P.J. van der; Sommeijer, B.P.; Wubs, F.W.
1990-01-01
A smoothing technique for the “preconditioning” of the right-hand side of semidiscrete partial differential equations is analyzed. For a parabolic and a hyperbolic model problem, optimal smoothing matrices are constructed which result in a substantial amplification of the maximal stable integration
Directory of Open Access Journals (Sweden)
A. Aghili
2011-12-01
Full Text Available In this work,we present new theorems on two-dimensional Laplace transformation. We also develop some applications based on these results. The two-dimensional Laplace transformation is useful in the solution of non-homogeneous partial differential equations. In the last section a boundary value problem is solved by using the double Laplace-Carson transform.
Introduction to partial differential equations from Fourier series to boundary-value problems
Broman, Arne
2010-01-01
This well-written, advanced-level text introduces students to Fourier analysis and some of its applications. The self-contained treatment covers Fourier series, orthogonal systems, Fourier and Laplace transforms, Bessel functions, and partial differential equations of the first and second orders. Over 260 exercises with solutions reinforce students' grasp of the material. 1970 edition.
Taylor, Lawrence W., Jr.; Rajiyah, H.
1991-01-01
Partial differential equations for modeling the structural dynamics and control systems of flexible spacecraft are applied here in order to facilitate systems analysis and optimization of these spacecraft. Example applications are given, including the structural dynamics of SCOLE, the Solar Array Flight Experiment, the Mini-MAST truss, and the LACE satellite. The development of related software is briefly addressed.
Existence and Stability for Stochastic Partial Differential Equations with Infinite Delay
Directory of Open Access Journals (Sweden)
Jing Cui
2014-01-01
Full Text Available We consider a class of neutral stochastic partial differential equations with infinite delay in real separable Hilbert spaces. We derive the existence and uniqueness of mild solutions under some local Carathéodory-type conditions and also exponential stability in mean square of mild solutions as well as its sample paths. Some known results are generalized and improved.
Variational iteration method for solving partial differential equations with variable coefficients
Energy Technology Data Exchange (ETDEWEB)
Ali, A.H.A. [Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom (Egypt)], E-mail: ahaali_49@yahoo.com; Raslan, K.R. [Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, Cairo (Egypt)], E-mail: kamal_raslan@yahoo.com
2009-05-15
An extremely simple and elementary but rigorous derivation of exact solutions of partial differential equations in different dimensions with variable coefficients is given using the variational iteration method. The efficiency of the considered method is illustrated by some examples. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.
Directory of Open Access Journals (Sweden)
Lucas Jódar
1992-01-01
Full Text Available In this paper coupled implicit initial-boundary value systems of second order partial differential equations are considered. Given a finite domain and an admissible error ϵ an analytic approximate solution whose error is upper bounded by ϵ in the given domain is constructed in terms of the data.
Directory of Open Access Journals (Sweden)
Huabin Chen
2011-01-01
Full Text Available By establishing two Lemmas, the exponential stability and the asymptotical stability for mild solution to the second-order neutral stochastic partial differential equations with infinite delay are obtained, respectively. Our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the obtained results.
Performance and scaling of locally-structured grid methods for partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Colella, Phillip; Bell, John; Keen, Noel; Ligocki, Terry; Lijewski, Michael; Straalen, Brian van [Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720 (United States)
2007-07-15
In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach.
Directory of Open Access Journals (Sweden)
Yi Shen
2013-01-01
Full Text Available We investigate a class of stochastic partial differential equations with Markovian switching. By using the Euler-Maruyama scheme both in time and in space of mild solutions, we derive sufficient conditions for the existence and uniqueness of the stationary distributions of numerical solutions. Finally, one example is given to illustrate the theory.
Taylor, Lawrence W., Jr.; Rajiyah, H.
Partial differential equations for modeling the structural dynamics and control systems of flexible spacecraft are applied here in order to facilitate systems analysis and optimization of these spacecraft. Example applications are given, including the structural dynamics of SCOLE, the Solar Array Flight Experiment, the Mini-MAST truss, and the LACE satellite. The development of related software is briefly addressed.
Rhebergen, S.; Bokhove, O.; Vegt, van der J.J.W.
2007-01-01
We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the formulation is that if the system of nonconservative partial differenti
Well-posedness of boundary-value problems for partial differential equations of even order
Directory of Open Access Journals (Sweden)
Djumaklych Amanov
2014-04-01
Full Text Available In this article, we establish the well-posedness of two boundary value problems for 2k-th order partial differential equations. It is shown that the solvability of these problems depends on the evenness and oddness of the number k.
Well-posedness of boundary-value problems for partial differential equations of even order
Djumaklych Amanov; Allaberen Ashyralyev
2014-01-01
In this article, we establish the well-posedness of two boundary value problems for 2k-th order partial differential equations. It is shown that the solvability of these problems depends on the evenness and oddness of the number k.
STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY MULTI-PARAMETER WHITE NOISE OF LÉVY PROCESSES
Øksendal, Bernt
2007-01-01
We give a short introduction to the white noise theory for multiparameter Lévy processes and its application to stochastic partial differential equations driven by such processes. Examples include temperature distribution with a Lévy white noise heat source, and heat propagation with a multiplicative Lévy white noise heat source.
The automatic solution of partial differential equations using a global spectral method
Townsend, Alex; Olver, Sheehan
2015-10-01
A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree (nx ,ny) is computed in O ((nxny) 3 / 2) operations. Partial differential operators of splitting rank ≥3 are solved via a linear system involving a block-banded matrix in O (min (nx3 ny ,nx ny3)) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under 60 seconds. An experimental implementation in the JULIA language can currently perform the same solve in 10 seconds.
Niang, Oumar; Thioune, Abdoulaye; El Gueirea, Mouhamed Cheikh; Deléchelle, Eric; Lemoine, Jacques
2012-09-01
The major problem with the empirical mode decomposition (EMD) algorithm is its lack of a theoretical framework. So, it is difficult to characterize and evaluate this approach. In this paper, we propose, in the 2-D case, the use of an alternative implementation to the algorithmic definition of the so-called "sifting process" used in the original Huang's EMD method. This approach, especially based on partial differential equations (PDEs), was presented by Niang in previous works, in 2005 and 2007, and relies on a nonlinear diffusion-based filtering process to solve the mean envelope estimation problem. In the 1-D case, the efficiency of the PDE-based method, compared to the original EMD algorithmic version, was also illustrated in a recent paper. Recently, several 2-D extensions of the EMD method have been proposed. Despite some effort, 2-D versions for EMD appear poorly performing and are very time consuming. So in this paper, an extension to the 2-D space of the PDE-based approach is extensively described. This approach has been applied in cases of both signal and image decomposition. The obtained results confirm the usefulness of the new PDE-based sifting process for the decomposition of various kinds of data. Some results have been provided in the case of image decomposition. The effectiveness of the approach encourages its use in a number of signal and image applications such as denoising, detrending, or texture analysis.
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Yunjiao Bai
2015-01-01
Full Text Available The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient. The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods.
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
Babuška, Ivo
2010-01-01
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
Component reuse in iterative solvers for the solution of fuzzy partial differential equations
Corveleyn, Samuel; Vandewalle, Stefan
2009-01-01
We consider elliptic partial differential equations with an uncertain diffusion parameter, where the uncertainty is modelled by fuzzy numbers or a fuzzy field. Our aim is to efficiently compute the fuzzy characteristics of the solution to the fuzzy equation. Using the so-called alpha-cut approach, it is possible to reformulate the fuzzy problems as a long sequence of global optimization problems. Function and gradient evaluations within these optimization problems, differ from each other thro...
On the solution of elliptic partial differential equations on regions with corners
Serkh, Kirill; Rokhlin, Vladimir
2016-01-01
In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.
A Holmgren type theorem for partial differential equations whose coefficients are Gevrey functions
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Masaki Kawagishi
2010-05-01
Full Text Available In this article, we consider a uniqueness theorem of Holmgren type for p-th order Kovalevskaja linear partial differential equations whose coefficients are Gevrey functions. We prove that the only $C^p$-solution to the zero initial-valued problem is the identically zero function. To prove this result we use the uniqueness theorem for higher-order ordinary differential equations in Banach scales.
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Asma Ali Elbeleze
2014-01-01
Full Text Available We are concerned here with singular partial differential equations of fractional order (FSPDEs. The variational iteration method (VIM is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.
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Fanwei Meng
2013-01-01
Full Text Available A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs. The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this method.
Component Reuse in Iterative Solvers for the Solution of Fuzzy Partial Differential Equations
Corveleyn, Samuel; Vandewalle, Stefan
2009-09-01
We consider elliptic partial differential equations with an uncertain diffusion parameter, where the uncertainty is modeled by fuzzy numbers or a fuzzy field. Our aim is to efficiently compute the fuzzy characteristics of the solution to the fuzzy equation. Using the so-called α-cut approach, it is possible to reformulate the fuzzy problem as a long sequence of global optimisation problems. Function and gradient evaluations within these optimisation problems, differ from each other through a possibly small change in one or more of the partial differential equation parameters. In order to reduce the computational complexity of the optimisation problems we consider component reuse in iterative solvers. We concentrate in particular on the reuse of the setup phase in an algebraic multigrid strategy and on reuse of initial approximations.
Mixing Monte-Carlo and Partial Differential Equations for Pricing Options
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Tobias LIPP; Grégoire LOEPER; Olivier PIRONNEAU
2013-01-01
There is a need for very fast option pricers when the financial objects are modeled by complex systems of stochastic differential equations.Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's.It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method,and pricing the underlying asset by a partial differential equation with random coefficients,derived by It(o) calculus.This strategy is investigated for vanilla options,barrier options and American options with stochastic volatilities and jumps optionally.
Parshad, Rana
2011-01-01
We consider a reduced form pricing model for mortgage backed securities, formulated as a non-linear partial differential equation. We prove that the model possesses a weak solution. We then show that under additional regularity assumptions on the initial data, we also have a mild solution. This mild solution is shown to be a strong solution via further regularity arguments. We also numerically solve the reduced model via a Fourier spectral method. Lastly, we compare our numerical solution to real market data. We observe interestingly that the reduced model captures a number of recent market trends in this data, that have escaped previous models.
Large Deviations for Stochastic Partial Differential Equations Driven by a Poisson Random Measure
Budhiraja, Amarjit; Dupuis, Paul
2012-01-01
Stochastic partial differential equations driven by Poisson random measures (PRM) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equations (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative func...
A matrix approach for partial differential equations with Riesz space fractional derivatives
Popolizio, M.
2013-09-01
Fractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach.