Fuzzy differential equations in various approaches
Gomes, Luciana Takata; Bede, Barnabas
2015-01-01
This book may be used as reference for graduate students interested in fuzzy differential equations and researchers working in fuzzy sets and systems, dynamical systems, uncertainty analysis, and applications of uncertain dynamical systems. Beginning with a historical overview and introduction to fundamental notions of fuzzy sets, including different possibilities of fuzzy differentiation and metric spaces, this book moves on to an overview of fuzzy calculus thorough exposition and comparison of different approaches. Innovative theories of fuzzy calculus and fuzzy differential equations using fuzzy bunches of functions are introduced and explored. Launching with a brief review of essential theories, this book investigates both well-known and novel approaches in this field; such as the Hukuhara differentiability and its generalizations as well as differential inclusions and Zadeh’s extension. Through a unique analysis, results of all these theories are examined and compared.
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...
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…
APPROACHED DECISION OF THE DIFFERENTIAL EQUATIONS
Oleksii B. Krasnozhon
2011-02-01
Full Text Available The urgency of the material stated in the article is caused by necessity of development, updating and improvements of methodical operating time on subject matters of issue "Calculus mathematics" which teaching is carried out in conditions of use of information-communication technologies. In the article the program realizations in Mathcad environment of Adams and Runge-Kutt methods of the approached decision of the differential equations are offered; examples on application of the specified methods are brought; the expediency of application of Mathcad environment during mathematical preparation of experts is proved. Perspective directions of the further scientific researches are methodical, mathematical and algorithmic aspects of creation of effective program realizations of numerical methods in Mathcad environment.
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
A New Approach for Solving Fractional Partial Differential Equations
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.
A neuro approach to solve fuzzy Riccati differential equations
Shahrir, Mohammad Shazri; Kumaresan, N.; Kamali, M. Z. M.; Ratnavelu, Kurunathan
2015-10-01
There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.
A neuro approach to solve fuzzy Riccati differential equations
Shahrir, Mohammad Shazri, E-mail: mshazri@gmail.com [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia); Telekom Malaysia, R& D TM Innovation Centre, LingkaranTeknokrat Timur, 63000 Cyberjaya, Selangor (Malaysia); Kumaresan, N., E-mail: drnk2008@gmail.com; Kamali, M. Z. M.; Ratnavelu, Kurunathan [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia)
2015-10-22
There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.
Symmetries of stochastic differential equations: A geometric approach
De Vecchi, Francesco C., E-mail: francesco.devecchi@unimi.it; Ugolini, Stefania, E-mail: stefania.ugolini@unimi.it [Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, Milano (Italy); Morando, Paola, E-mail: paola.morando@unimi.it [DISAA, Università degli Studi di Milano, via Celoria 2, Milano (Italy)
2016-06-15
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.
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
Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach
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.
Stochastic differential equation approach for waves in a random medium.
Dimitropoulos, Dimitris; Jalali, Bahram
2009-03-01
We present a mathematical approach that simplifies the theoretical treatment of electromagnetic localization in random media and leads to closed-form analytical solutions. Starting with the assumption that the dielectric permittivity of the medium has delta-correlated spatial fluctuations, and using Ito's lemma, we derive a linear stochastic differential equation for a one-dimensional random medium. The equation leads to localized wave solutions. The localized wave solutions have a localization length that scales as L approximately omega(-2) for low frequencies whereas in the high-frequency regime this length behaves as L approximately omega(-2/3) .
The Pentabox Master Integrals with the Simplified Differential Equations approach
Papadopoulos, Costas G; Wever, Christopher
2015-01-01
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many $2\\to 3$ amplitudes of interest at the LHC, as for instance three jet production, $\\gamma, V, H +2$ jets etc., based on the Simplified Differential Equations approach.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
Junaid Ali Khan; Muhammad Asif Zahoor Raja; Ijaz Mansoor Qureshi
2011-01-01
@@ We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs).The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error.The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique.The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations.We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods.The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy.With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.%We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
Nonlinear eigenvalue approach to differential Riccati equations for contraction analysis
Kawano, Yu; Ohtsuka, Toshiyuki
2017-01-01
In this paper, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian ma
Barbu, Viorel
2016-01-01
This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.
Fokker-Planck approach to stochastic delay differential equations
Guillouzic, Steve
2001-10-01
Models written in terms of stochastic delay differential equations (SDDE's) have recently appeared in a number of fields, such as physiology, optics, and climatology. Unfortunately, the development of a Fokker-Planck approach for these equations is being hampered by their non-Markovian nature. In this thesis, an exact Fokker- Planck equation (FPE) is formulated for univariate SDDE's involving Gaussian white noise. Although this FPE is not self-sufficient, it is found to be helpful in at least two different contexts: with a short delay approximation and under an appropriate separation of time scales. In the short delay approximation, a Taylor expansion is applied to an SDDE with nondelayed diffusion and yields a nondelayed stochastic differential equation. The aforementioned FPE then allows the derivation of an alternate and complementary approximation of the original SDDE. This method is illustrated with linear and logistic SDDE's. Under the separation of time scales assumption, the FPE of a bistable system is reduced to a form that is uniquely determined by the steady-state probability density when the diffusion term of the SDDE is nondelayed. In the context of an overdamped particle with delayed coupling to a symmetrical and stochastically driven potential, the resulting FPE is used with standard techniques to express the transition rate between wells in terms of the noise amplitude and of the steady-state probability density. The same is also accomplished for the mean first passage time from one point to another. This whole approach is then applied to the case of a quartic potential, for which all realisations eventually stabilise on an oscillatory trajectory with an ever increasing amplitude. Although this latter phenomenon prevents the existence of a steady-state limit, a pseudo- steady-state probability density can be defined and used instead of the non-existent steady-state one when the transition rate to these unbounded oscillatory trajectories is
Tricomi, FG
2013-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff
Numerical Aspects of Solving Differential Equations: Laboratory Approach for Students.
Witt, Ana
1997-01-01
Describes three labs designed to help students in a first course on ordinary differential equations with three of the most common numerical difficulties they might encounter when solving initial value problems with a numerical software package. The goal of these labs is to help students advance to independent work on common numerical anomalies.…
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...
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.
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.
G. Darmani; S. Setayeshi; H. Ramezanpour
2012-01-01
In this paper an efficient computational method based on extending the sensitivity approach （SA） is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations （DDEs）, the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach.
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
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
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
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.
Chen, Guiling
2013-01-01
This thesis studies asymptotic behavior and stability of determinsitic and stochastic delay differential equations. The approach used in this thesis is based on fixed point theory, which does not resort to any Liapunov function or Liapunov functional. The main contribution of this thesis is to study
Modified differential equations
Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles
2007-01-01
Motivated by the theory of modified differential equations (backward error analysis) an approach for the construction of high order numerical integrators that preserve geometric properties of the exact flow is developed. This summarises a talk presented in honour of Michel Crouzeix.
Biala, T A; Jator, S N
2015-01-01
In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.
A Novel Approach to Modeling of Hydrogeologic Systems Using Fuzzy Differential Equations
Faybishenko, B. A.
2003-12-01
The many simultaneously occurring processes in unsaturated-saturated heterogeneous soils and fractured rocks can cause field observations to become imprecise and incomplete. Consequently, the results of predictions using deterministic and stochastic mathematical models are often uncertain, vague or "fuzzy." One of the alternative approaches to modeling hydrogeologic systems is the application of a fuzzy-systems approach, which is already widely used in such fields as engineering, physics, chemistry, and biology. After presenting a hydrogeologic system as a fuzzy system, the author presents a fuzzy form of Darcy's equation. Based on this equation, second-order fuzzy partial differential equations of the elliptic type (analogous to the Laplace equation) and the parabolic type (analogous to the Richards equation) are derived. These equations are then approximated as fuzzy-difference equations and solved using the basic principles of fuzzy arithmetic. The solutions for the fuzzy-difference equations take the form of fuzzy membership functions for each observation point (node). The author gives examples of the solutions of these equations for flow in unsaturated and saturated media and then compares them with those obtained using deterministic and stochastic methods.
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...
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.
Matrix approach to discrete fractional calculus II: Partial fractional differential equations
Podlubny, Igor; Chechkin, Aleksei; Skovranek, Tomas; Chen, YangQuan; Vinagre Jara, Blas M.
2009-05-01
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach [I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis 3 (4) (2000) 359-386]. Four examples of numerical solution of fractional diffusion equation with various combinations of time-/space-fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
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.
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.
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
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...
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
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Rafiq, Arif [Department of Mathematics, COMSATS Institute of Information Technology, Islamabad (Pakistan)], E-mail: arafiq@comsats.edu.pk; Ahmed, Munshoor [Department of Mathematics, COMSATS Institute of Information Technology, Islamabad (Pakistan)], E-mail: ahmed.manshoor@gmail.com; Hussain, Sifat [CASPAM, Bahauddin Zakariya University, Multan (Pakistan)], E-mail: siffat2002@gmail.com
2008-07-21
Homotopy perturbation method is used to solve specific second order ordinary differential equations and tested for different examples. The results obtained demonstrate efficiency of the proposed method.
Introduction to differential equations
Taylor, Michael E
2011-01-01
The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis. The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponen
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
Fractional Differential Equations
Jianping Zhao
2012-01-01
Full Text Available An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
Qingxue Huang
2017-01-01
Full Text Available In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.
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.
Xu, Mingdong; Wu, Fan; Leung, Henry
2009-09-01
Based on the stochastic delay differential equation (SDDE) modeling of neural networks, we propose an effective signal transmission approach along the neurons in such a network. Utilizing the linear relationship between the delay time and the variance of the SDDE system output, the transmitting side encodes a message as a modulation of the delay time and the receiving end decodes the message by tracking the delay time, which is equivalent to estimating the variance of the received signal. This signal transmission approach turns out to follow the principle of the spread spectrum technique used in wireless and wireline wideband communications but in the analog domain rather than digital. We hope the proposed method might help to explain some activities in biological systems. The idea can further be extended to engineering applications. The error performance of the communication scheme is also evaluated here.
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...
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
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
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
A novel approach to construct numerical methods for stochastic differential equations
Halidias, Nikolaos
2013-01-01
In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.
Geometry of differential equations
Khovanskiĭ, A; Vassiliev, V
1998-01-01
This volume contains articles written by V. I. Arnold's colleagues on the occasion of his 60th birthday. The articles are mostly devoted to various aspects of geometry of differential equations and relations to global analysis and Hamiltonian mechanics.
Problems in differential equations
Brenner, J L
2013-01-01
More than 900 problems and answers explore applications of differential equations to vibrations, electrical engineering, mechanics, and physics. Problem types include both routine and nonroutine, and stars indicate advanced problems. 1963 edition.
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.
Chen, G; de Figueiredo, R P
1993-01-01
The unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed-form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal-spatial. The unknown image is reconstructed from a finite set of sampled data in such a way that a mean-square error is minimized by first expressing the solution in terms of the reproducing kernel of a related Hilbert space, and then constructing this kernel using the fundamental solution of an induced linear partial differential equation, or the Green's function of the corresponding self-adjoint operator. It is proved that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first- or second-order linear partial differential operator. An efficient method for obtaining the corresponding closed-form fundamental solutions (or Green's functions) of the operators is presented. A computer simulation demonstrates the reconstruction procedure.
A novel approach for solving fractional Fisher equation using differential transform method
MIRZAZADEH M
2016-05-01
In the present paper, an analytic solution of nonlinear fractional Fisher equation is deduced with the help of the powerful differential transform method (DTM). To illustrate the method, two examples have been prepared. The method for this equation has led to an exact solution. The reliability, simplicity and cost-effectiveness of the method are confirmed by applying this method on different forms of functional equations.
Differential Equation of Equilibrium
user
than the classical method in the solution of the aforementioned differential equation. Keywords: ... present a successful approximation of shell ... displacement function. .... only applicable to cylindrical shell subject to ..... (cos. 4. 4. 4. 3 β. + β. + β. -. = β. - β x x e ex. AL. xA w. Substituting equations (29); (30) and (31) into.
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Muhammed Çetin
2015-01-01
Full Text Available An approximation method based on Lucas polynomials is presented for the solution of the system of high-order linear differential equations with variable coefficients under the mixed conditions. This method transforms the system of ordinary differential equations (ODEs to the linear algebraic equations system by expanding the approximate solutions in terms of the Lucas polynomials with unknown coefficients and by using the matrix operations and collocation points. In addition, the error analysis based on residual function is developed for present method. To demonstrate the efficiency and accuracy of the method, numerical examples are given with the help of computer programmes written in Maple and Matlab.
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
Sun, Shuyu
2012-06-02
A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.
Theory of differential equations
Gel'fand, I M
1967-01-01
Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations.This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cau
Analytical approach to linear fractional partial differential equations arising in fluid mechanics
Momani, Shaher [Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak (Jordan)]. 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
2006-07-10
In this Letter, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear fractional partial differential equations arising in fluid mechanics. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these methods, the solution takes the form of a convergent series with easily computable components. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of the two methods.
Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay
Hassane Bouzahir
2007-02-01
Full Text Available We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.
Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay
Bouzahir Hassane
2007-01-01
Full Text Available We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.
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
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...
Parametric inference for stochastic differential equations: a smooth and match approach
Gugushvili, S.; Spreij, P.
2012-01-01
We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the
Analytical lie group approach for solving fractional integro-differential equations
Pashayi, S.; Hashemi, M. S.; Shahmorad, S.
2017-10-01
This study is concerned with the Lie symmetry group analysis of Fractional Integro-Differential Equations (FIDEs) with nonlocal structures based on a new development of prolongation formula. A new prolongation for FIDEs is extracted and invariant solutions are finally presented for some illustrative examples.
Parametric inference for stochastic differential equations: a smooth and match approach
Gugushvili, S.; Spreij, P.
2012-01-01
We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the
Differential Equations as Actions
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
A new analytical approach to solve some of the fractional-order partial differential equations
Manafian, Jalil; Lakestani, Mehrdad
2017-03-01
The aim of the present paper is to present an analytical method for the time fractional biological population model, time fractional Burgers, time fractional Cahn-Hilliard, space-time fractional Whitham-Broer-Kaup, space-time fractional Fokas equations by using the generalized tanh-coth method. The fractional derivative is described in the sense of the modified Riemann-Liouville derivatives. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. We have obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these fractional equations to ordinary differential equations which subsequently resulted into number of exact solutions.
A new analytical approach to solve some of the fractional-order partial differential equations
Manafian, Jalil; Lakestani, Mehrdad
2016-09-01
The aim of the present paper is to present an analytical method for the time fractional biological population model, time fractional Burgers, time fractional Cahn-Hilliard, space-time fractional Whitham-Broer-Kaup, space-time fractional Fokas equations by using the generalized tanh-coth method. The fractional derivative is described in the sense of the modified Riemann-Liouville derivatives. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. We have obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these fractional equations to ordinary differential equations which subsequently resulted into number of exact solutions.
LIU Hong-Zhun; PAN Zu-Liang; LI Peng
2006-01-01
In this article, we will derive an equality, where the Taylor series expansion around ε = 0for any asymptotical analytical solution of the perturbed partial differential equation (PDE) with perturbing parameter ε must be admitted.By making use of the equality, we may obtain a transformation, which directly map the analytical solutions of a given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The notion of Lie-B(a)cklund symmetries is introduced in order to obtain more transformations. Hence, we can directly create more transformations in virtue of known Lie-B(a)cklund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries (KdV) equation are used as examples.
Stochastic differential equations and applications
Friedman, Avner
2006-01-01
This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic es
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
Nielsen number and differential equations
Andres Jan
2005-01-01
Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial -structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.
The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics
Estabrook, F. B.; Wahlquist, H. D.
1975-01-01
The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.
Mohammed, Mogtaba; Sango, Mamadou
2016-07-01
This paper deals with the homogenization of a linear hyperbolic stochastic partial differential equation (SPDE) with highly oscillating periodic coefficients. We use Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized linear hyperbolic SPDE with constant coefficients. We also prove the convergence of the associated energies.
Corveleyn, Samuel; Vandewalle, Stefan
2011-01-01
Uncertain parameters in mathematical models of physical phenomena are typically modeled by means of random numbers, random fields or random processes. If these uncertainties are of the epistemic kind, calculations with random parameters can lead to very unexpected and unreliable results. Fuzzy set theory was introduced as an alternative to probability theory for a better modeling of epistemic uncertainty. We consider the solution of elliptic partial differential equations with a fuzzy diffus...
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
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...
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.
Roller Coasters without Differential Equations--A Newtonian Approach to Constrained Motion
Muller, Rainer
2010-01-01
Within the context of Newton's equation, we present a simple approach to the constrained motion of a body forced to move along a specified trajectory. Because the formalism uses a local frame of reference, it is simpler than other methods, making more complicated geometries accessible. No Lagrangian multipliers are necessary to determine the…
Novel Approach for Dealing with Partial Differential Equations with Mixed Derivatives
Abdon Atangana
2014-01-01
Full Text Available We propose a powerful iteration scheme for solving analytically a class of partial equations with mixed derivatives. Our approach is based upon the Lagrange multiplier in two-dimensional spaces. The local convergence and uniqueness of the proposed method are analyzed. In order to demonstrate the applicability of our method, we present an algorithm to compute the solution for two examples.
Roller Coasters without Differential Equations--A Newtonian Approach to Constrained Motion
Muller, Rainer
2010-01-01
Within the context of Newton's equation, we present a simple approach to the constrained motion of a body forced to move along a specified trajectory. Because the formalism uses a local frame of reference, it is simpler than other methods, making more complicated geometries accessible. No Lagrangian multipliers are necessary to determine the…
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.
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.
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.
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Brezzi, Franco; Cangiani, Andrea; Georgoulis, Emmanuil
2016-01-01
This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques.
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
Pure Gaussian state generation via dissipation: a quantum stochastic differential equation approach.
Yamamoto, Naoki
2012-11-28
Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.
Stability of linear delay differential equations a numerical approach with Matlab
Breda, Dimitri; Vermiglio, Rossana
2015-01-01
This book presents the authors' recent work on the numerical methods for the stability analysis of linear autonomous and periodic delay differential equations, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator and in using the eigenvalues of the resulting matrices to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from population dynamics and engineering applications, and Matlab programs implementing the proposed numerical methods. A number of proofs is given to furnish a solid foundation, but the emphasis is on the (unifying) idea of the pseudospectral technique for the stability analysis of DDEs. It is aimed at advanced students and researchers in applied mathematics, in dynamical systems and in various fields of science and engineering, concerned with delay systems. A relevant feature of the book is t...
Scaling of differential equations
Langtangen, Hans Petter
2016-01-01
The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...
Norris, Scott A; Brenner, Michael P; Aziz, Michael J [Harvard School of Engineering and Applied Sciences, Cambridge MA 02138 (United States)
2009-06-03
We develop a methodology for deriving continuum partial differential equations for the evolution of large-scale surface morphology directly from molecular dynamics simulations of the craters formed from individual ion impacts. Our formalism relies on the separation between the length scale of ion impact and the characteristic scale of pattern formation, and expresses the surface evolution in terms of the moments of the crater function. We demonstrate that the formalism reproduces the classical Bradley-Harper results, as well as ballistic atomic drift, under the appropriate simplifying assumptions. Given an actual set of converged molecular dynamics moments and their derivatives with respect to the incidence angle, our approach can be applied directly to predict the presence and absence of surface morphological instabilities. This analysis represents the first work systematically connecting molecular dynamics simulations of ion bombardment to partial differential equations that govern topographic pattern-forming instabilities.
Norris, Scott A; Brenner, Michael P; Aziz, Michael J
2009-06-03
We develop a methodology for deriving continuum partial differential equations for the evolution of large-scale surface morphology directly from molecular dynamics simulations of the craters formed from individual ion impacts. Our formalism relies on the separation between the length scale of ion impact and the characteristic scale of pattern formation, and expresses the surface evolution in terms of the moments of the crater function. We demonstrate that the formalism reproduces the classical Bradley-Harper results, as well as ballistic atomic drift, under the appropriate simplifying assumptions. Given an actual set of converged molecular dynamics moments and their derivatives with respect to the incidence angle, our approach can be applied directly to predict the presence and absence of surface morphological instabilities. This analysis represents the first work systematically connecting molecular dynamics simulations of ion bombardment to partial differential equations that govern topographic pattern-forming instabilities.
Deng, Chenhui; Plan, Elodie L; Karlsson, Mats O
2016-06-01
Parameter variation in pharmacometric analysis studies can be characterized as within subject parameter variability (WSV) in pharmacometric models. WSV has previously been successfully modeled using inter-occasion variability (IOV), but also stochastic differential equations (SDEs). In this study, two approaches, dynamic inter-occasion variability (dIOV) and adapted stochastic differential equations, were proposed to investigate WSV in pharmacometric count data analysis. These approaches were applied to published count models for seizure counts and Likert pain scores. Both approaches improved the model fits significantly. In addition, stochastic simulation and estimation were used to explore further the capability of the two approaches to diagnose and improve models where existing WSV is not recognized. The results of simulations confirmed the gain in introducing WSV as dIOV and SDEs when parameters vary randomly over time. Further, the approaches were also informative as diagnostics of model misspecification, when parameters changed systematically over time but this was not recognized in the structural model. The proposed approaches in this study offer strategies to characterize WSV and are not restricted to count data.
Nielsen number and differential equations
Jan Andres
2005-06-01
Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via PoincarÃƒÂ©'s translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial RÃŽÂ´-structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.
Roller coasters without differential equations-a Newtonian approach to constrained motion
Mueller, Rainer, E-mail: rainer.mueller@tu-bs.d [Technische Universitaet Braunschweig, Physikdidaktik, Pockelsstr. 11, D-38106 Braunschweig (Germany)
2010-07-15
Within the context of Newton's equation, we present a simple approach to the constrained motion of a body forced to move along a specified trajectory. Because the formalism uses a local frame of reference, it is simpler than other methods, making more complicated geometries accessible. No Lagrangian multipliers are necessary to determine the constraining forces. Although the method is able to deal with friction, it becomes particularly simple for conservative systems. We give an analytic expression for the constraining force for any two-dimensional frictionless trajectory that can be written in the form y = f(x). The approach is illustrated with examples from roller coaster physics, e.g. the camelback or the clothoid loop. It is possible to find analytic expressions for the constraining force in both cases.
Solving systems of fractional differential equations using differential transform method
Erturk, Vedat Suat; Momani, Shaher
2008-05-01
This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.
Solving Differential Equations Using Modified Picard Iteration
Robin, W. A.
2010-01-01
Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…
Differential equations extended to superspace
Torres, J. [Instituto de Fisica, Universidad de Guanajuato, A.P. E-143, Leon, Guanajuato (Mexico); Rosu, H.C. [Instituto Potosino de Investigacion Cientifica y Tecnologica, A.P. 3-74, Tangamanga, San Luis Potosi (Mexico)
2003-07-01
We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)
The dynamics of the HIV infection: a time-delay differential equation approach
Bacelar, Flora S; Santos, Rita M Zorzenon dos
2010-01-01
In this work we introduce a differential equation model with time-delay that describes the three-stage dynamics and the two time scales observed in HIV infection. Assuming that the virus has high mutation and rapid reproduction rates that stress the immune system throughout the successive activation of new responses to new undetectable strains, the delay term describes the time interval necessary to mount new specific immune responses. This single term increases the number of possible solutions and changes the phase space dynamics if compared to the model without time delay. We observe very slow transits near the unstable fixed point, corresponding to a healthy state, and long time decay to the stable fixed point that corresponds to the infected state. In contrast to the results obtained for models using regular ODE, which only allow for partial descriptions of the course of the infection, our model describes the entire course of infection observed in infected patients: the primary infection, the latency peri...
Lectures on differential equations for Feynman integrals
Henn, Johannes M
2014-01-01
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the differential equations. Finally, as an application of the differential equations method we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a differential equation.
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.
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
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...
Viana, A., E-mail: Antoine.Viana@g2elab.grenoble-inp.f [Grenoble Electrical Engineering Lab (CNRS UMR5269), Universite de Grenoble, ENSE3, BP 46, 38402 Saint Martin d' Heres (France); Coulomb, J.-L.; Rouve, L.-L.; Cauffet, G. [Grenoble Electrical Engineering Lab (CNRS UMR5269), Universite de Grenoble, ENSE3, BP 46, 38402 Saint Martin d' Heres (France)
2010-01-15
The Langevin equation is classically used to model the anhysteretic magnetization curve. A modified version of this equation has been introduced by Jiles to take into account the effects of magnetostriction on the anhysteretic magnetization behavior when a ferromagnetic material undergoes mechanical stresses. The numerical resolution of the modified Langevin equation is usually performed with a root-finding algorithm. In this paper, a differential form of the modified Langevin equation is proposed, allowing a faster numerical resolution.
A new approach to investigation of evolution differential equations in Banach spaces
Alber, Y I
1993-01-01
and that $B$ is dense in $H$. The stabilization of solutions of evolution equations has been proven either in the sense of weak convergence in $B$ or in the norm of $H$ space, and only asymptotic estimates of stabilization rate have been obtained [15]. In the present paper we consider equations of type (0.1) without conditions (0.2) and establish stabilization with both
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.
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Asymptotic integration of differential and difference equations
Bodine, Sigrun
2015-01-01
This book presents the theory of asymptotic integration for both linear differential and difference equations. This type of asymptotic analysis is based on some fundamental principles by Norman Levinson. While he applied them to a special class of differential equations, subsequent work has shown that the same principles lead to asymptotic results for much wider classes of differential and also difference equations. After discussing asymptotic integration in a unified approach, this book studies how the application of these methods provides several new insights and frequent improvements to results found in earlier literature. It then continues with a brief introduction to the relatively new field of asymptotic integration for dynamic equations on time scales. Asymptotic Integration of Differential and Difference Equations is a self-contained and clearly structured presentation of some of the most important results in asymptotic integration and the techniques used in this field. It will appeal to researchers i...
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...
Solving Differential Equations in R: Package deSolve
Soetaert, K.E.R.; Petzoldt, T.; Setzer, R.W.
2010-01-01
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The differenti
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...
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.
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.
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.
Differential equations a concise course
Bear, H S
2011-01-01
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
Vijay Nehra
2014-01-01
.... The present paper addresses different approaches used to derive mathematical models of first and second order system, developing MATLAB script implementation and building a corresponding Simulink model...
Functional methods in differential equations
Hokkanen, Veli-Matti
2002-01-01
In recent years, functional methods have become central to the study of theoretical and applied mathematical problems. As demonstrated in this Research Note, functional methods can not only provide more generality, but they can also unify results and techniques and lead to better results than those obtained by classical methods. Presenting entirely original results, the authors use functional methods to explore a broad range of elliptic, parabolic, and hyperbolic boundary value problems and various classes of abstract differential and integral equations. They show that while it is crucial to choose an appropriate functional framework, this approach can lead to mathematical models that better describe concrete physical phenomena. In particular, they reach a concordance between the physical sense and the mathematical sense for the solutions of some special models. Beyond its importance as a survey of the primary techniques used in the area, the results illuminated in this volume will prove valuable in a wealth ...
An introduction to differential equations
Ladde, Anil G
2012-01-01
This is a twenty-first century book designed to meet the challenges of understanding and solving interdisciplinary problems. The book creatively incorporates "cutting-edge" research ideas and techniques at the undergraduate level. The book also is a unique research resource for undergraduate/graduate students and interdisciplinary researchers. It emphasizes and exhibits the importance of conceptual understandings and its symbiotic relationship in the problem solving process. The book is proactive in preparing for the modeling of dynamic processes in various disciplines. It introduces a "break-down-the problem" type of approach in a way that creates "fun" and "excitement". The book presents many learning tools like "step-by-step procedures (critical thinking)", the concept of "math" being a language, applied examples from diverse fields, frequent recaps, flowcharts and exercises. Uniquely, this book introduces an innovative and unified method of solving nonlinear scalar differential equations. This is called ...
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
Perturbative self-interacting scalar field theory: a differential equation approach
Rocha, R; Coimbra-Araujo, C H
2005-01-01
We revisit the investigation about the partition function related to a \\phi^4-scalar field theory on a n-dimensional Minkowski spacetime, which is shown to be a self-interacting scalar field theory at least in 4-dimensional Minkowski spacetime. After rederiving the analytical calculation of the perturbative expansion coefficients and also the approximate values for suitable limits using Stirling's formulae, which consists of Witten's proposed questions, solved by P. Deligne, D. Freed, L. Jeffrey, and S. Wu, we investigate a spherically symmetric scalar field in a n-dimensional Minkowski spacetime. For the first perturbative expansion coefficient it is shown how it can be derived a modified Bessel equation (MBE), which solutions are investigated in one, four, and eleven-dimensional Minkowski spacetime. The solutions of MBE are the first expansion coefficient of the series associated with the partition function of \\phi^4-scalar field theory.
Boolean differential equations
Steinbach, Bernd
2013-01-01
The Boolean Differential Calculus (BDC) is a very powerful theory that extends the structure of a Boolean Algebra significantly. Based on a small number of definitions, many theorems have been proven. The available operations have been efficiently implemented in several software packages. There is a very wide field of applications. While a Boolean Algebra is focused on values of logic functions, the BDC allows the evaluation of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. Due to the same basic data structures, the BDC can
Introductory course on differential equations
Gorain, Ganesh C
2014-01-01
Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.
Fractional complex transform for fractional differential equations
Lİ, Zheng Biao; HE, Ji Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily...
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 ...
VIM-based dynamic sparse grid approach to partial differential equations.
Mei, Shu-Li
2014-01-01
Combining the variational iteration method (VIM) with the sparse grid theory, a dynamic sparse grid approach for nonlinear PDEs is proposed in this paper. In this method, a multilevel interpolation operator is constructed based on the sparse grids theory firstly. The operator is based on the linear combination of the basic functions and independent of them. Second, by means of the precise integration method (PIM), the VIM is developed to solve the nonlinear system of ODEs which is obtained from the discretization of the PDEs. In addition, a dynamic choice scheme on both of the inner and external grid points is proposed. It is different from the traditional interval wavelet collocation method in which the choice of both of the inner and external grid points is dynamic. The numerical experiments show that our method is better than the traditional wavelet collocation method, especially in solving the PDEs with the Nuemann boundary conditions.
Group analysis of differential equations
Ovsiannikov, L V
1982-01-01
Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g
Differential equations and mathematical biology
Jones, DS; Sleeman, BD
2009-01-01
""… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.""-SIAM Review, Sept. 2010, Vol. 52, No. 3""… Where this text stands out i
Lectures on ordinary differential equations
Hurewicz, Witold
2014-01-01
Hailed by The American Mathematical Monthly as ""a rigorous and lively introduction,"" this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown funct
Loop equations from differential systems
Eynard, Bertrand; Marchal, Olivier
2016-01-01
To any differential system $d\\Psi=\\Phi\\Psi$ where $\\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\\Phi$ is a Lie algebra $\\mathfrak g$ valued 1-form on a Riemann surface $\\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.
Qian Guo
2014-01-01
Full Text Available Almost sure exponential stability of the split-step backward Euler (SSBE method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.
Introduction to Piecewise Differentiable Equations
Scholtes, Stefan
2012-01-01
This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the non smooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop
Stochastic nonlinear differential equations. I
Heilmann, O.J.; Kampen, N.G. van
1974-01-01
A solution method is developed for nonlinear differential equations having the following two properties. Their coefficients are stochastic through their dependence on a Markov process. The magnitude of the fluctuations, multiplied with their auto-correlation time, is a small quantity. Under these co
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
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.
Differential Equations for Morphological Amoebas
Welk, Martin; Breuß, Michael; Vogel, Oliver
This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential equation related to self-snakes and the well-known (mean) curvature motion equation. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.
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...
Cox, S.G.
2012-01-01
The thesis deals with various aspects of the study of stochastic partial differential equations driven by Gaussian noise. The approach taken is functional analytic rather than probabilistic: the stochastic partial differential equation is interpreted as an ordinary stochastic differential equation i
Algebrization of Nonautonomous Differential Equations
María Aracelia Alcorta-García
2015-01-01
Full Text Available Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w, conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w=H(te,w and the maps H1(τ=H(τ,ξ and H2(ξ=H(τ,ξ are Lorch differentiable with respect to A for all (τ,ξ∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ of the differential equation dξ/dτ=H(τ,ξ over A define solutions (x(t,y(t=ξ(te of the planar system.
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
Schiesser, William E
2014-01-01
Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-worldODE 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 andextend the comp
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
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
Stability theory of differential equations
Bellman, Richard
2008-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
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: ·...
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...
Partial differential equations possessing Frobenius integrable decompositions
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.
Polynomial normal forms of constrained differential equations with three parameters
Jardon-Kojakhmetov, H.; Broer, Henk W.
2014-01-01
We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local
Introduction to computation and modeling for differential equations
Edsberg, Lennart
2008-01-01
An introduction to scientific computing for differential equationsIntroduction to Computation and Modeling for Differential Equations provides a unified and integrated view of numerical analysis, mathematical modeling in applications, and programming to solve differential equations, which is essential in problem-solving across many disciplines, such as engineering, physics, and economics. This book successfully introduces readers to the subject through a unique ""Five-M"" approach: Modeling, Mathematics, Methods, MATLAB, and Multiphysics. This approach facilitates a thorough understanding of h
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
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
Conservation Laws of Differential Equations in Finance
QIN Mao-Chang; MEI Feng-Xiang; SHANG Mei
2005-01-01
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Stochastic Runge-Kutta Software Package for Stochastic Differential Equations
Gevorkyan, M N; Korolkova, A V; Kulyabov, D S; Sevastyanov, L A
2016-01-01
As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker--Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge--Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black--Scholes two-dimensional model are used. To illustrate the stochastic "predator--prey" type model is us...
Functional differential equations of third order
Tuncay Candan
2005-04-01
Full Text Available In this paper, we consider the third-order neutral functional differential equation with distributed deviating arguments. We give sufficient conditions for the oscillatory behavior of this functional differential equation.
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.
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...
The Differential Equation Algorithm for General Deformed Swept Volumes
汪国平; 华宣积; 孙家广
2000-01-01
The differential equation approach for characterizing swept volume boundaries is extended to include objects experiencing deformation. For deformed swept volume, it is found that the structure and algorithm of sweep-envelope differential equation (SEDE) are similar between the deformed and the rigid swept volumes. The efficiency of SEDE approach for deformed swept volume is proved with an example.
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Auxiliary equation method for solving nonlinear partial differential equations
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.
A reformulation of an ordinary differential equation
Barraza, Oscar A.
2013-01-01
The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able to open a new way in order to calculate or approximate the solution of an ordinary differential equation. Some examples are presented.
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.
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.
Modelling conjugation with stochastic differential equations.
Philipsen, K R; Christiansen, L E; Hasman, H; Madsen, H
2010-03-07
Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared to the model without plate conjugation. The modelling approach described in this article can be applied generally when modelling dynamical systems.
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
Sobolev gradients and differential equations
Neuberger, J W
2010-01-01
A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair p...
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
Linearized asymptotic stability for fractional differential equations
Nguyen Cong
2016-06-01
Full Text Available We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector $\\{\\lambda \\in \\mathbb{C} : |\\arg \\lambda| > \\frac{\\alpha \\pi}{2}\\}$ where $\\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.
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.
Introduction to differential equations with dynamical systems
Campbell, Stephen L
2011-01-01
Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Cam
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
Solving Partial Differential Equations Using a New Differential Evolution Algorithm
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.
Sobolev gradients and differential equations
Neuberger, John William
1997-01-01
A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.
Approximating chaotic saddles for delay differential equations
Taylor, S. Richard; Campbell, Sue Ann
2007-04-01
Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a “logistic” delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.
Modeling and Prediction Using Stochastic Differential Equations
Juhl, Rune; Møller, Jan Kloppenborg; Jørgensen, John Bagterp
2016-01-01
Pharmacokinetic/pharmakodynamic (PK/PD) modeling for a single subject is most often performed using nonlinear models based on deterministic ordinary differential equations (ODEs), and the variation between subjects in a population of subjects is described using a population (mixed effects) setup...... that describes the variation between subjects. The ODE setup implies that the variation for a single subject is described by a single parameter (or vector), namely the variance (covariance) of the residuals. Furthermore the prediction of the states is given as the solution to the ODEs and hence assumed...... deterministic and can predict the future perfectly. A more realistic approach would be to allow for randomness in the model due to e.g., the model be too simple or errors in input. We describe a modeling and prediction setup which better reflects reality and suggests stochastic differential equations (SDEs...
Ordinary differential equations and mechanical systems
Awrejcewicz, Jan
2014-01-01
This book applies a step-by-step treatment of the current state-of-the-art of ordinary differential equations used in modeling of engineering systems/processes and beyond. It covers systematically ordered problems, beginning with first and second order ODEs, linear and higher-order ODEs of polynomial form, theory and criteria of similarity, modeling approaches, phase plane and phase space concepts, stability optimization, and ending on chaos and synchronization. Presenting both an overview of the theory of the introductory differential equations in the context of applicability and a systematic treatment of modeling of numerous engineering and physical problems through linear and non-linear ODEs, the volume is self-contained, yet serves both scientific and engineering interests. The presentation relies on a general treatment, analytical and numerical methods, concrete examples, and engineering intuition. The scientific background used is well balanced between elementary and advanced level, making it as a uniqu...
The myth about nonlinear differential equations
Radhakrishnan, C.
2002-01-01
Taking the example of Koretweg--de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation. Even the ordinary one dimensional linear partial differential equation can produce a soliton.
Wilsonian renormalization, differential equations and Hopf algebras
Thomas, Krajewski
2008-01-01
In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Finally, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally...
Kurzweil, J
1986-01-01
The author, Professor Kurzweil, is one of the world's top experts in the area of ordinary differential equations - a fact fully reflected in this book. Unlike many classical texts which concentrate primarily on methods of integration of differential equations, this book pursues a modern approach: the topic is discussed in full generality which, at the same time, permits us to gain a deep insight into the theory and to develop a fruitful intuition. The basic framework of the theory is expanded by considering further important topics like stability, dependence of a solution on a parameter, Car
Differential operator multiplication method for fractional differential equations
Tang, Shaoqiang; Ying, Yuping; Lian, Yanping; Lin, Stephen; Yang, Yibo; Wagner, Gregory J.; Liu, Wing Kam
2016-08-01
Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction-advection-diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations.
Reduced minimax filtering by means of differential-algebraic equations
Mallet, V.; Zhuk, S.
2011-01-01
A reduced minimax state estimation approach is proposed for high-dimensional models. It is based on the reduction of the ordinary differential equation with high state space dimension to the low-dimensional Differential-Algebraic Equation (DAE) and on the subsequent application of the minimax state
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
A fifth order differential equation for charged perfect fluids
Kweyama, M C; Maharaj, S D
2013-01-01
We investigate the master nonlinear partial differential equation that governs the evolution of shear-free spherically symmetric charged fluids. We use an approach which has not been considered previously for the underlying equation in shear-free spherically symmetric spacetimes. We derive a fifth order purely differential equation that must be satisfied for the underlying equation to admit a Lie point symmetry. We then perform a comprehensive analysis of this equation utilising the Lie symmetry analysis and direct integration. This enables us to reduce the fifth order equation to quadratures. Earlier results are shown to be contained in our general treatment.
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…
Introduction to complex theory of differential equations
Savin, Anton
2017-01-01
This book discusses the complex theory of differential equations or more precisely, the theory of differential equations on complex-analytic manifolds. Although the theory of differential equations on real manifolds is well known – it is described in thousands of papers and its usefulness requires no comments or explanations – to date specialists on differential equations have not focused on the complex theory of partial differential equations. However, as well as being remarkably beautiful, this theory can be used to solve a number of problems in real theory, for instance, the Poincaré balayage problem and the mother body problem in geophysics. The monograph does not require readers to be familiar with advanced notions in complex analysis, differential equations, or topology. With its numerous examples and exercises, it appeals to advanced undergraduate and graduate students, and also to researchers wanting to familiarize themselves with the subject.
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
Solving Partial Differential Equations on Overlapping Grids
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.
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.
On a complex differential Riccati equation
Khmelnytskaya, Kira V; Kravchenko, Vladislav V [Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 Mexico (Mexico)], E-mail: vkravchenko@qro.cinvestav.mx
2008-02-29
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schroedinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical 'one-dimensional' results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem.
Solutions manual to accompany Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Solving Operator Equation Based on Expansion Approach
A. Aminataei
2014-01-01
Full Text Available To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form Lu+Nu=g(x,x∈Γ, with respect to boundary condition Bu=λ, where L, N and B are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, Γ is the domain of approximation, λ is an arbitrary constant, and g(x∈L2(Γ is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in L2(Γ norm.
Henriques, David; Rocha, Miguel; Saez-Rodriguez, Julio; Banga, Julio R
2015-09-15
Systems biology models can be used to test new hypotheses formulated on the basis of previous knowledge or new experimental data, contradictory with a previously existing model. New hypotheses often come in the shape of a set of possible regulatory mechanisms. This search is usually not limited to finding a single regulation link, but rather a combination of links subject to great uncertainty or no information about the kinetic parameters. In this work, we combine a logic-based formalism, to describe all the possible regulatory structures for a given dynamic model of a pathway, with mixed-integer dynamic optimization (MIDO). This framework aims to simultaneously identify the regulatory structure (represented by binary parameters) and the real-valued parameters that are consistent with the available experimental data, resulting in a logic-based differential equation model. The alternative to this would be to perform real-valued parameter estimation for each possible model structure, which is not tractable for models of the size presented in this work. The performance of the method presented here is illustrated with several case studies: a synthetic pathway problem of signaling regulation, a two-component signal transduction pathway in bacterial homeostasis, and a signaling network in liver cancer cells. Supplementary data are available at Bioinformatics online. julio@iim.csic.es or saezrodriguez@ebi.ac.uk. © The Author 2015. Published by Oxford University Press.
Nth-order Fuzzy Differential Equations Under Generalized Differentiability
Soheil Salahshour
2011-11-01
Full Text Available In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space.
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
A Geometric Treatment of Implicit Differential-Algebraic Equations
Rabier, P. J.; Rheinboldt, W. C.
A differential-geometric approach for proving the existence and uniqueness of implicit differential-algebraic equations is presented. It provides for a significant improvement of an earlier theory developed by the authors as well as for a completely intrinsic definition of the index of such problems. The differential-algebraic equation is transformed into an explicit ordinary differential equation by a reduction process that can be abstractly defined for specific submanifolds of tangent bundles here called reducible π-submanifolds. Local existence and uniqueness results for differential-algebraic equations then follow directly from the final stage of this reduction by means of an application of the standard theory of ordinary differential equations.
Numerical methods for ordinary differential equations
Butcher, John C
2008-01-01
In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author''s pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book includeIntroductory work on differential and difference equations.A comprehensive introduction to the theory and practice of solving ordinary differential equations numeri...
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 $r\\ge 1$.
A practical course in differential equations and mathematical modeling
Ibragimov , Nail H
2009-01-01
A Practical Course in Differential Equations and Mathematical Modelling is a unique blend of the traditional methods of ordinary and partial differential equations with Lie group analysis enriched by the author's own theoretical developments. The book which aims to present new mathematical curricula based on symmetry and invariance principles is tailored to develop analytic skills and working knowledge in both classical and Lie's methods for solving linear and nonlinear equations. This approach helps to make courses in differential equations, mathematical modelling, distributions and fundame
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
Moh’d Khier Al-Srihin
2017-01-01
Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.
Symposium on Differential Geometry and Differential Equations
Berger, Marcel; Bryant, Robert
1987-01-01
The DD6 Symposium was, like its predecessors DD1 to DD5 both a research symposium and a summer seminar and concentrated on differential geometry. This volume contains a selection of the invited papers and some additional contributions. They cover recent advances and principal trends in current research in differential geometry.
Strict Stability of Impulsive Differential Equations
Yu ZHANG; Ji Tao SUN
2006-01-01
In this paper, we will extend the strict stability to impulsive differential equations. By using Lyapunov functions, we will get some criteria for the strict stability of impulsive differential equations, and we can see that impulses do contribute to the system's strict stability behavior. An example is also given in this paper to illustrate the efficiency of the obtained results.
Selected papers on analysis and differential equations
Society, American Mathematical
2010-01-01
This volume contains translations of papers that originally appeared in the Japanese journal Sūgaku. These papers range over a variety of topics in ordinary and partial differential equations, and in analysis. Many of them are survey papers presenting new results obtained in the last few years. This volume is suitable for graduate students and research mathematicians interested in analysis and differential equations.
Topologies for neutral functional differential equations.
Melvin, W. R.
1973-01-01
Bounded topologies are considered for functional differential equations of the neutral type in which present dynamics of the system are influenced by its past behavior. A special bounded topology is generated on a collection of absolutely continuous functions with essentially bounded derivatives, and an application to a class of nonlinear neutral functional differential equations due to Driver (1965) is presented.
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.
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
ON ALGEBRICO-DIFFERENTIAL EQUATIONS-SOLVING
WU Wenjun(Wu Wen-tsun)
2004-01-01
The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations. As an illustration of the method, the Devil's Problem of Pommaret is solved in details.
Exponentially Convergent Algorithms for Abstract Differential Equations
Gavrilyuk, Ivan; Vasylyk, Vitalii
2011-01-01
This book presents new accurate and efficient exponentially convergent methods for abstract differential equations with unbounded operator coefficients in Banach space. These methods are highly relevant for the practical scientific computing since the equations under consideration can be seen as the meta-models of systems of ordinary differential equations (ODE) as well as the partial differential equations (PDEs) describing various applied problems. The framework of functional analysis allows one to obtain very general but at the same time transparent algorithms and mathematical results which
Statistical Methods for Stochastic Differential Equations
Kessler, Mathieu; Sorensen, Michael
2012-01-01
The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a sp
On implicit abstract neutral nonlinear differential equations
Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)
2016-04-15
In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.
Stochastic differential equation model to Prendiville processes
Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Stochastic differential equation model to Prendiville processes
Granita, Bahar, Arifah
2015-10-01
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
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
The Numerical Approximation of Functional Differential Equations
Venturi, Daniele
2016-01-01
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equations), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective action methods). However, no effective numerical method has yet been developed to compute their solution. The purpose of this manuscript is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations. The proposed methods will be described and demonstrated in various examples.
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.
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.
Differential equations inverse and direct problems
Favini, Angelo
2006-01-01
DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA
Backward stochastic differential equations from linear to fully nonlinear theory
Zhang, Jianfeng
2017-01-01
This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.
The Riccati Differential Equation and a Diffusion-Type Equation
Suazo, Erwin; Vega-Guzman, Jose M
2008-01-01
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equation with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.
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...
Differential geometric formulation of the Cauchy Navier equations
Schadt, Frank
2011-01-01
The paper presents a reformulation of some of the most basic entities and equations of linear elasticity - the stress and strain tensor, the Cauchy Navier equilibrium equations, material equations for linear isotropic bodies - in a modern differential geometric language using differential forms and lie derivatives. Similar steps have been done successfully in general relativity, quantum physics and electrodynamics and are of great use in those fields. In Elasticity Theory, however, such a modern differential geometric approach is much less common. Furthermore, existing reformulations demand a vast knowledge of differential geometry, including nonstandard entities such as vector valued differential forms and the like. This paper presents a less general but more easily accessible approach to using modern differential geometry in elasticity theory than those published up to now.
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
Stability analysis of impulsive functional differential equations
Stamova, Ivanka
2009-01-01
This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equationsis under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied research
Quantum algorithms for solving linear differential equations
Berry, Dominic W
2010-01-01
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by homogeneous linear differential equations that produce only oscillating terms. Here we extend quantum simulation algorithms to general inhomogeneous linear differential equations, which can include exponential terms as well as oscillating terms in their solution. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state. The algorithm does not give the explicit solution, but it is possible to extract global features of the solution.
A Simple Stochastic Differential Equation with Discontinuous Drift
Simonsen, Maria; Leth, John-Josef; Schiøler, Henrik
2013-01-01
In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density f...
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.
Differential invariants of second-order ordinary differential equations
Rosado Maria, Maria Eugenia
2011-01-01
The notion of a differential invariant for systems of second-order differential equations on a manifold M with respect to the group of vertical automorphisms of the projection is de?ned and the Chern connection attached to a SODE allows one to determine a basis for second-order differential invariants of a SODE.
Discretizing a backward stochastic differential equation
Yinnan Zhang; Weian Zheng
2002-01-01
We show a simple method to discretize Pardoux-Peng's nonlinear backward stochastic differential equation. This discretization scheme also gives a numerical method to solve a class of semi-linear PDEs.
Trends in differential equations and applications
Neble, María; Galván, José
2016-01-01
This work collects the most important results presented at the Congress on Differential Equations and Applications/Congress on Applied Mathematics (CEDYA/CMA) in Cádiz (Spain) in 2015. It supports further research in differential equations, numerical analysis, mechanics, control and optimization. In particular, it helps readers gain an overview of specific problems of interest in the current mathematical research related to different branches of applied mathematics. This includes the analysis of nonlinear partial differential equations, exact solutions techniques for ordinary differential equations, numerical analysis and numerical simulation of some models arising in experimental sciences and engineering, control and optimization, and also trending topics on numerical linear Algebra, dynamical systems, and applied mathematics for Industry. This volume is mainly addressed to any researcher interested in the applications of mathematics, especially in any subject mentioned above. It may be also useful to PhD s...
An introduction to differential equations using MATLAB
Butt, Rizwan
2016-01-01
An Introduction to Differential Equations using MATLAB exploits the symbolic, numerical, and graphical capabilitiesof MATLAB to develop a thorough understanding of differential equations algorithms. This book provides the readerwith numerous applications, m-files, and practical examples to problems. Balancing theoretical concepts withcomputational speed and accuracy, the book includes numerous short programs in MATLAB that can be used to solveproblems involving first-and higher-order differential equations, Laplace transforms, linear systems of differentialequations, numerical solutions of differential equations, computer graphics, and more. The author emphasizes thebasic ideas of analytical and numerical techniques and the uses of modern mathematical software (MATLAB) ratherthan relying only on complex mathematical derivations to engineers, mathematician, computer scientists, andphysicists or for use as a textbook in applied or computational courses.A CD-ROM with all the figures, codes, solutions, appendices...
Fractional complex transforms for fractional differential equations
Ibrahim, Rabha W
2012-01-01
The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk...
Selected papers on analysis and differential equations
Nomizu, Katsumi
2003-01-01
This volume contains translations of papers that originally appeared in the Japanese journal, Sugaku. The papers range over a variety of topics, including nonlinear partial differential equations, C^*-algebras, and Schrödinger operators.
Approximate Method for Solving the Linear Fuzzy Delay Differential Equations
S. Narayanamoorthy
2015-01-01
Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.
Rough differential equations driven by signals in Besov spaces
Prömel, David J.; Trabs, Mathias
2016-03-01
Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in Hölder and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by Gubinelli, Imkeller and Perkowski [24] to analyze singular stochastic PDEs, is extended from Hölder to Besov spaces. As an application we solve stochastic differential equations driven by random functions in Besov spaces and Gaussian processes in a pathwise sense.
Liouvillian propagators, Riccati equation and differential Galois theory
Acosta-Humánez, Primitivo; Suazo, Erwin
2013-11-01
In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.
Monotone Semiflows Generated by Functional Differential Equations,
1986-02-01
These results have been applied to ordinary differential equations in Rn (see e.g. [10,23]) where the well-known Kamke theorem applies and to nonlinear...sufficient condition (H) Whenever 0 - 0 and ,i(0) = i( 0) it follows that fi(0) ( fi(O). For those familiar with the Kamke (quasimonotone) condition for...ordinary differential equations, (H) will seem quite natural, it reduces to the Kamke condition. The order preserving property of a semiflow is not
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.
Symmetrized solutions for nonlinear stochastic differential equations
G. Adomian
1981-01-01
Full Text Available Solutions of nonlinear stochastic differential equations in series form can be put into convenient symmetrized forms which are easily calculable. This paper investigates such forms for polynomial nonlinearities, i.e., equations of the form Ly+ym=x where x is a stochastic process and L is a linear stochastic operator.
On Fractional Order Hybrid Differential Equations
Mohamed A. E. Herzallah
2014-01-01
Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Time-course window estimator for ordinary differential equations linear in the parameters
Vujacic, Ivan; Dattner, Itai; Gonzalez, Javier; Wit, Ernst
2015-01-01
In many applications obtaining ordinary differential equation descriptions of dynamic processes is scientifically important. In both, Bayesian and likelihood approaches for estimating parameters of ordinary differential equations, the speed and the convergence of the estimation procedure may crucial
Time-course window estimator for ordinary differential equations linear in the parameters
Vujacic, Ivan; Dattner, Itai; Gonzalez, Javier; Wit, Ernst
2015-01-01
In many applications obtaining ordinary differential equation descriptions of dynamic processes is scientifically important. In both, Bayesian and likelihood approaches for estimating parameters of ordinary differential equations, the speed and the convergence of the estimation procedure may
Adaptive finite element methods for differential equations
Bangerth, Wolfgang
2003-01-01
These Lecture Notes discuss concepts of `self-adaptivity' in the numerical solution of differential equations, with emphasis on Galerkin finite element methods. The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order ...
Campoamor-Stursberg, R.
2016-08-01
Using the general solution of the differential equation x¨(t) +g1(t) x˙ +g2(t) x = 0 , a generic basis of the point-symmetry algebra sl(3 , R) is constructed. Deriving the equation from a time-dependent Lagrangian, the basis elements corresponding to Noether symmetries are deduced. The generalized Lewis invariant is constructed explicitly using a linear combination of Noether symmetries. The procedure is generalized to the case of systems of second-order ordinary differential equations with maximal sl(n + 2 , R) -symmetry, and its possible adaptation to the inhomogeneous non-linear case illustrated by an example.
A Unified Introduction to Ordinary Differential Equations
Lutzer, Carl V.
2006-01-01
This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)
Nonlinear differentiation equation and analytic function spaces
Li, Hao; Li, Songxiao
2015-01-01
In this paper we consider the nonlinear complex differential equation $$(f^{(k)})^{n_{k}}+A_{k-1}(z)(f^{(k-1)})^{n_{k-1}}+\\cdot\\cdot\\cdot+A_{1}(z)(f')^{n_{1}}+A_{0}(z)f^{n_{0}}=0, $$where $ A_{j}(z)$, $ j=0, \\cdots, k-1 $, are analytic in the unit disk $ \\mathbb{D} $, $ n_{j}\\in R^{+} $ for all $ j=0, \\cdots, k $. We investigate this nonlinear differential equation from two aspects. On one hand, we provide some sufficient conditions on coefficients such that all solutions of this equation bel...
Existence theorems for ordinary differential equations
Murray, Francis J
2007-01-01
Theorems stating the existence of an object-such as the solution to a problem or equation-are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations.The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, th
Stochastic Differential Equations and Kondratiev Spaces
Vaage, G.
1995-05-01
The purpose of this mathematical thesis was to improve the understanding of physical processes such as fluid flow in porous media. An example is oil flowing in a reservoir. In the first of five included papers, Hilbert space methods for elliptic boundary value problems are used to prove the existence and uniqueness of a large family of elliptic differential equations with additive noise without using the Hermite transform. The ideas are then extended to the multidimensional case and used to prove existence and uniqueness of solution of the Stokes equations with additive noise. The second paper uses functional analytic methods for partial differential equations and presents a general framework for proving existence and uniqueness of solutions to stochastic partial differential equations with multiplicative noise, for a large family of noises. The methods are applied to equations of elliptic, parabolic as well as hyperbolic type. The framework presented can be extended to the multidimensional case. The third paper shows how the ideas from the second paper can be extended to study the moving boundary value problem associated with the stochastic pressure equation. The fourth paper discusses a set of stochastic differential equations. The fifth paper studies the relationship between the two families of Kondratiev spaces used in the thesis. 102 refs.
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...
On new solutions of fuzzy differential equations
Chalco-Cano, Y. [Departamento de Matematica, Universidad de Tarapaca, Casilla 7D, Arica (Chile)], E-mail: ychalco@uta.cl; Roman-Flores, H. [Instituto de Investigacion, Universidad de Tarapaca, Casilla 7D, Arica (Chile)
2008-10-15
We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made.
Symmetries of Differential equations and Applications in Relativistic Physics
Paliathanasis, Andronikos
2015-01-01
In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order the two and three dimensional Newtonian dynamical systems to be classified in relation to the point symmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian spaces; to study the symmetries between classical and quantum systems and to investigate the geometric origin of the Type II hidden symmetries for the homogeneous heat equation and for the Laplace equation in Riemannian spaces. At last but not least, we apply this geometric approach in order to determine the dark energy models by use the Noether symmetries as a geometric criterion in modified theories of gra...
An introduction to neural network methods for differential equations
Yadav, Neha; Kumar, Manoj
2015-01-01
This book introduces a variety of neural network methods for solving differential equations arising in science and engineering. The emphasis is placed on a deep understanding of the neural network techniques, which has been presented in a mostly heuristic and intuitive manner. This approach will enable the reader to understand the working, efficiency and shortcomings of each neural network technique for solving differential equations. The objective of this book is to provide the reader with a sound understanding of the foundations of neural networks, and a comprehensive introduction to neural network methods for solving differential equations together with recent developments in the techniques and their applications. The book comprises four major sections. Section I consists of a brief overview of differential equations and the relevant physical problems arising in science and engineering. Section II illustrates the history of neural networks starting from their beginnings in the 1940s through to the renewed...
Modelling conjugation with stochastic differential equations
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik
2010-01-01
Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...... using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared...
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.
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.
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.
Homogenizing atomic dynamics by fractional differential equations
Tang, Shaoqiang; Ying, Yuping
2017-10-01
In this paper, we propose two ways to construct fractional differential equations (FDE) for approximating atomic chain dynamics. Taking harmonic chain as an example, we add a power function of fractional order to Taylor expansion of the dispersion relation, and determine the parameters by matching two selected wave numbers. This approximate function leads to an FDE after considering both directions for wave propagation. As an alternative, we consider the symbol of the force term, and approximate it by a similar function. It also induces an FDE. Both approaches produce excellent agreement with the harmonic chain dynamics. The accuracy may be improved by optimizing the selected wave numbers, or starting with higher order Taylor expansions. When resolved in the lattice constant, the resulting FDE's faithfully reproduce the lattice dynamics. When resolved in a coarse grid instead, they systematically generate homogenized algorithms. Numerical tests are performed to verify the proposed approaches. Moreover, FDE's are also constructed for diatomic chain and anharmonic lattice, to illustrate the generality of the proposed approaches.
Abstract methods in differential equations
Amann, H
2003-01-01
This is an expanded version, enriched by references, of my inaugural speech held on November 7, 2001 at the Real Academia de Ciencias Exactas, Físicas y Naturales in Madrid. It explains in a nontechnical way, accessible to a general scientific community, some of the motivation and basic ideas of my research of the last twenty years on a functional-analytical approach to nonlinear parabolic problems.
Solving constant-coefficient differential equations with dielectric metamaterials
Zhang, Weixuan; Qu, Che; Zhang, Xiangdong
2016-07-01
Recently, the concept of metamaterial analog computing has been proposed (Silva et al 2014 Science 343 160-3). Some mathematical operations such as spatial differentiation, integration, and convolution, have been performed by using designed metamaterial blocks. Motivated by this work, we propose a practical approach based on dielectric metamaterial to solve differential equations. The ordinary differential equation can be solved accurately by the correctly designed metamaterial system. The numerical simulations using well-established numerical routines have been performed to successfully verify all theoretical analyses.
Analytical solution of linear ordinary differential equations by differential transfer matrix method
Sina Khorasani
2003-08-01
Full Text Available We report a new analytical method for finding the exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the $n$th-order differential equation to a system of $n$ linear differential equations with unity order. The full analytical solution is then found by the perturbation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the applicability of the method.
Paul, Wolfgang; Koeppe, Jeanette [Institut fuer Physik, Martin Luther Universitaet, 06099 Halle (Germany); Grecksch, Wilfried [Institut fuer Mathematik, Martin Luther Universitaet, 06099 Halle (Germany)
2016-07-01
The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schroedinger equation. We show a way to go around it. This way is based on the derivation of the Schroedinger equation from conservative diffusion processes and the establishment of (several) stochastic variational principles leading to the Schroedinger equation under the assumption of a kinematics described by Nelson's diffusion processes. Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelson's stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schroedinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schroedinger equation and exemplify the approach for a simple 1d problem.
A short course in ordinary differential equations
Kong, Qingkai
2014-01-01
This text is a rigorous treatment of the basic qualitative theory of ordinary differential equations, at the beginning graduate level. Designed as a flexible one-semester course but offering enough material for two semesters, A Short Course covers core topics such as initial value problems, linear differential equations, Lyapunov stability, dynamical systems and the Poincaré—Bendixson theorem, and bifurcation theory, and second-order topics including oscillation theory, boundary value problems, and Sturm—Liouville problems. The presentation is clear and easy-to-understand, with figures and copious examples illustrating the meaning of and motivation behind definitions, hypotheses, and general theorems. A thoughtfully conceived selection of exercises together with answers and hints reinforce the reader's understanding of the material. Prerequisites are limited to advanced calculus and the elementary theory of differential equations and linear algebra, making the text suitable for senior undergraduates as w...
A first course in differential equations
Logan, J David
2015-01-01
The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market. It presents a thorough treatment of the standard topics in an accessible, easy-to-read, format. The overarching perspective of the text conveys that differential equations are about applications. This book illuminates the mathematical theory in the text with a wide variety of applications that will appeal to students in physics, engineering, the biosciences, economics and mathematics. Instructors are likely to find that the first four or five chapters are suitable for a first course in the subject. This edition contains a healthy increase over earlier editions in the number of worked examples and exercises, particularly those routine in nature. Two appendices include a review with practice problems, and a MATLAB® supplement that gives basic codes and commands for solving differential equations. MATLAB® is not required; students are encouraged t...
Stochastic Differential Equation of Earthquakes Series
Mariani, Maria C.; Tweneboah, Osei K.; Gonzalez-Huizar, Hector; Serpa, Laura
2016-07-01
This work is devoted to modeling earthquake time series. We propose a stochastic differential equation based on the superposition of independent Ornstein-Uhlenbeck processes driven by a Γ (α, β ) process. Superposition of independent Γ (α, β ) Ornstein-Uhlenbeck processes offer analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to the study of earthquakes by fitting the superposed Γ (α, β ) Ornstein-Uhlenbeck model to earthquake sequences in South America containing very large events (Mw ≥ 8). We obtained very good fit of the observed magnitudes of the earthquakes with the stochastic differential equations, which supports the use of this methodology for the study of earthquakes sequence.
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations
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.
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
An introduction to stochastic differential equations
Evans, Lawrence C
2014-01-01
These notes provide a concise introduction to stochastic differential equations and their application to the study of financial markets and as a basis for modeling diverse physical phenomena. They are accessible to non-specialists and make a valuable addition to the collection of texts on the topic. -Srinivasa Varadhan, New York University This is a handy and very useful text for studying stochastic differential equations. There is enough mathematical detail so that the reader can benefit from this introduction with only a basic background in mathematical analysis and probability. -George Papa
Surveys in differential-algebraic equations III
Reis, Timo
2015-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Flexibility of DAE formulations - Reachability analysis and deterministic global optimization - Numerical linear algebra methods - Boundary value problems The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
Surveys in differential-algebraic equations II
Reis, Timo
2015-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Observers for DAEs - DAEs in chemical processes - Optimal control of DAEs - DAEs from a functional-analytic viewpoint - Algebraic methods for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
Asymptotic analysis for functional stochastic differential equations
Bao, Jianhai; Yuan, Chenggui
2016-01-01
This brief treats dynamical systems that involve delays and random disturbances. The study is motivated by a wide variety of systems in real life in which random noise has to be taken into consideration and the effect of delays cannot be ignored. Concentrating on such systems that are described by functional stochastic differential equations, this work focuses on the study of large time behavior, in particular, ergodicity. This brief is written for probabilists, applied mathematicians, engineers, and scientists who need to use delay systems and functional stochastic differential equations in their work. Selected topics from the brief can also be used in a graduate level topics course in probability and stochastic processes.
Surveys in differential-algebraic equations IV
Reis, Timo
2017-01-01
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs) which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - History of DAEs - DAE aspects of mechanical multibody systems - Model reduction of DAEs - Observability for DAEs - Numerical Analysis for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
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.
Stochastic versus deterministic systems of differential equations
Ladde, G S
2003-01-01
This peerless reference/text unfurls a unified and systematic study of the two types of mathematical models of dynamic processes-stochastic and deterministic-as placed in the context of systems of stochastic differential equations. Using the tools of variational comparison, generalized variation of constants, and probability distribution as its methodological backbone, Stochastic Versus Deterministic Systems of Differential Equations addresses questions relating to the need for a stochastic mathematical model and the between-model contrast that arises in the absence of random disturbances/flu
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
Combat modeling with partial differential equations
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.
Laplace transform of fractional order differential equations
Song Liang
2015-05-01
Full Text Available In this article, we show that Laplace transform can be applied to fractional system. To this end, solutions of linear fractional-order equations are first derived by a direct method, without using Laplace transform. Then the solutions of fractional-order differential equations are estimated by employing Gronwall and Holder inequalities. They are showed be to of exponential order, which are necessary to apply the Laplace transform. Based on the estimates of solutions, the fractional-order and the integer-order derivatives of solutions are all estimated to be exponential order. As a result, the Laplace transform is proved to be valid in fractional equations.
Differential equations and applications recent advances
2014-01-01
Differential Equations and Applications : Recent Advances focus on the latest developments in Nonlinear Dynamical Systems, Neural Networks, Fluid Dynamics, Fractional Differential Systems, Mathematical Modelling and Qualitative Theory. Different aspects such as Existence, Stability, Controllability, Viscosity and Numerical Analysis for different systems have been discussed in this book. This book will be of great interest and use to researchers in Applied Mathematics, Engineering and Mathematical Physics.
Counting Coloured Planar Maps: Differential Equations
Bernardi, Olivier; Bousquet-Mélou, Mireille
2017-08-01
We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic, that is, satisfies a non-trivial polynomial differential equation with respect to the edge variable. We give explicitly a differential system that characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. In statistical physics terms, we solve the q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating function was proved to be algebraic for certain values of q, including {q=1, 2} and 3. It is known to be transcendental in general. In contrast, our differential system holds for an indeterminate q. For certain special cases of combinatorial interest (four colours; proper q-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.
Inverse problems in ordinary differential equations and applications
Llibre, Jaume
2016-01-01
This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.
A Collocation Method for Solving Fractional Riccati Differential Equation
Yalçın Öztürk
2013-01-01
Full Text Available We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.
Differential-algebraic solutions of the heat equation
Buchstaber, Victor M.; Netay, Elena Yu.
2014-01-01
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.
Difference and differential equations with applications in queueing theory
Haghighi, Aliakbar Montazer
2013-01-01
A Useful Guide to the Interrelated Areas of Differential Equations, Difference Equations, and Queueing Models Difference and Differential Equations with Applications in Queueing Theory presents the unique connections between the methods and applications of differential equations, difference equations, and Markovian queues. Featuring a comprehensive collection of
A first course in differential equations, modeling, and simulation
Smith, Carlos A
2011-01-01
IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn Example Antidifferentiation: Technique for Solving First-Order Ordinary Differential EquationsBack to Section 2-1Another ExampleSeparation of Variables: Technique for Solving First-Order Ordinary Differential Equations Back to Section 2-5Equations, Unknowns, and Degrees of FreedomClassical Solutions of Ordinary Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating Factor MethodCharacteristic Equation
International Conference on Differential and Difference Equations with Applications
Došlá, Zuzana; Došlý, Ondrej; Kloeden, Peter
2016-01-01
Aimed at the community of mathematicians working on ordinary and partial differential equations, difference equations, and functional equations, this book contains selected papers based on the presentations at the International Conference on Differential and Difference Equations and Applications (ICDDEA) 2015, dedicated to the memory of Professor Georg Sell. Contributions include new trends in the field of differential and difference equations, applications of differential and difference equations, as well as high-level survey results. The main aim of this recurring conference series is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential and difference equations are represented, with special emphasis on applications.
Delay differential equations with homogeneous integral conditions
Abdur Raheem
2013-03-01
Full Text Available In this article we prove the existence and uniqueness of a strong solution of a delay differential equation with homogenous integral conditions using the method of semidiscretization in time. As an application, we include an example that illustrates the main result.
Efficient Estimating Functions for Stochastic Differential Equations
Jakobsen, Nina Munkholt
The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over...
Jensen's Inequality for Backward Stochastic Differential Equations
Long JIANG
2006-01-01
Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen's inequality holds for backward stochastic differential equations ith generator g if and only ifg is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen's inequality for gexpectation in [4, 7-9].
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
Nonstandard finite difference schemes for differential equations
Mohammad Mehdizadeh Khalsaraei
2014-12-01
Full Text Available In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs. Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with standard methods.
The geometry of differential difference equations
Helminck, G.F.; Post, G.F.
1994-01-01
To each maximal commuting subalgebra h of glm(C) is associated a system of differential difference equations, generalizing several known systems. Starting from a Grassmann manifold, solutions are constructed, their properties are discussed and the relation with other systems is given. Finally it is shown how to express these solutions in T-functions.
Neutral Operator and Neutral Differential Equation
Jingli Ren
2011-01-01
Full Text Available In this paper, we discuss the properties of the neutral operator (Ax(t=x(t−cx(t−δ(t, and by applying coincidence degree theory and fixed point index theory, we obtain sufficient conditions for the existence, multiplicity, and nonexistence of (positive periodic solutions to two kinds of second-order differential equations with the prescribed neutral operator.
Strong monotonicity for analytic ordinary differential equations
Sebastian Walcher
2009-09-01
Full Text Available We present a necessary and sufficient criterion for the flow of an analytic ordinary differential equation to be strongly monotone; equivalently, strongly order-preserving. The criterion is given in terms of the reducibility set of the derivative of the right-hand side. Some applications to systems relevant in biology and ecology, including nonlinear compartmental systems, are discussed.
Singular Linear Differential Equations in Two Variables
Braaksma, B.L.J.; Put, M. van der
2008-01-01
The formal and analytic classification of integrable singular linear differential equations has been studied among others by R. Gerard and Y. Sibuya. We provide a simple proof of their main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no
Fractional Order Differential Equations Involving Caputo Derivative
Zoubir Dahmani
2014-04-01
Full Text Available In this paper, the Banach contraction principle and Schaefer theorem are applied to establish new results for the existence and uniqueness of solutions for some Caputo fractional differential equations. Some examples are also discussed to illustrate the main results.
Stochastic differential equations used to model conjugation
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
Stochastic differential equations (SDEs) are used to model horizontal transfer of antibiotic resis- tance by conjugation. The model describes the concentration of donor, recipient, transconjugants and substrate. The strength of the SDE model over the traditional ODE models is that the noise can...
A simple derivation of Kepler's laws without solving differential equations
Provost, J. -P.; Bracco, C.
2008-01-01
Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, ...
Rough differential equations with unbounded drift term
Riedel, S.; Scheutzow, M.
2017-01-01
We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply strong completeness and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin-Wentzell-type large deviation result.
Ordinary differential equations a graduate text
Bhamra, K S
2015-01-01
ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. The book discusses qualitative and quantitative aspects of the Strum - Liouville problems, Green's function, integral equations, Laplace transform and is supported by a number of worked-out examples in each lesson to make the concepts clear. A lot of stress on stability theory is laid down, especially on Lyapunov and Poincare stability theory. A numerous figures in various lessons (in particular lessons dealing with stability theory) have been added to clarify the key concepts in DE theory. Nonlinear oscillation in conservative systems and Hamiltonian systems highlights basic nature of the systems considered. Perturbation techniques lesson deals in fairly d...
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.
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 ...
Introduction to numerical methods for time dependent differential equations
Kreiss, Heinz-Otto
2014-01-01
Introduces both the fundamentals of time dependent differential equations and their numerical solutions Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs). Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the t
The example of modeling of logistics processes using differential equations
Ryczyński, Jacek
2017-07-01
The article describes the use of differential calculus to determine the form of differential equations family of curves. Form of differential equations obtained by eliminating the parameters of the equations describing the different family of curves. Elimination of the parameters has been performed several times by differentiation starting equations. Received appropriate form of differential equations for the case of family circles, family of curves of the second degree and the families of the logistic function.
Electrocardiogram classification using delay differential equations
Lainscsek, Claudia; Sejnowski, Terrence J.
2013-06-01
Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.
Regularized Semiparametric Estimation for Ordinary Differential Equations.
Li, Yun; Zhu, Ji; Wang, Naisyin
2015-07-01
Ordinary differential equations (ODEs) are widely used in modeling dynamic systems and have ample applications in the fields of physics, engineering, economics and biological sciences. The ODE parameters often possess physiological meanings and can help scientists gain better understanding of the system. One key interest is thus to well estimate these parameters. Ideally, constant parameters are preferred due to their easy interpretation. In reality, however, constant parameters can be too restrictive such that even after incorporating error terms, there could still be unknown sources of disturbance that lead to poor agreement between observed data and the estimated ODE system. In this paper, we address this issue and accommodate short-term interferences by allowing parameters to vary with time. We propose a new regularized estimation procedure on the time-varying parameters of an ODE system so that these parameters could change with time during transitions but remain constants within stable stages. We found, through simulation studies, that the proposed method performs well and tends to have less variation in comparison to the non-regularized approach. On the theoretical front, we derive finite-sample estimation error bounds for the proposed method. Applications of the proposed method to modeling the hare-lynx relationship and the measles incidence dynamic in Ontario, Canada lead to satisfactory and meaningful results.
A textbook on ordinary differential equations
Ahmad, Shair
2014-01-01
The book is a primer of the theory of Ordinary Differential Equations. Each chapter is completed by a broad set of exercises; the reader will also find a set of solutions of selected exercises. The book contains many interesting examples as well (like the equations for the electric circuits, the pendium equation, the logistic equation, the Lotka-Volterra system, and many other) which introduce the reader to some interesting aspects of the theory and its applications. The work is mainly addressed to students of Mathematics, Physics, Engineering, Statistics, Computer Sciences, with knowledge of Calculus and Linear Algebra, and contains more advanced topics for further developments, such as Laplace transform; Stability theory and existence of solutions to Boundary Value problems. The authors are preparing a complete solutions manual, containing solutions to all the exercises published in the book. The manual will be available Summer 2014. Instructors who wish to adopt the book may request the manual by writing...
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...
New Fractional Complex Transform for Conformable Fractional Partial Differential Equations
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.
On a perturbation method for partial differential equations
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)
Parameter Estimates in Differential Equation Models for Population Growth
Winkel, Brian J.
2011-01-01
We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer "Mathematica" code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist,…
Differential equations of the shell suspension. Three-dimensional problem
Гнатейко, Нонна Валентинівна
2016-01-01
Differential equations of suspension of gyroscope are constructed as a shell of rotation. Analytical material well-being of analysis of properties of suspension as a system with part parameters is created. Three-dimensional approach gives possibility for the comprehensive study of errors of devices of inertial navigation from position impedance status.
A Simple Derivation of Kepler's Laws without Solving Differential Equations
Provost, J.-P.; Bracco, C.
2009-01-01
Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…
A RATIONAL SEPCTRAL METHOD FOR SINGULAR DIFFERENTIAL EQUATIONS
王中庆; 王立联; 郭本瑜
2003-01-01
An orthogonal system of rational functions is derived from the mapped Laguerre polynomials,which is used for numerical solution of singular differential equations.A model problem is considered.A multiple-step algorithm is developed to implement this method.Numerical results show the efficiency of this new approach.
Initial and boundary value problems for partial functional differential equations
Ntouyas , S. K.; P. Ch Tsamatos
1997-01-01
In this paper we study the existence of solutions to initial and boundary value problems of partial functional differential equations via a fixed-point analysis approach. Using the topological transversality theorem we derive conditions under which an initial or a boundary value problem has a solution.
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Modification of Ordinary Differential Equations MATLAB Solver
E. Cocherova
2003-12-01
Full Text Available Various linear or nonlinear electronic circuits can be described bythe set of ordinary differential equations (ODEs. The ordinarydifferential equations can be solved in the MATLAB environment inanalytical (symbolic toolbox or numerical way. The set of nonlinearODEs with high complexity can be usually solved only by use ofnumerical integrator (solver. The modification of ode23 MATLABnumerical solver has been suggested in this article for the applicationin solution of some special cases of ODEs. The main feature of thismodification is that the solution is found at every prescribed point,in which the special behavior of system is anticipated. Theextrapolation of solution is not allowed in those points.
An introduction to ordinary differential equations
Coddington, Earl A
1989-01-01
""Written in an admirably cleancut and economical style."" - Mathematical Reviews. This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented. The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solu
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.
Asymptotic stability of singularly perturbed differential equations
Artstein, Zvi
2017-02-01
Asymptotic stability is examined for singularly perturbed ordinary differential equations that may not possess a natural split into fast and slow motions. Rather, the right hand side of the equation is comprised of a singularly perturbed component and a regular one. The limit dynamics consists then of Young measures, with values being invariant measures of the fast contribution, drifted by the slow one. Relations between the asymptotic stability of the perturbed system and the limit dynamics are examined, and a Lyapunov functions criterion, based on averaging, is established.
Small delay approximation of stochastic delay differential equations
Guillouzic, Steve; L'heureux, Ivan; Longtin, André
1999-04-01
Delay differential equations evolve in an infinite-dimensional phase space. In this paper, we consider the effect of external fluctuations (noise) on delay differential equations involving one variable, thus leading to univariate stochastic delay differential equations (SDDE's). For small delays, a univariate nondelayed stochastic differential equation approximating such a SDDE is presented. Another approximation, complementary to the first, is also obtained using an average of the SDDE's drift term over the delayed dynamical variable, which defines a conditional average drift. This second approximation is characterized by the fact that the diffusion term is identical to that of the original SDDE. For small delays, our approach yields a steady-state probability density and a conditional average drift which are in close agreement with numerical simulations of the original SDDE. We illustrate this scheme with the delayed linear Langevin equation and a stochastic version of the delayed logistic equation. The technique can be used with any type of noise, and is easily generalized to multiple delays.
Numerical diagnostics of solution blowup in differential equations
Belov, A. A.
2017-01-01
New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.
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.
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.
Linear measure functional differential equations with infinite delay
Monteiro, G.; Slavík, A.
2014-01-01
We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.
A Birkhoff-Noether method of solving differential equations
Shang Mei; Guo Yong-Xin; Mei Feng-Xiang
2007-01-01
In this paper, a Birkhoff-Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoff's equations. The first integrals for differential equations can be found by using the Noether theory for Birkhoffian systems. Two examples are given to illustrate the application of the method.
Parameter estimation in stochastic differential equations
Bishwal, Jaya P N
2008-01-01
Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modelling complex phenomena and making beautiful decisions. The subject has attracted researchers from several areas of mathematics and other related fields like economics and finance. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods. Useful because of the current availability of high frequency data is the study of refined asymptotic properties of several estimators when the observation time length is large and the observation time interval is small. Also space time white noise driven models, useful for spatial data, and more sophisticated non-Markovian and non-semimartingale models like fractional diffusions that model the long memory phenomena are examined in this volume.
Stochastic differential equations and a biological system
Wang, Chunyan
1994-01-01
on experimental data is considered. As an example, the growth of bacteria Pseudomonas fluorescens is taken. Due to the specific features of stochastic differential equations, namely that their solutions do not exist in the general sense, two new integrals - the Ito integral and the Stratonovich integral - have......The purpose of this Ph.D. study is to explore the property of a growth process. The study includes solving and simulating of the growth process which is described in terms of stochastic differential equations. The identification of the growth and variability parameters of the process based......, Milstein and Runge-Kutta methods are used. Because of the specific feature of the model for the growth process, that its solution does not exist in the general sense, we combine these numerical integration methods with a transformation technique, and the solutions are derived in the Ito sense...
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.
Stochastic Functional Differential Equation under Regime Switching
Ling Bai
2012-01-01
Full Text Available We discuss stochastic functional differential equation under regime switching dx(t=f(xt,r(t,tdt+q(r(tx(tdW1(t+σ(r(t|x(t|βx(tdW2(t. We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system.
Stationary conditions for stochastic differential equations
Adomian, G.; Walker, W. W.
1972-01-01
This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.
Integrability Estimates for Gaussian Rough Differential Equations
Cass, Thomas; Lyons, Terry
2011-01-01
We derive explicit tail-estimates for the Jacobian of the solution flow of stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems.
Stability of the Stochastic Differential Equations
Klimešová, M.
2015-01-01
Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. The method of Lyapunov functions for the analysis of qualitative behavior of SDEs provide some very powerful instruments in the study of stability properties for concrete stochastic dynamical systems, conditions of existence the stationary solutions of SDEs and related problems. The study of exponential stability of the moments makes natural the conside...
Desingularization of implicit analytic differential equations
Cendra, Hernan [Universidad Nacional del Sur, Av. Alem 1253, 8000 BahIa Blanca and CONICET (Argentina); Etchechoury, MarIa [Laboratorio de Electronica Industrial, Control e Instrumentacion, Facultad de IngenierIa, Universidad Nacional de La Plata, La Plata (Argentina)
2006-09-01
The question of finding solutions to a given implicit differential equation (IDE) is an important one, in part because it appears very naturally in several problems in physics, engineering and many other fields. In this work, we show how to reduce a given analytic IDE to an analytic IDE of locally constant rank. This can be done by using some fundamental results on subanalytic subsets and desingularization of closed subanalytic subsets. An example from nonholonomic mechanics is studied using these methods.
Ordinary differential equations introduction and qualitative theory
Cronin, Jane
2007-01-01
… a classic treatment of many of the topics an instructor would want in such a course, with particular emphasis on those aspects of the qualitative theory that are important for applications to mathematical biology. … A nice feature of this edition is an extended and unified treatment of the perturbation problem for periodic solutions. … a solid graduate-level introduction to ordinary differential equations, especially for applications. …-MAA Reviews, August 2010
Underdetermined systems of partial differential equations
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.
Ordinary differential equations in affine geometry
Salvador Gigena
1996-05-01
Full Text Available The method of qualitative analysis is used, as applied to a class of fourth order, nonlinear ordinary differential equations, in order to classify, both locally and globally, two classes of hypersurfaces of decomposable type in affine geometry: those with constant unimodular affine mean curvature L , and those with constant Riemannian scalar curvature R. This allows to provide a large number of new examples of hypersurfaces in affine geometry.
Ordinary differential equations in affine geometry
Salvador Gigena
1996-01-01
The method of qualitative analysis is used, as applied to a class of fourth order, nonlinear ordinary differential equations, in order to classify, both locally and globally, two classes of hypersurfaces of decomposable type in affine geometry: those with constant unimodular affine mean curvature L , and those with constant Riemannian scalar curvature R. This allows to provide a large number of new examples of hypersurfaces in affine geometry.
Ambit processes and stochastic partial differential equations
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....
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...
Synchronization with propagation - The functional differential equations
Rǎsvan, Vladimir
2016-06-01
The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators. On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems. The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.
A textbook on ordinary differential equations
Ahmad, Shair
2015-01-01
This book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, whic...
A New Expanded Method for Solving Nonlinear Differential-difference Equation
ZHANG Shan-qing
2008-01-01
A new expanded approach is presented to find exact solutions of nonlinear differential-difference equations. As its application, the soliton solutions and periodic solutions of a lattice equation are obtained.
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
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.
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
Effective action for stochastic partial differential equations
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
Numerical analysis of systems of ordinary and stochastic differential equations
Artemiev, S S
1997-01-01
This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).
On oscillatory solutions of third order differential equation with quasiderivatives
Miroslav Bartusek
2000-07-01
Full Text Available This paper gives sufficient conditions under which all oscillatory solutions of a third order nonlinear differential equation with quasiderivatives vanish at infinity. Applications to third order differentials equation with a middle term are also given.
Oscillation of third order functional differential equations with delay
Tuncay Candan
2003-02-01
Full Text Available We consider third order functional differential equations with discrete and continuous delay. We then develop several theorems related to the oscillatory behavior of these differential equations.
Bounded solutions for fuzzy differential and integral equations
Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es
2006-03-01
We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.
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 ...
Singular solutions of a singular differential equation
Naito Manabu
2000-01-01
Full Text Available An attempt is made to study the problem of existence of singular solutions to singular differential equations of the type which have never been touched in the literature. Here and are positive constants and is a positive continuous function on . A solution with initial conditions given at is called singular if it ceases to exist at some finite point . Remarkably enough, it is observed that the equation may admit, in addition to a usual blowing-up singular solution, a completely new type of singular solution with the property that Such a solution is named a black hole solution in view of its specific behavior at . It is shown in particular that there does exist a situation in which all solutions of are black hole solutions.
Differential equations, associators, and recurrences for amplitudes
Georg Puhlfürst
2016-01-01
Full Text Available We provide new methods to straightforwardly obtain compact and analytic expressions for ϵ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ϵ-orders of a power series solution in ϵ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ϵ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ϵ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system. Finally, we set up our methods to systematically get compact and explicit α′-expansions of tree-level superstring amplitudes to any order in α′.
Differential equations, associators, and recurrences for amplitudes
Puhlfürst, Georg; Stieberger, Stephan
2016-01-01
We provide new methods to straightforwardly obtain compact and analytic expressions for ɛ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ɛ-orders of a power series solution in ɛ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ɛ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ɛ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we set up our methods to systematically get compact and explicit α‧-expansions of tree-level superstring amplitudes to any order in α‧.
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.
Elliptic differential equations theory and numerical treatment
Hackbusch, Wolfgang
2017-01-01
This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics.
Differential Equations, Associators, and Recurrences for Amplitudes
Puhlfuerst, Georg
2015-01-01
We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different epsilon-orders of a power series solution in epsilon of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the epsilon-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also setup up our tools for computing epsilon-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system). Finally, we apply our methods to systematically get compact and explicit alpha'-expansions of tree-level superstring amplitudes to any order in alpha'.
Algorithm refinement for stochastic partial differential equations.
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.
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
Optimizing second-order differential equation systems
Tamas Hajba
2011-03-01
Full Text Available In this article we study some continuous versions of the Fletcher-Reeves iteration for minimization described by a system of second-order differential equations. This problem has been studied in earlier papers [19, 20] under the assumption that the minimizing function is strongly convex. Now instead of the strong convexity, only the convexity of the minimizing function will be required. We will use the Tikhonov regularization [28, 29] to obtain the minimal norm solution as the asymptotically stable limit point of the trajectories.
Stochastic differential equations and diffusion processes
Ikeda, N
1989-01-01
Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sectio
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
ASYMPTOTIC STABILITIES OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
SHEN Yi; JIANG Ming-hui; LIAO Xiao-xin
2006-01-01
Asymptotic characteristic of solution of the stochastic functional differential equation was discussed and sufficient condition was established by multiple Lyapunov functions for locating the limit set of t he solution. Moreover, from them many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in application, were obtained. The results show that the wellknown classical theorem on stochastic asymptotic stability is a special case of our more general results. In the end, application in stochastic Hopfield neural networks is given to verify our results.
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
The Approach to Equilibrium: Detailed Balance and the Master Equation
Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.
2011-01-01
The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…
Differential equations a primer for scientists and engineers
Constanda, Christian
2017-01-01
This textbook is designed with the needs of today’s student in mind. It is the ideal textbook for a first course in elementary differential equations for future engineers and scientists, including mathematicians. This book is accessible to anyone who has a basic knowledge of precalculus algebra and differential and integral calculus. Its carefully crafted text adopts a concise, simple, no-frills approach to differential equations, which helps students acquire a solid experience in many classical solution techniques. With a lighter accent on the physical interpretation of the results, a more manageable page count than comparable texts, a highly readable style, and over 1000 exercises designed to be solved without a calculating device, this book emphasizes the understanding and practice of essential topics in a succinct yet fully rigorous fashion. Apart from several other enhancements, the second edition contains one new chapter on numerical methods of solution. The book formally splits the "pure" and "applie...
Positive periodic solutions for third-order nonlinear differential equations
Jingli Ren
2011-05-01
Full Text Available For several classes of third-order constant coefficient linear differential equations we obtain existence and uniqueness of periodic solutions utilizing explicit Green's functions. We discuss an iteration method for constant coefficient nonlinear differential equations and provide new conditions for the existence of periodic positive solutions for third-order time-varying nonlinear and neutral differential equations.
Probability Measures for Numerical Solutions of Differential Equations
Conrad, Patrick R.; Girolami, Mark; Särkkä, Simo; Stuart, Andrew; Zygalakis, Konstantinos
2015-01-01
In this paper, we present a formal quantification of epistemic uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it is therefore important to explicitly...
GLOBAL LINEARIZATION OF DIFFERENTIAL EQUATIONS WITH SPECIAL STRUCTURES
无
2011-01-01
This paper introduces the global linearization of the differential equations with special structures.The function in the differential equation is unbounded.We prove that the differential equation with unbounded function can be topologically linearlized if it has a special structure.
Abstract Operators and Higher-order Linear Partial Differential Equation
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.
Complex Transforms for Systems of Fractional Differential Equations
Rabha W. Ibrahim
2012-01-01
Full Text Available We provide a complex transform that maps the complex fractional differential equation into a system of fractional differential equations. The homogeneous and nonhomogeneous cases for equivalence equations are discussed and also nonequivalence equations are studied. Moreover, the existence and uniqueness of solutions are established and applications are illustrated.
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.
A NEW APPROACH TO SOLVE SYSTEMS OF LINEAR EQUATIONS
Luis Vázquez; José L. Vázquez-Poletti
2001-01-01
We propose a new iterative approach to solve systems of linear equations. The new strategy integrates the algebraic basis of the problem with elements from classical mechanics and the finite difference method. The approach defines two families of convergent iterative methods. Each family is characterized by a linear differential equation, and every method is obtained from a suitable finite difference scheme to integrate the associated differential equation. The methods are general and depend on neither the matrix dimension nor the matrix structure. In this preliminary work, we present the basic features of the method with a simple application to a low dimensional system.
Some recent advances in the numerical solution of differential equations
D'Ambrosio, Raffaele
2016-06-01
The purpose of the talk is the presentation of some recent advances in the numerical solution of differential equations, with special emphasis to reaction-diffusion problems, Hamiltonian problems and ordinary differential equations with discontinuous right-hand side. As a special case, in this short paper we focus on the solution of reaction-diffusion problems by means of special purpose numerical methods particularly adapted to the problem: indeed, following a problem oriented approach, we propose a modified method of lines based on the employ of finite differences shaped on the qualitative behavior of the solutions. Constructive issues and a brief analysis are presented, together with some numerical experiments showing the effectiveness of the approach and a comparison with existing solvers.
Numerical and asymptotic studies of delay differential equations
Adhikari, Mohit Hemchandra
Two classes of differential delay equations exhibiting diverse phenomena are studied. The first one is a singularly perturbed delay differential equation which is used to model selected physical systems involving feedback where relaxation effects are combined with nonlinear driving from the past. In the limit of fast relaxation, the differential equation reduces to a difference equation or a map, due to the presence of the delay. A basic question in this field is how the behavior of the map is reflected in the behavior of the solutions of the delay differential equation. In this work, a generic logistic form is used for the underlying map and the above question is studied in the first period-doubling regime of the map. Using an efficient numerical algorithm, the shape and the period of the corresponding asymptotically stable periodic solution is studied first, for various values of the delay. In the limit of large delay, these solutions resemble square-waves of period close to twice the value of the delay, with sharp transition layers joining flat plateau-like regions. A Poincare-Lindstedt method involving a two-parameter perturbation expansion is applied to solve equations representing these layers and accurate expressions for the shape and the period of these solutions, in terms of Jacobi elliptic functions, are obtained. A similar approach is used to obtain leading order expressions for sub-harmonic solutions of shorter periods, but it is shown that while they are extremely long-lived for large values of delay, they eventually decay to the fundamental solutions mentioned above. The spectral algorithm used for the numerical integration is tested by comparing its accuracy and efficiency in obtaining stiff solutions of linear delay equations, with that of a current state-of-the-art time-stepping algorithm for integrating delay equations. Effect of delay on the synchronization of two nerve impulses traveling along two parallel nerve fibers, is the second question
Analytic solutions of a class of nonlinear partial differential equations
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.
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
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
Differential Forms and Wave Equations for General Relativity
Lau, S R
1996-01-01
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's 2-form, which in the ``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The tensor-index version of our wave e...
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.
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.
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.
Computing spacetime curvature via differential-algebraic equations
Ashby, S.F. [Lawrence Livermore National Lab., CA (United States); Lee, S.L. [Oak Ridge National Lab., TN (United States); Petzold, L.R. [Minnesota Univ., Minneapolis, MN (United States). Dept. of Computer Science; Saylor, P.E.; Seidel, E. [Illinois Univ., Urbana, IL (United States)
1996-01-01
The equations that govern the behavior of physical systems can often solved numerically using a method of lines approach and differential-algebraic equation (DAE) solvers. For example, such an approach can be used to solve the Einstein field equations of general relativity, and thereby simulate significant astrophysical events. In this paper, we describe some preliminary work in which two model problems in general relativity are formulated, spatially discretized, and then numerically solved as a DAE. In particular, we seek to reproduce the solution to the spherically symmetric Schwarzschild spacetime. This is an important testbed calculation in numerical relativity since the solution is the steady-state for the collision of two (or more) non-rotating black holes. Moreover, analytic late-time properties of the Schwarzschild spacetime are well known and can be used the accuracy of the simulation.
Numerical Methods for Stochastic Partial Differential Equations
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.
Lattice quantum chromodynamics equation of state: A better differential method
Rajiv V Gavai; Sourendu Gupta; Swagato Mukherjee
2008-09-01
We propose a better differential method for the computation of the equation of state of QCD from lattice simulations. In contrast to the earlier differential method, our technique yields positive pressure for all temperatures including the temperatures in the transition region. Employing it on temporal lattices of 8, 10 and 12 sites and by extrapolating to zero lattice spacing we obtained the pressure, energy density, entropy density, specific heat and speed of sound in quenched QCD for 0.9 ≤ /c ≤ 3. At high temperatures comparisons of our results are made with those from the dimensional reduction approach and also with those from a conformal symmetric theory.
Exact periodic wave solutions for some nonlinear partial differential equations
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.
Ullah, Saif; Farooq, Muhammad; Ahmad, Latif; Abdullah, Saleem
2014-01-01
Fuzzy partial integro-differential equations have a major role in the fields of science and engineering. In this paper, we propose the solution of fuzzy partial Volterra integro-differential equation with convolution type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with few examples to show the ability of the proposed method.
A note on the Lie symmetries of complex partial differential equations and their split real systems
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.
Function Substitution in Partial Differential Equations: Nonhomogeneous Boundary Conditions
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
Tornøe, Christoffer Wenzel; Overgaard, Rune Viig; Agerso, H.;
2005-01-01
Purpose. The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling. Methods. The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types...
Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility
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.
Exact solutions for nonlinear partial fractional differential equations
Khaled A.Gepreel; Saleh Omran
2012-01-01
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved (G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
An Implementation Solution for Fractional Partial Differential Equations
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.
Fast methods for static Hamilton-Jacobi Partial Differential Equations
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
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.
Advances in differential equations and applications
Martínez, Vicente
2014-01-01
The book contains a selection of contributions given at the 23rd Congress on Differential Equations and Applications (CEDYA) / 13th Congress of Applied Mathematics (CMA) that took place at Castellon, Spain, in 2013. CEDYA is renowned as the congress of the Spanish Society of Applied Mathematics (SEMA) and constitutes the main forum and meeting point for applied mathematicians in Spain. The papers included in this book have been selected after a thorough refereeing process and provide a good summary of the recent activity developed by different groups working mainly in Spain on applications of mathematics to several fields of science and technology. The purpose is to provide a useful reference of academic and industrial researchers working in the area of numerical analysis and its applications.
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.
Ordinary differential equations basics and beyond
Schaeffer, David G
2016-01-01
This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos). While proofs are rigorous, the exposition is reader-friendly, aiming for the informality of face-to-face interactions. A unique feature of this book is the integration of rigorous theory with numerous applications of scientific interest. Besides providing motivation, this synthesis clarifies the theory and enhances scientific literacy. Other features include: (i) a wealth of exercises at various levels, along with commentary that explains why they matter; (ii) figures with consistent color conventions to identify nullclines, periodic orbits, stable and unstable manifolds; and (iii) a dedicated website with software templates, problem solutions, and other resources supporting the text. Given its many applications, the book may be used comfortably in sc...
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.
Stability and Control of Functional Differential Equations
Peet, M M
2006-01-01
This thesis addresses the question of stability of systems defined by differential equations which contain nonlinearity and delay. In particular, we analyze the stability of a well-known delayed nonlinear implementation of a certain Internet congestion control protocol. We also describe a generalized methodology for proving stability of time-delay systems through the use of semidefinite programming. In Chapters 4 and 5, we consider an Internet congestion control protocol based on the decentralized gradient projection algorithm. For a certain class of utility function, this algorithm was shown to be globally convergent for some sufficiently small value of a gain parameter. Later work gave an explicit bound on this gain for a linearized version of the system. This thesis proves that this bound also implies stability of the original system. In Chapter 7, we describe a general methodology for proving stability of linear time-delay systems by computing solutions to an operator-theoretic version of the Lyapunov ine...
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.
Differential equations of my young years
Maz'ya, Vladimir
2014-01-01
Vladimir Maz'ya (born 1937) is an outstanding mathematician who systematically made fundamental contributions to a wide array of areas in mathematical analysis and in the theory of partial differential equations. In this fascinating book he describes the first thirty years of his life in Leningrad (now St. Petersburg). He starts with the story of his family, speaks about his childhood, the high school and university years, and recalls his formative years as a mathematician. Behind the author's personal recollections, with his own joys, sorrows and hopes, one sees a vivid picture of those times in the former Sovjet Union. He speaks warmly about his friends, both outside and inside the world of mathematics, about discovering his passion for mathematics and his early achievements, and about a number of mathematicians who influenced his professional life. The book is written in a highly readable and inviting style, spiced with the occasional touch of humor.
Hilbert space methods for partial differential equations
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...
An axiomatic approach to Maxwell's equations
Heras, José A
2016-01-01
This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.
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
Camporesi, Roberto
2011-06-01
We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.
Modelling biochemical reaction systems by stochastic differential equations with reflection.
Niu, Yuanling; Burrage, Kevin; Chen, Luonan
2016-05-07
In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach.
Partial Differential Equations in General Relativity
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)
Introduction to inverse problems for differential equations
Hasanov Hasanoğlu, Alemdar
2017-01-01
This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering. The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations. In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here a...
Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition
Marek T. Malinowski
2016-01-01
Full Text Available We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic 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.
Relations between Stochastic and Partial Differential Equations in Hilbert Spaces
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.
NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS
无
2006-01-01
In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h).
Time Reversal of Volterra Processes Driven Stochastic Differential Equations
L. Decreusefond
2013-01-01
Full Text Available We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.
Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach
DAI Chao-Qing; ZHANG Jie-Fang
2006-01-01
In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.
Compatible Spatial Discretizations for Partial Differential Equations
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
A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS
Ishtiaq Ali; Hermann Brunner; Tao Tang
2009-01-01
We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the given pantograph delay differential equation are smooth.
Asymptotic behavior and stability of second order neutral delay differential equations
Chen, G.L.; van Gaans, O.W.; Verduyn Lunel, Sjoerd
2014-01-01
We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed
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.
Exact solutions for some nonlinear partial differential equations
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.
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Structure-preserving algorithms for oscillatory differential equations
Wu, Xinyuan; Wang, Bin
2013-01-01
Structure-Preserving Algorithms for Oscillatory Differential Equations describes a large number of highly effective and efficient structure-preserving algorithms for second-order oscillatory differential equations by using theoretical analysis and numerical validation. Structure-preserving algorithms for differential equations, especially for oscillatory differential equations, play an important role in the accurate simulation of oscillatory problems in applied sciences and engineering. The book discusses novel advances in the ARKN, ERKN, two-step ERKN, Falkner-type and energy-preserving methods, etc. for oscillatory differential equations. The work is intended for scientists, engineers, teachers and students who are interested in structure-preserving algorithms for differential equations. Xinyuan Wu is a professor at Nanjing University; Xiong You is an associate professor at Nanjing Agricultural University; Bin Wang is a joint Ph.D student of Nanjing University and University of Cambridge.
FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOPPING TIME
吴臻
2004-01-01
The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.
Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations
Gugushvili, S.; Spreij, P.
2014-01-01
We consider nonparametric Bayesian estimation of the drift coefficient of a multidimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions, we establish posterior consistency in this context.
An introduction to differential equations and their applications
Farlow, Stanley J
2006-01-01
This introductory text explores 1st- and 2nd-order differential equations, series solutions, the Laplace transform, difference equations, much more. Numerous figures, problems with solutions, notes. 1994 edition. Includes 268 figures and 23 tables.
Limit theorems for solutions of stochastic differential equation problems
J. Vom Scheidt
1980-01-01
Full Text Available In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.
STRICT STABILITY OF IMPULSIVE SET VALUED DIFFERENTIAL EQUATIONS
无
2011-01-01
In this paper, we develop strict stability concepts of ODE to impulsive hybrid set valued differential equations. By Lyapunov’s original method, we get some basic strict stability criteria of impulsive hybrid set valued equations.
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.
Modeling Tree Crown Dynamics with 3D Partial Differential Equations
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.
RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
Zhou Shaobo; Hu Shigeng
2009-01-01
The stability of stochastic functional differential equation with Markovian switching was studied by several authors, but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching. The aim of this article is to close this gap. The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching, and those without Markovian switching.
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
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
Hamilton Jacobi method for solving ordinary differential equations
Mei, Feng-Xiang; Wu, Hui-Bin; Zhang, Yong-Fa
2006-08-01
The Hamilton-Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton-Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.
On Volatility Induced Stationarity for Stochastic Differential Equations
Albin, J.M.P.; Astrup Jensen, Bjarne; Muszta, Anders;
2006-01-01
This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples.......This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples....
Local behavior of autonomous neutral functional differential equations.
Hale, J. K.
1972-01-01
Basic problems for a special class of neutral functional differential equations (NFDE) are formulated, and some contributions to a general qualitative theory in the neighborhood of an equilibrium point are indicated. The properties of a NFDE (G,f) are examined to determine in what sense these properties are insensitive to small changes in (G,f) in the topology G x F. The special class of equations that is introduced includes retarded functional differential equations and difference equations.
Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
Li, Lei; Liu, Jian-Guo; Lu, Jianfeng
2017-09-01
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.
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.
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.
Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Khaled A. Gepreel
2013-01-01
Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.
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.
Ondřej Došlý
2012-01-01
Full Text Available We investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.
Solving Differential Equations in R: Package deSolve
Karline Soetaert
2010-02-01
Full Text Available In this paper we present the R package deSolve to solve initial value problems (IVP written as ordinary differential equations (ODE, differential algebraic equations (DAE of index 0 or 1 and partial differential equations (PDE, the latter solved using the method of lines approach. The differential equations can be represented in R code or as compiled code. In the latter case, R is used as a tool to trigger the integration and post-process the results, which facilitates model development and application, whilst the compiled code significantly increases simulation speed. The methods implemented are efficient, robust, and well documented public-domain Fortran routines. They include four integrators from the ODEPACK package (LSODE, LSODES, LSODA, LSODAR, DVODE and DASPK2.0. In addition, a suite of Runge-Kutta integrators and special-purpose solvers to efficiently integrate 1-, 2- and 3-dimensional partial differential equations are available. The routines solve both stiff and non-stiff systems, and include many options, e.g., to deal in an efficient way with the sparsity of the Jacobian matrix, or finding the root of equations. In this article, our objectives are threefold: (1 to demonstrate the potential of using R for dynamic modeling, (2 to highlight typical uses of the different methods implemented and (3 to compare the performance of models specified in R code and in compiled code for a number of test cases. These comparisons demonstrate that, if the use of loops is avoided, R code can efficiently integrate problems comprising several thousands of state variables. Nevertheless, the same problem may be solved from 2 to more than 50 times faster by using compiled code compared to an implementation using only R code. Still, amongst the benefits of R are a more flexible and interactive implementation, better readability of the code, and access to R’s high-level procedures. deSolve is the successor of package odesolve which will be deprecated in
Simple equation method for nonlinear partial differential equations and its applications
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.
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.
A Method for Image Decontamination Based on Partial Differential Equation
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.
Calculation of similarity solutions of partial differential equations
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/.
Ibragimov, Nail H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Meleshko, Sergey V [School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000 (Thailand); Rudenko, Oleg V, E-mail: nib@bth.se, E-mail: sergey@math.sut.ac.th, E-mail: rudenko@acs366.phys.msu.ru [Department of Physics, Moscow State University, 119991 Moscow (Russian Federation)
2011-08-05
The paper deals with an evolutionary integro-differential equation describing nonlinear waves. A particular choice of the kernel in the integral leads to well-known equations such as the Khokhlov-Zabolotskaya equation, the Kadomtsev-Petviashvili equation and others. Since the solutions of these equations describe many physical phenomena, the analysis of the general model studied in this paper is important. One of the methods for obtaining solutions of differential equations is provided by the Lie group analysis. However, this method is not applicable to integro-differential equations. Therefore, we discuss new approaches developed in modern group analysis and apply them to the general model considered in this paper. Reduced equations and exact solutions are also presented.
Discretization of Fractional Differential Equations by a Piecewise Constant Approximation
Angstmann, Christopher N; McGann, Anna V
2016-01-01
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be eq...
Differential and difference equations a comparison of methods of solution
Maximon, Leonard C
2016-01-01
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associat...
Probabilistic Forecasts of Solar Irradiance by Stochastic Differential Equations
Iversen, Jan Emil Banning; Morales González, Juan Miguel; Møller, Jan Kloppenborg
2014-01-01
approach allows for characterizing both the interdependence structure of prediction errors of short-term solar irradiance and their predictive distribution. Three different stochastic differential equation models are first fitted to a training data set and subsequently evaluated on a one-year test set......Probabilistic forecasts of renewable energy production provide users with valuable information about the uncertainty associated with the expected generation. Current state-of-the-art forecasts for solar irradiance have focused on producing reliable point forecasts. The additional information...
Lie groups, differential equations, and geometry advances and surveys
2017-01-01
This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
A note on the auxiliary equation method for solving nonlinear partial differential equations
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.
Quantifying uncertainty, variability and likelihood for ordinary differential equation models
Weisse, Andrea Y
2010-10-28
Abstract Background In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space. Results The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability. Conclusions While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.
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.
Stochastic fuzzy differential equations of a nonincreasing type
Malinowski, Marek T.
2016-04-01
Stochastic fuzzy differential equations constitute an apparatus in modeling dynamic systems operating in fuzzy environment and governed by stochastic noises. In this paper we introduce a new kind of such the equations. Namely, the stochastic fuzzy differential of nonincreasing type are considered. The fuzzy stochastic processes which are solutions to these equations have trajectories with nonincreasing fuzziness in their values. In our previous papers, as a first natural extension of crisp stochastic differential equations, stochastic fuzzy differential equations of nondecreasing type were studied. In this paper we show that under suitable conditions each of the equations has a unique solution which possesses property of continuous dependence on data of the equation. To prove existence of the solutions we use sequences of successive approximate solutions. An estimation of an error of the approximate solution is established as well. Some examples of equations are solved and their solutions are simulated to illustrate the theory of stochastic fuzzy differential equations. All the achieved results apply to stochastic set-valued differential equations.
Sourcing for Parameter Estimation and Study of Logistic Differential Equation
Winkel, Brian J.
2012-01-01
This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…
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.
On the $psi$-dichotomy for homogeneous linear differential equations
Pham Ngoc Boi
2006-03-01
Full Text Available In this article we present some conditions for the $psi$-dichotomy of the homogeneous linear differential equation $x'=A(tx$. Under our condition every $psi$-integrally bounded function $f$ the nonhomogeneous linear differential equation $x'=A(tx +f(t$ has at least one $psi$-bounded solution on $(0,+infty$.
Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications
Lakshman Mahto
2013-01-01
Full Text Available We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.
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.
Sourcing for Parameter Estimation and Study of Logistic Differential Equation
Winkel, Brian J.
2012-01-01
This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…
Differential equations and folding of $n$-mani-folds
I. Mousa
2005-09-01
Full Text Available In this paper we will describe some topological and geometric characters of $n$-manifold by using the properties of differential equations. The folding and unfolding of $n$-manifold into itself will be deduced from viewpoint of the differential equations.
On Exact Controllability of First-Order Impulsive Differential Equations
Juan J. Nieto
2010-01-01
Full Text Available Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work, we present some new results concerning the exact controllability of a nonlinear ordinary differential equation with impulses.
Undergraduate Students' Mental Operations in Systems of Differential Equations
Whitehead, Karen; Rasmussen, Chris
2003-01-01
This paper reports on research conducted to understand undergraduate students' ways of reasoning about systems of differential equations (SDEs). As part of a semester long classroom teaching experiment in a first course in differential equations, we conducted task-based interviews with six students after their study of first order differential…
Exact solutions for some nonlinear systems of partial differential equations
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.
Lagrangian vector field and Lagrangian formulation of partial differential equations
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
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.
Analysis of Caputo impulsive fractional order differential equations with applications
Mahto, Lakshman; Favini, Angelo
2012-01-01
We use Sadavoskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order \\alpha between 0 and 1 with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and complement the existing one.
Using StarLogo To Introduce Differential Equations.
Anderson, Philip; Seaquist, Carl R.
Massively parallel programming languages, like StarLogo, provide a rich environment for introducing differential equations to students with an unsophisticated mathematical background. This paper describes the basic software for stimulating and monitoring various population dynamics. Simple differential equations that describe the observed dynamics…
Charles François Sturm and Differential Equations
Lützen, Jesper; Mingarelli, Angelo
2008-01-01
An analysis of Sturm's works on differential equations, in particular Sturm-Liouville theory. The historical connection to Sturm's theorem about real roots of polynomials is established......An analysis of Sturm's works on differential equations, in particular Sturm-Liouville theory. The historical connection to Sturm's theorem about real roots of polynomials is established...
One Dimensional Quasi-Exactly Solvable Differential Equations
Fasihi, Mohammad A.
2006-01-01
In this paper by means of similarity transformation we find some one-dimensional quasi-exactly solvable differential equations and their related Hamiltonians which appear in physical problems. We have provided also two examples with application of these differential equations.
Numbers of Subnormal Solutions for Higher Order Periodic Differential Equations
Zong Xuan CHEN; Kwang Ho SHON
2011-01-01
In this paper,we estimate the number of subnormal solutions for higher order linear periodic differential equations,and estimate the growth of subnormal solutions and all other solutions.We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.
Camporesi, Roberto
2011-01-01
We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…
Camporesi, Roberto
2011-01-01
We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…
On the hierarchy of partially invariant submodels of differential equations
Golovin, Sergey V
2007-01-01
It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given.
On the hierarchy of partially invariant submodels of differential equations
Golovin, Sergey V [Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk 630090 (Russian Federation)], E-mail: sergey@hydro.nsc.ru
2008-07-04
It is noted that the partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PISs of the higher rank. This introduces a hierarchic structure in the set of all PISs of a given system of differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. The hierarchy simplifies the process of enumeration and analysis of partially invariant submodels to the given system of differential equations. In this framework, the complete classification of regular partially invariant solutions of ideal MHD equations is given.
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.
The Radially Symmetric Euler Equations as an Exterior Differential System
Baty, Roy; Ramsey, Scott; Schmidt, Joseph
2016-11-01
This work develops the Euler equations as an exterior differential system in radially symmetric coordinates. The Euler equations are studied for unsteady, compressible, inviscid fluids in one-dimensional, converging flow fields with a general equation of state. The basic geometrical constructions (for example, the differential forms, tangent planes, jet space, and differential ideal) used to define and analyze differential equations as systems of exterior forms are reviewed and discussed for converging flows. Application of the Frobenius theorem to the question of the existence of solutions to radially symmetric converging flows is also reviewed and discussed. The exterior differential system is further applied to derive and analyze the general family of characteristic vector fields associated with the one-dimensional inviscid flow equations.
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...
Field Method for Integrating the First Order Differential Equation
JIA Li-qun; ZHENG Shi-wang; ZHANG Yao-yu
2007-01-01
An important modern method in analytical mechanics for finding the integral, which is called the field-method, is used to research the solution of a differential equation of the first order. First, by introducing an intermediate variable, a more complicated differential equation of the first order can be expressed by two simple differential equations of the first order, then the field-method in analytical mechanics is introduced for solving the two differential equations of the first order. The conclusion shows that the field-method in analytical mechanics can be fully used to find the solutions of a differential equation of the first order, thus a new method for finding the solutions of the first order is provided.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
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.
Hyperbolic function method for solving nonlinear differential-different equations
Zhu Jia-Min
2005-01-01
An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation, and successfully constructed some explicit and exact travelling wave solutions.
A Survey on Oscillation of Impulsive Ordinary Differential Equations
Fatma Karakoç
2010-01-01
Full Text Available This paper summarizes a series of results on the oscillation of impulsive ordinary differential equations. We consider linear, half-linear, super-half-linear, and nonlinear equations. Several oscillation criteria are given. The Sturmian comparison theory for linear and half linear equations is also included.