Boundary Value Problems and Approximate Solutions ...
African Journals Online (AJOL)
In this paper, we discuss about some basic things of boundary value problems. Secondly, we study boundary conditions involving derivatives and obtain finite difference approximations of partial derivatives of boundary value problems. The last section is devoted to determine an approximate solution for boundary value ...
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
Fourier analysis and boundary value problems
Gonzalez-Velasco, Enrique A
1996-01-01
Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics.A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.Key Features* Topics are covered from a historical perspective with biographical information on key contributors to the field* The text contains more than 500 exercises* Includes practical applicati...
Boundary Value Problems Arising in Kalman Filtering
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Bashirov Agamirza
2008-01-01
Full Text Available The classic Kalman filtering equations for independent and correlated white noises are ordinary differential equations (deterministic or stochastic with the respective initial conditions. Changing the noise processes by taking them to be more realistic wide band noises or delayed white noises creates challenging partial differential equations with initial and boundary conditions. In this paper, we are aimed to give a survey of this connection between Kalman filtering and boundary value problems, bringing them into the attention of mathematicians as well as engineers dealing with Kalman filtering and boundary value problems.
Boundary Value Problems Arising in Kalman Filtering
Directory of Open Access Journals (Sweden)
Sinem Ertürk
2009-01-01
Full Text Available The classic Kalman filtering equations for independent and correlated white noises are ordinary differential equations (deterministic or stochastic with the respective initial conditions. Changing the noise processes by taking them to be more realistic wide band noises or delayed white noises creates challenging partial differential equations with initial and boundary conditions. In this paper, we are aimed to give a survey of this connection between Kalman filtering and boundary value problems, bringing them into the attention of mathematicians as well as engineers dealing with Kalman filtering and boundary value problems.
Nonlocal boundary value problem for telegraph equations
Ashyralyev, Allaberen; Modanli, Mahmut
2015-09-01
In this work, the nonlocal boundary value problem for a telegraph equation in a Hilbert space is conceived. Stability estimates for the solution of this problem are obtained. The first and second order of accuracy difference schemes for the approximate solution of this problem are constructed. Stability estimates for the solution of these difference schemes are established. In implementations, two mixed problems for telegraph partial differential equations are investigated. The methods are showed by numerical experiments.
Initial boundary value problems in mathematical physics
Leis, Rolf
2013-01-01
Based on the author's lectures at the University of Bonn in 1983-84, this book introduces classical scattering theory and the time-dependent theory of linear equations in mathematical physics. Topics include proof of the existence of wave operators, some special equations of mathematical physics, exterior boundary value problems, radiation conditions, and limiting absorption principles. 1986 edition.
Boundary Value Problems and Approximate Solutions
African Journals Online (AJOL)
Tadesse
2. METHODOLOGY. The finite difference method for the solution of a two point boundary value problem consists in replacing the derivatives present in the differential equation and the boundary conditions with the help of finite difference approximations and then solving the resulting linear system of equations by a standard ...
Hierarchies of DIFFdifference boundary value problems II ...
African Journals Online (AJOL)
This paper provides an illustration of the work done in [14] where a hierarchy of difference boundary value problems was developed. In particular, we studied the effect of applying a Crum-type transformation to a weighted second order difference equation with general -dependent boundary conditions at the end points, ...
Superlinear singular fractional boundary-value problems
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Imed Bachar
2016-04-01
Full Text Available In this article, we study the superlinear fractional boundary-value problem $$\\displaylines{ D^{\\alpha }u(x =u(xg(x,u(x,\\quad 00$. The function $g(x,u\\in C((0,1\\times [ 0,\\infty ,[0,\\infty$ that may be singular at x=0 and x=1 is required to satisfy convenient hypotheses to be stated later.
Homology in Electromagnetic Boundary Value Problems
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Pellikka Matti
2010-01-01
Full Text Available We discuss how homology computation can be exploited in computational electromagnetism. We represent various cellular mesh reduction techniques, which enable the computation of generators of homology spaces in an acceptable time. Furthermore, we show how the generators can be used for setting up and analysis of an electromagnetic boundary value problem. The aim is to provide a rationale for homology computation in electromagnetic modeling software.
Fully nonlinear boundary value problems with impulse
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Paul Eloe
2011-04-01
Full Text Available An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression - expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth as $t\\rightarrow\\infty$ or $t\\rightarrow 0^+$. The nonlinear impulse terms and the nonlinear boundary terms are assumed to satisfy the analogous asymptotic behavior.
Group invariance in engineering boundary value problems
Seshadri, R
1985-01-01
REFEREN CES . 156 9 Transforma.tion of a Boundary Value Problem to an Initial Value Problem . 157 9.0 Introduction . 157 9.1 Blasius Equation in Boundary Layer Flow . 157 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods . 163 9.3 Summary . 168 REFERENCES . . . . . . . . . . . . . . . . . . 168 . 10 From Nonlinear to Linear Differential Equa.tions Using Transformation Groups. . . . . . . . . . . . . . 169 . 10.1 From Nonlinear to Linear Differential Equations . 170 10.2 Application to Ordinary Differential Equations -Bernoulli's Equation . . . . . . . . . . . 173 10.3 Application to Partial Differential Equations -A Nonlinear Chemical Exchange Process . 178 10.4 Limitations of the Inspectional Group Method . 187 10.5 Summary . 188 REFERENCES . . . . 188 11 Miscellaneous Topics . 190 11.1 Reduction of Differential Equations to Algebraic Equations 190 11.2 Reduction of Order of an Ordinary Differential Equation . 191 11.3 Transformat.ion From Ordinary to Partial Differential Equations-Search for First Inte...
Mixed Boundary Value Problem on Hypersurfaces
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R. DuDuchava
2014-01-01
Full Text Available The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation divC(A∇Cφ=f on a smooth hypersurface C with the boundary Γ=∂C in Rn. A(x is an n×n bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts Γ=ΓD∪ΓN and on ΓD the Dirichlet boundary conditions are prescribed, while on ΓN the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to divS(A∇S is proved, which is interpreted as the invertibility of this operator in the setting Hp,#s(S→Hp,#s-2(S, where Hp,#s(S is a subspace of the Bessel potential space and consists of functions with mean value zero.
To the boundary value problem of ordinary differential equations
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Serikbai Aisagaliev
2015-09-01
Full Text Available Method for solving of a boundary value problem for ordinary differential equations with boundary conditions at phase and integral constraints is proposed. The base of the method is an immersion principle based on the general solution of the first order Fredholm integral equation which allows to reduce the original boundary value problem to the special problem of the optimal equation.
Lyapunov-type inequalities for fractional boundary-value problems
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Mohamed Jleli
2015-04-01
Full Text Available In this article, we establish some Lyapunov-type inequalities for fractional boundary-value problems under Sturm-Liouville boundary conditions. As applications, we obtain intervals where linear combinations of certain Mittag-Leffler functions have no real zeros. We deduce also nonexistence results for some fractional boundary-value problems.
On the solvability of initial boundary value problems for nonlinear ...
African Journals Online (AJOL)
In this paper, we study the initial boundary value problems for a non-linear time dependent Schrödinger equation with Dirichlet and Neumann boundary conditions, respectively. We prove the existence and uniqueness of solutions of the initial boundary value problems by using Galerkin's method. Keywords: Initial boundary ...
Spline solutions for nonlinear two point boundary value problems
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Riaz A. Usmani
1980-01-01
Full Text Available Necessary formulas are developed for obtaining cubic, quartic, quintic, and sextic spline solutions of nonlinear boundary value problems. These methods enable us to approximate the solution of the boundary value problems, as well as their successive derivatives smoothly. Numerical evidence is included to demonstrate the relative performance of these four techniques.
Existence results for anisotropic discrete boundary value problems
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Avci Avci
2016-06-01
Full Text Available In this article, we prove the existence of nontrivial weak solutions for a class of discrete boundary value problems. The main tools used here are the variational principle and critical point theory.
Initial-boundary value problems for the wave equation
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Tynysbek Sh. Kalmenov
2014-02-01
Full Text Available In this work we consider an initial-boundary value problem for the one-dimensional wave equation. We prove the uniqueness of the solution and show that the solution coincides with the wave potential.
Boundary value problems and Fourier expansions
MacCluer, Charles R
2004-01-01
Based on modern Sobolev methods, this text for advanced undergraduates and graduate students is highly physical in its orientation. It integrates numerical methods and symbolic manipulation into an elegant viewpoint that is consonant with implementation by digital computer. The first five sections form an informal introduction that develops students' physical and mathematical intuition. The following section introduces Hilbert space in its natural environment, and the next six sections pose and solve the standard problems. The final seven sections feature concise introductions to selected topi
Nonlinear eigenvalue problems for higher order Lidstone boundary value problems
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Paul Eloe
2000-01-01
Full Text Available In this paper, we consider the Lidstone boundary value problem $y^{(2m}(t = \\lambda a(tf(y(t, \\dots, y^{(2j}(t, \\dots y^{(2(m-1}(t, 0 0$ and $a$ is nonnegative. Growth conditions are imposed on $f$ and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a $\\lambda$ interval on which there exists a nontrivial solution in a cone for each $\\lambda$ in that interval. The methods of the paper are known. The emphasis here is that $f$ depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth.
Nonlinear second-order multivalued boundary value problems
Indian Academy of Sciences (India)
Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operatory theory and from multivalued analysis, we ...
Nonlinear boundary value problems with p-Laplace operator
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WANG Yingbo
2013-04-01
Full Text Available We study the second-order three-point boundary value problem with a p-Laplacian operator,and give the expressions of the Green's function for the boundary problems. By the monotone iterative method,sufficient conditions for extreme solutions are obtained.An example is given to illuminate the effectiveness of the main result.
Numerical solution of fuzzy boundary value problems using Galerkin ...
Indian Academy of Sciences (India)
Abstract. This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the.
Solutions of boundary-value problems in discretized volumes
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Mihaly Makai
2002-01-01
Full Text Available The solution of a boundary-value problem in a volume discretized by finitely many copies of a tile is obtained via a Green's function. The algorithm for constructing the solution exploits results from graph and group theory. This technique produces integral equations on the internal and external boundaries of the volume and demonstrates that two permutation matrices characterize the symmetries of the volume. We determine the number of linearly independent solutions required over the tile and the conditions needed for two boundary-value problems to be isospectral. Our method applies group theoretical considerations to asymmetric volumes.
Periodic solutions of a certain nonlinear boundary value problem ...
African Journals Online (AJOL)
... differential equation formed the basis for a theorem for existence of periodic solutions for the nonlinear boundary value problem of a fourth order differential equation. The proof of the theorem is by the Leray-Schauder fixed point technique with the use of integrated equation as the mode for estimating the a priori bounds.
Fractional extensions of some boundary value problems in oil strata
Indian Academy of Sciences (India)
In the present paper, we solve three boundary value problems related to the temperature field in oil strata - the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo ...
Fourth-order discrete anisotropic boundary-value problems
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Maciej Leszczynski
2015-09-01
Full Text Available In this article we consider the fourth-order discrete anisotropic boundary value problem with both advance and retardation. We apply the direct method of the calculus of variations and the mountain pass technique to prove the existence of at least one and at least two solutions. Non-existence of non-trivial solutions is also undertaken.
Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems
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Boglaev Igor
2009-01-01
Full Text Available This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated, and convergence rates are estimated. Numerical experiments complement the theoretical results.
Existence theory for nonlinear functional boundary value problems
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Bapurao Dhage
2004-01-01
Full Text Available In this paper the existence of a solution of a general nonlinear functional two point boundary value problem is proved under mixed generalized Lipschitz and Carath\\'eodory conditions. An existence theorem for extremal solutions is also proved under certain monotonicity and weaker continuity conditions. Examples are provided to illustrate the theory developed in this paper.
Nonlinear initial boundary-value problems with Riesz fractional derivative
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Martin P. Arciga-Alejandre
2015-11-01
Full Text Available We consider an initial boundary-value problem for a nonlinear partial differential equation with fractional derivative of Riesz type on a half-line. We study local and global existence of solutions in time, as well as the asymptotic behavior of solutions for large time.
Positive Solutions for Some Beam Equation Boundary Value Problems
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Xu Weiya
2009-01-01
Full Text Available A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative where is continuous.
Numerical solutions of fifth order boundary value problems using ...
African Journals Online (AJOL)
Mamadu-Njoseh polynomials are polynomials constructed in the interval [-1,1] with respect to the weight function () = 2 + 1. This paper aims at applying these polynomials, as trial functions satisfying the boundary conditions, in a numerical approach for the solution of fifth order boundary value problems. For this, these ...
Fractional extensions of some boundary value problems in oil strata
Indian Academy of Sciences (India)
Abstract. In the present paper, we solve three boundary value problems related to the temperature field in oil strata – the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the ...
A boundary value problem for the wave equation
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Nezam Iraniparast
1999-01-01
Full Text Available Traditionally, boundary value problems have been studied for elliptic differential equations. The mathematical systems described in these cases turn out to be “well posed”. However, it is also important, both mathematically and physically, to investigate the question of boundary value problems for hyperbolic partial differential equations. In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of special interest. The most recent works in this area have resulted in a number of interesting discoveries [3, 4, 5, 7, 8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.
On weak solvability of boundary value problems for elliptic systems
Ponce, Felipe; Lebedev, Leonid,; Rendón, Leonardo,
2013-01-01
This paper concerns with existence and uniqueness of a weak solution for elliptic systems of partial differential equations with mixed boundary conditions. The proof is based on establishing the coerciveness of bilinear forms, related with the system of equations, which depend on first-order derivatives of vector functions in Rn. The condition of coerciveness relates to Korn's type inequalities. The result is illustrated by an example of boundary value problems for a class of elliptic equatio...
Eigenvalues of boundary value problems for higher order differential equations
Wong, Patricia J. Y.; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Right focal boundary value problems for difference equations
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Johnny Henderson
2010-01-01
Full Text Available An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.
Multiple Solutions for a Class of Fractional Boundary Value Problems
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Ge Bin
2012-01-01
Full Text Available We study the multiplicity of solutions for the following fractional boundary value problem: where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a real number, is a given function, and is the gradient of at . The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.
Solution of higher order boundary value problems by spline methods
Chaurasia, Anju; Srivastava, P. C.; Gupta, Yogesh
2017-10-01
Spline solution of Boundary Value Problems has received much attention in recent years. It has proven to be a powerful tool due to the ease of use and quality of results. This paper concerns with the survey of methods that try to approximate the solution of higher order BVPs using various spline functions. The purpose of this article is to thrash out the problems as well as conclusions, reached by the numerous authors in the related field. We critically assess many important relevant papers, published in reputed journals during last six years.
Periodic boundary value problems of second order random differential equations
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Bapurao Dhage
2009-04-01
Full Text Available In this paper, an existence and the existence of extremal random solutions are proved for a periodic boundary value problem of second order ordinary random differential equations. Our investigations have been placed in the space of real-valued functions defined and continuous on closed and bounded intervals of real line together with the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem of Dhage. An example is also indicated for demonstrating the realizations of the abstract theory developed in this paper.
Partial differential equations and boundary-value problems with applications
Pinsky, Mark A
2011-01-01
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems-rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate th
Eigenvalue characterization for a class of boundary value problems
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C. J. Chyan
1999-01-01
Full Text Available We consider the $n$'th order ordinary differential equation $(-1^{n-k} y^{(n}=\\lambda a(t f(y$, $t\\in[0,1]$, $n\\geq 3$ together with the boundary condition $y^{(i}(0=0$, $0\\leq i\\leq k-1$ and $y^{(l}=0$, $j\\leq l\\leq j+n-k-1$, for $1\\leq j\\leq k-1$ fixed. Values of $\\lambda$ are characterized so that the boundary value problem has a positive solution.
Solution of Boundary-Value Problems using Kantorovich Method
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Gusev A.A.
2016-01-01
Full Text Available We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum.
Chebyshev Finite Difference Method for Fractional Boundary Value Problems
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Boundary
2015-09-01
Full Text Available This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivatives are described in the Caputo sense. Numerical results show that this method is of high accuracy and is more convenient and efficient for solving boundary value problems involving fractional ordinary differential equations. AMS Subject Classification: 34A08 Keywords and Phrases: Chebyshev polynomials, Gauss-Lobatto points, fractional differential equation, finite difference 1. Introduction The idea of a derivative which interpolates between the familiar integer order derivatives was introduced many years ago and has gained increasing importance only in recent years due to the development of mathematical models of a certain situations in engineering, materials science, control theory, polymer modelling etc. For example see [20, 22, 25, 26]. Most fractional order differential equations describing real life situations, in general do not have exact analytical solutions. Several numerical and approximate analytical methods for ordinary differential equation Received: December 2014; Accepted: March 2015 57 Journal of Mathematical Extension Vol. 9, No. 3, (2015, 57-71 ISSN: 1735-8299 URL: http://www.ijmex.com Chebyshev Finite Difference Method for Fractional Boundary Value Problems H. Azizi Taft Branch, Islamic Azad University Abstract. This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivative
On a Fourth-Order Boundary Value Problem at Resonance
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Man Xu
2017-01-01
Full Text Available We investigate the spectrum structure of the eigenvalue problem u4x=λux, x∈0,1; u0=u1=u′0=u′1=0. As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance -u4x+λ1ux+gx,ux=hx, x∈0,1; u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.
Nonsmooth critical point theory and nonlinear boundary value problems
Gasinski, Leszek
2004-01-01
Starting in the early 1980s, people using the tools of nonsmooth analysis developed some remarkable nonsmooth extensions of the existing critical point theory. Until now, however, no one had gathered these tools and results together into a unified, systematic survey of these advances.This book fills that gap. It provides a complete presentation of nonsmooth critical point theory, then goes beyond it to study nonlinear second order boundary value problems. The authors do not limit their treatment to problems in variational form. They also examine in detail equations driven by the p-Laplacian, its generalizations, and their spectral properties, studying a wide variety of problems and illustrating the powerful tools of modern nonlinear analysis. The presentation includes many recent results, including some that were previously unpublished. Detailed appendices outline the fundamental mathematical tools used in the book, and a rich bibliography forms a guide to the relevant literature.Most books addressing critica...
Initial boundary value problems for some damped nonlinear conservation laws
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Manoj Yadav
2015-11-01
Full Text Available In this paper, we study the non-negative solutions of initial boundary value problems for some damped nonlinear conservation laws on the half line modelled by first order nonlinear hyperbolic PDEs. We consider the class of initial profile which are non-negative, bounded and compactly supported. Using the method of characteristics and Rankine-Hugoniot jump condition, an entropy solution is constructed subject to a top-hat initial profile. Then the large time behaviour of the constructed entropy solution is obtained. Finally, taking recourse to some comparison principles and the method of super and sub solutions the large time behaviour of entropy solutions subject to the general class of bounded and compactly supported initial profiles are established as the large time behaviour of the entropy solution subject to top-hat initial profiles.
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...
Vertical and horizontal spheroidal boundary-value problems
Šprlák, Michal; Tangdamrongsub, Natthachet
2017-12-01
Vertical and horizontal spheroidal boundary-value problems (BVPs), i.e., determination of the external gravitational potential from the components of the gravitational gradient on the spheroid, are discussed in this article. The gravitational gradient is decomposed into the series of the vertical and horizontal vector spheroidal harmonics, before being orthogonalized in a weighted sense by two different approaches. The vertical and horizontal spheroidal BVPs are then formulated and solved in the spectral and spatial domains. Both orthogonalization methods provide the same analytical solutions for the vertical spheroidal BVP, and give distinct, but equivalent, analytical solutions for the horizontal spheroidal BVP. A closed-loop simulation is performed to test the correctness of the analytical solutions, and we investigate analytical properties of the sub-integral kernels. The systematic treatment of the spheroidal BVPs and the resulting mathematical equations extend the theoretical apparatus of geodesy and of the potential theory.
An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems
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Mohammad Maleki
2012-01-01
Full Text Available An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE. By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.
The boundary value problem for discrete analytic functions
Skopenkov, Mikhail
2013-06-01
This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd.
Dirichlet-Neumann bracketing for boundary-value problems on graphs
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Sonja Currie
2005-08-01
Full Text Available We consider the spectral structure of second order boundary-value problems on graphs. A variational formulation for boundary-value problems on graphs is given. As a consequence we can formulate an analogue of Dirichlet-Neumann bracketing for boundary-value problems on graphs. This in turn gives rise to eigenvalue and eigenfunction asymptotic approximations.
A class of renormalised meshless Laplacians for boundary value problems
Basic, Josip; Degiuli, Nastia; Ban, Dario
2018-02-01
A meshless approach to approximating spatial derivatives on scattered point arrangements is presented in this paper. Three various derivations of approximate discrete Laplace operator formulations are produced using the Taylor series expansion and renormalised least-squares correction of the first spatial derivatives. Numerical analyses are performed for the introduced Laplacian formulations, and their convergence rate and computational efficiency are examined. The tests are conducted on regular and highly irregular scattered point arrangements. The results are compared to those obtained by the smoothed particle hydrodynamics method and the finite differences method on a regular grid. Finally, the strong form of various Poisson and diffusion equations with Dirichlet or Robin boundary conditions are solved in two and three dimensions by making use of the introduced operators in order to examine their stability and accuracy for boundary value problems. The introduced Laplacian operators perform well for highly irregular point distribution and offer adequate accuracy for mesh and mesh-free numerical methods that require frequent movement of the grid or point cloud.
Solvability of some Neumann-type boundary value problems for biharmonic equations
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Valery Karachik
2017-09-01
Full Text Available We study some boundary-value problems for inhomogeneous biharmonic equation with periodic boundary conditions. These problems are generalization to periodic data of the Neumann-type boundary-value problems considered before by the authors. We obtain existence and uniqueness of solutions for the problems under consideration.
A numerical solution of a singular boundary value problem arising in boundary layer theory.
Hu, Jiancheng
2016-01-01
In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner-Skan equation. And the one-dimensional third-order boundary value problem on interval [Formula: see text] is equivalently transformed into a second-order boundary value problem on finite interval [Formula: see text]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.
Directory of Open Access Journals (Sweden)
Liu Yuji
2008-01-01
Full Text Available Abstract This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: . Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.
Existence of Solutions for Nonlinear Four-Point -Laplacian Boundary Value Problems on Time Scales
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Topal SGulsan
2009-01-01
Full Text Available We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with a -Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result for -Laplacian boundary value problem is also given by the monotone method.
Boundary value problems for third order differential equations by solution matching
Directory of Open Access Journals (Sweden)
Johnny Henderson
2009-10-01
Full Text Available For the ordinary differential equation, $y''' = f(x,y,y',$ $y'',$ solutions of 3-point boundary value problems on $[a,b]$ are matched with solutions of 3-point boundary value problems on $[b,c]$ to obtain solutions satisfying 5-point boundary conditions on $[a,c]$.
Boundary value problems on the half line in the theory of colloids
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Ravi P. Agarwal
2002-01-01
Full Text Available We present existence results for some boundary value problems defined on infinite intervals. In particular our discussion includes a problem which arises in the theory of colloids.
Electromagnetic wave theory for boundary-value problems an advanced course on analytical methods
Eom, Hyo J
2004-01-01
Electromagnetic wave theory is based on Maxwell's equations, and electromagnetic boundary-value problems must be solved to understand electromagnetic scattering, propagation, and radiation. Electromagnetic theory finds practical applications in wireless telecommunications and microwave engineering. This book is written as a text for a two-semester graduate course on electromagnetic wave theory. As such, Electromagnetic Wave Theory for Boundary-Value Problems is intended to help students enhance analytic skills by solving pertinent boundary-value problems. In particular, the techniques of Fourier transform, mode matching, and residue calculus are utilized to solve some canonical scattering and radiation problems.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
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Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
A new type of shooting method for nonlinear boundary value problems
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Muhammad Ahsan
2013-12-01
Full Text Available In this article we introduce a new type of iterative method for initial value problems (IVPs. We enhance this method by using shooting techniques and interpolation for the boundary value problems. Our method is more accurate and applicable than built in methods used in different software packages. We solved several examples for initial value problems and linear and non-linear boundary value problems and compared results to those obtained using MATLAB.
Zou, Li; Liang, Songxin; Li, Yawei; Jeffrey, David J.
2017-03-01
Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.
Energy Technology Data Exchange (ETDEWEB)
Zou, Li [Dalian Univ. of Technology, Dalian City (China). State Key Lab. of Structural Analysis for Industrial Equipment; Liang, Songxin; Li, Yawei [Dalian Univ. of Technology, Dalian City (China). School of Mathematical Sciences; Jeffrey, David J. [Univ. of Western Ontario, London (Canada). Dept. of Applied Mathematics
2017-06-01
Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.
Laplace Boundary-Value Problem in Paraboloidal Coordinates
Duggen, L.; Willatzen, M.; Voon, L. C. Lew Yan
2012-01-01
This paper illustrates both a problem in mathematical physics, whereby the method of separation of variables, while applicable, leads to three ordinary differential equations that remain fully coupled via two separation constants and a five-term recurrence relation for series solutions, and an exactly solvable problem in electrostatics, as a…
Boundary value problems for a nonlinear elliptic equation
Egorov, Yu. V.
2017-06-01
It is proved that the Dirichlet and Neumann problems for a nonlinear second-order elliptic equation have infinitely many solutions. The spectrum of these problems is studied and the weak convergence of the normed eigenfunctions to zero is established. Bibliography: 10 titles.
A Boundary Value Problem for Hermitian Monogenic Functions
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Ricardo Abreu Blaya
2008-02-01
Full Text Available We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in Ã¢Â„Âm,Ã¢Â€Â‰m=2n. Necessary and sufficient conditions for the solvability of this problem are obtained.
numerical solutions of fifth order boundary value problems using ...
African Journals Online (AJOL)
Dr A.B.Ahmed
solving these problems by employing polynomials as trial functions in the ... numerical solution of Volterra integral equations by Galerkin method. Caglar et .... and continuous on [0,1], i α ,. 2,1,0. = i and i β ,. ,1,0. = i are finite real constants. Transforming (10) – (11) to systems of ordinary differential equations, we have. 1 yy. =.
Positive solutions of singular boundary value problem of negative ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
may be singular at t = 0,t = 1. When λ 0, [6] shows the existence and uniqueness to (1) and (2) in the case of β = δ = 0 by means of the shooting method. For the following problem u + p(t)u−λ(t) + q(t)u−m(t) = 0, 0
Global solution branches of two point boundary value problems
Schaaf, Renate
1990-01-01
The book deals with parameter dependent problems of the form u"+*f(u)=0 on an interval with homogeneous Dirichlet or Neuman boundary conditions. These problems have a family of solution curves in the (u,*)-space. By examining the so-called time maps of the problem the shape of these curves is obtained which in turn leads to information about the number of solutions, the dimension of their unstable manifolds (regarded as stationary solutions of the corresponding parabolic prob- lem) as well as possible orbit connections between them. The methods used also yield results for the period map of certain Hamiltonian systems in the plane. The book will be of interest to researchers working in ordinary differential equations, partial differential equations and various fields of applications. By virtue of the elementary nature of the analytical tools used it can also be used as a text for undergraduate and graduate students with a good background in the theory of ordinary differential equations.
Existence Results for Higher-Order Boundary Value Problems on Time Scales
Sang Yanbin; Liu Jian
2009-01-01
By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales , ; , ; , ; , , where , , , , , , , and is rd-continuous.
Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems
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Moulay Rchid Sidi Ammi
2013-01-01
Full Text Available We obtain, by using the Leggett-Williams fixed point theorem, sufficient conditions that ensure the existence of at least three positive solutions to some p-Laplacian boundary value problems on time scales.
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Jianwei Dong
2005-11-01
Full Text Available We show the existence of solutions for mixed boundary-value problems that model quantum hydrodynamics in thermal equilibrium. Also we find the semi-classical limit of the solutions.
On the Solvability of Discrete Nonlinear Two-Point Boundary Value Problems
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Blaise Kone
2012-01-01
Full Text Available We prove the existence and uniqueness of solutions for a family of discrete boundary value problems by using discrete's Wirtinger inequality. The boundary condition is a combination of Dirichlet and Neumann boundary conditions.
Existence and uniqueness of solutions for a Neumann boundary-value problem
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Safia Benmansour
2011-09-01
Full Text Available In this article, we show the existence and uniqueness of positive solutions for perturbed Neumann boundary-value problems of second-order differential equations. We use a fixed point theorem for general $alpha$-concave operators.
Variational methods for boundary value problems for systems of elliptic equations
Lavrent'ev, M A
2012-01-01
Famous monograph by a distinguished mathematician presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli. 1963 edition.
Initial boundary value problems for second order parabolic systems in cylinders with polyhedral base
National Research Council Canada - National Science Library
Luong, Vu Trong; Loi, Do Van
2011-01-01
The purpose of this article is to establish the well posedness and the regularity of the solution of the initial boundary value problem with Dirichlet boundary conditions for second-order parabolic...
Asymptotic Solution of the Theory of Shells Boundary Value Problem
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I. V. Andrianov
2007-01-01
Full Text Available This paper provides a state-of-the-art review of asymptotic methods in the theory of plates and shells. Asymptotic methods of solving problems related to theory of plates and shells have been developed by many authors. The main features of our paper are: (i it is devoted to the fundamental principles of asymptotic approaches, and (ii it deals with both traditional approaches, and less widely used, new approaches. The authors have paid special attention to examples and discussion of results rather than to burying the ideas in formalism, notation, and technical details.
Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problems
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Muhammad Aslam Noor
2006-12-01
Full Text Available We apply the homotopy perturbation method for solving the fourth-order boundary value problems. The analytical results of the boundary value problems have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method. Comparisons are made to confirm the reliability of the method. Homotopy method can be considered an alternative method to Adomian decomposition method and its variant forms.
Non-homogeneous boundary-value problems of higher order differential equations with p-Laplacian
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Yuji Liu
2008-02-01
Full Text Available We establish sufficient conditions for the existence of positive solutions to five multi-point boundary value problems. These problems have a common equation (in different function domains and different boundary conditions. It is interesting note that the methods for solving all these problems and most of the reference are based on the Mawhin's coincidence degree theory. First, we present a survey of multi-point boundary-value problems and the motivation of this paper. Then we present the main results which generalize and improve results in the references. We conclude this article with examples of problems that can not solved by methods known so far.
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Hussein A. H. Salem
2013-01-01
Full Text Available The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.
Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces
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Yevheniia Hnyp
2017-03-01
Full Text Available We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.
THE USE OF DIFFERENTIAL TRANSFORMATIONS FOR SOLVING NON-LINEAR BOUNDARY VALUE PROBLEMS
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Viacheslav Gusynin
2016-12-01
Full Text Available Purpose: The aim of our study is comparison of method applications based on differential transformations for solving boundary value problems which are described by non-linear ordinary differential equations. Methods: This article reviews two approaches based on differential transformations for solving non-linear boundary value problems: the modified differential transform method and the system-analogue simulation method. Results: In this paper, we present results of the numerical solution of non-linear boundary value problem by methods based on differential transformations for demonstration the effectiveness and applicability of techniques. The relative error for given solutions, obtained with using first 6 discretes of differential spectra is presented. Discussion: Comparison of numerical solutions obtained by modified differential transform method and system-analogue simulation method with exact solution shows that both methods have good agreement with exact solution of non-linear boundary value problem for small intervals. However, application of system-analogue simulation method is preferential for big intervals, on which the boundary value problem is solved.
Solvability of boundary-value problems for Poisson equations with Hadamard type boundary operator
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Batirkhan Turmetov
2016-06-01
Full Text Available In this article we study properties of some integro-differential operators of fractional order. As an application of the properties of these operators for Poisson equation we examine questions on solvability of a fractional analogue of Neumann problem and analogues of periodic boundary-value problems for circular domains. The exact conditions for solvability of these problems are found.
An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics
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A. Barari
2008-01-01
Full Text Available Variational iteration method (VIM is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM. The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.
Numerical Analysis of Forth-Order Boundary Value Problems in Fluid Mechanics and Mathematics
DEFF Research Database (Denmark)
Hosseinzadeh, E.; Barari, Amin; Fouladi, F.
2011-01-01
In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed o...
Numerical analysis of fourth-order boundary value problems in fluid mechanics and mathematics
DEFF Research Database (Denmark)
Hosseinzadeh, Elham; Barari, Amin; Fouladi, Fama
2010-01-01
In this paper He's variational iteration method is used to solve some examples of linear and non-linear forth-order boundary value problems. The first problem compared with homotopy analysis method solution and the other ones with the exact solution. The results show the high accuracy and speed o...
A generalized Lyapunov inequality for a higher-order fractional boundary value problem
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Dexiang Ma
2016-10-01
Full Text Available Abstract In the paper, we establish a Lyapunov inequality and two Lyapunov-type inequalities for a higher-order fractional boundary value problem with a controllable nonlinear term. Two applications are discussed. One concerns an eigenvalue problem, the other a Mittag-Leffler function.
Yousef, Hamood Mohammed; Ismail, Ahmad Izani
2017-11-01
In this paper, Laplace Adomian decomposition method (LADM) was applied to solve Delay differential equations with Boundary Value Problems. The solution is in the form of a convergent series which is easy to compute. This approach is tested on two test problem. The findings obtained exhibit the reliability and efficiency of the proposed method.
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Hossein Jafari
2014-01-01
Full Text Available We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example.
On an initial-boundary value problem for the nonlinear Schrödinger equation
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Herbert Gajewski
1979-01-01
Full Text Available We study an initial-boundary value problem for the nonlinear Schrödinger equation, a simple mathematical model for the interaction between electromagnetic waves and a plasma layer. We prove a global existence and uniqueness theorem and establish a Galerkin method for solving numerically the problem.
Numerical solutions of a three-point boundary value problem with an ...
African Journals Online (AJOL)
Numerical solutions of a three-point boundary value problem with an integral condition for a third-order partial differential equation by using Laplace transform method Solutions numeriques d'un probleme pour une classe d'equations differentielles d'ordr.
Solution of fourth order three-point boundary value problem using ADM and RKM
Directory of Open Access Journals (Sweden)
Ghazala Akram
2016-06-01
Full Text Available In this paper, a computational method is proposed, for solving linear and nonlinear fourth order three-point boundary value problem (BVP and the system of nonlinear BVP. This method is based on the Adomian decomposition method (ADM and the reproducing kernel method (RKM. The solution of linear fourth order three-point boundary value problem (BVP is determined by the reproducing kernel method, and the solution of nonlinear fourth order three-point BVP is determined using the combination of Adomian decomposition method and reproducing kernel method. The approximate solutions are given in the form of series. Numerical results are shown to illustrate the accuracy of the present method.
Mapping physical problems on fractals onto boundary value problems within continuum framework
Balankin, Alexander S.
2018-01-01
In this Letter, we emphasize that methods of fractal homogenization should take into account a loop structure of the fractal, as well as its connectivity and geodesic metric. The fractal attributes can be quantified by a set of dimension numbers. Accordingly, physical problems on fractals can be mapped onto the boundary values problems in the fractional-dimensional space with metric induced by the fractal topology. The solutions of these problems represent analytical envelopes of non-analytical functions defined on the fractal. Some examples are briefly discussed. The interplay between effects of fractal connectivity, loop structure, and mass distributions on electromagnetic fields in fractal media is highlighted. The effects of fractal connectivity, geodesic metric, and loop structure are outlined.
On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem
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Yumei Zou
2017-01-01
Full Text Available By using the method of order reduction and the fixed point index, the existence of positive solutions for a fourth-order boundary value problem is studied. We provide conditions under which the existence results hold. Such conditions are related to the first eigenvalue corresponding to the relevant linear differential equation with dependence on the derivatives of unknown function.
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Yumei Zou
2017-01-01
Full Text Available This paper deals with the integral boundary value problems of fractional differential equations at resonance. By Mawhin’s coincidence degree theory, we present some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance. An example is also included to illustrate the main results.
On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem
Yumei Zou
2017-01-01
By using the method of order reduction and the fixed point index, the existence of positive solutions for a fourth-order boundary value problem is studied. We provide conditions under which the existence results hold. Such conditions are related to the first eigenvalue corresponding to the relevant linear differential equation with dependence on the derivatives of unknown function.
On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem
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Ozgur Yildirim
2015-01-01
Full Text Available This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.
Ateş, I.; Zegeling, P. A.|info:eu-repo/dai/nl/073634433
2017-01-01
In this paper we describe the application of the homotopy perturbation method (HPM) to two-point boundary-value problems with fractional-order derivatives of Caputo-type. We show that HPM is equivalent to the semi-analytical Adomian decomposition method when applied to a class of nonlinear
Initial boundary value problem for a system in elastodynamics with viscosity
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Kayyunnapara Thomas Joseph
2005-12-01
Full Text Available In this paper we prove existence of global solutions to boundary-value problems for two systems with a small viscosity coefficient and derive estimates uniform in the viscosity parameter. We do not assume any smallness conditions on the data.
Bifurcation from infinity and multiple solutions for first-order periodic boundary-value problems
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Zhenyan Wang
2011-10-01
Full Text Available In this article, we study the existence and multiplicity of solutions for the first-order periodic boundary-value problem $$displaylines{ u'(t-a(tu(t=lambda u(t+g(u(t-h(t, quad tin (0, T,cr u(0=u(T. }$$
Eigenvalues for Iterative Systems of (n,p-Type Fractional Order Boundary Value Problems
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K. R. Prasad
2014-04-01
Full Text Available In this paper, we determine the eigenvalue intervals of λ1, λ2, ..., λn for which the iterative system of (n,p-type fractional order two-point boundary value problem has a positive solution by an application of Guo-Krasnosel’skii fixed point theorem on a cone.
Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems
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Shi Haiping
2011-01-01
Full Text Available By using critical point theory, Lyapunov-Schmidt reduction method, and characterization of the Brouwer degree of critical points, sufficient conditions to guarantee the existence of five or six solutions together with their sign properties to discrete second-order two-point boundary value problem are obtained. An example is also given to demonstrate our main result.
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Mohamed Jleli
2014-01-01
Full Text Available A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.
Directory of Open Access Journals (Sweden)
An Yukun
2011-01-01
Full Text Available Abstract This paper deals with the periodic boundary value problems where is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping.
Three symmetric positive solutions of fourth-order singular nonlocal boundary value problems
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Fuyi Xu
2011-12-01
Full Text Available In this paper, we study the existence of three positive solutions of fourth-order singular nonlocal boundary value problems. We show that there exist triple symmetric positive solutions by using Leggett-Williams fixed-point theorem. The conclusions in this paper essentially extend and improve some known results.
Positive solutions for second-order boundary-value problems with sign changing Green's functions
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Alberto Cabada
2017-10-01
Full Text Available In this article we analyze some possibilities of finding positive solutions for second-order boundary-value problems with the Dirichlet and periodic boundary conditions, for which the corresponding Green's functions change sign. The obtained results can also be adapted to Neumann and mixed boundary conditions.
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Yuji Liu
2004-01-01
Full Text Available A new fixed point theorem on cones is applied to obtain the existence of at least two positive solutions of a higher-order three-point boundary value problem for the differential equation subject to a class ofboundary value conditions. The associated Green's function is given. Some results obtained recently are generalized.
Uniqueness in some higher order elliptic boundary value problems in n dimensional domains
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C.-P. Danet
2011-07-01
Full Text Available We develop maximum principles for several P functions which are defined on solutions to equations of fourth and sixth order (including a equation which arises in plate theory and bending of cylindrical shells. As a consequence, we obtain uniqueness results for fourth and sixth order boundary value problems in arbitrary n dimensional domains.
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Nieto JuanJ
2009-01-01
Full Text Available This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.
Positive solutions of multi-point boundary value problem of fractional differential equation
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De-xiang Ma
2015-07-01
Full Text Available By means of two fixed-point theorems on a cone in Banach spaces, some existence and multiplicity results of positive solutions of a nonlinear fractional differential equation boundary value problem are obtained. The proofs are based upon some properties of Green’s function, which are also the key of the paper.
A smart nonstandard finite difference scheme for second order nonlinear boundary value problems
Erdogan, Utku; Ozis, Turgut
2011-01-01
A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed.
Directory of Open Access Journals (Sweden)
Jian Liu
2013-09-01
Full Text Available In this article, we consider the free boundary value problem for one-dimensional compressible bipolar Navier-Stokes-Possion (BNSP equations with density-dependent viscosities. For general initial data with finite energy and the density connecting with vacuum continuously, we prove the global existence of the weak solution. This extends the previous results for compressible NS [27] to NSP.
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John Graef
2013-09-01
Full Text Available The authors consider a nonlinear fractional boundary value problem with the Dirichlet boundary condition. An associated Green's function is constructed as a series of functions by applying spectral theory. Criteria for the existence and uniqueness of solutions are obtained based on it.
Solution Matching for a Second Order Boundary Value Problem on Time Scales
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Aprillya Lanz
2012-01-01
Full Text Available Let be a time scale such that <;,∈. We will show the existence and uniqueness of solutions for the second-order boundary value problem ΔΔ(=(,(,Δ(,∈[,],(=,(=, by matching a solution of the first equation satisfying boundary conditions on [,] with a solution of the first equation satisfying boundary conditions on [,], where ∈(,.
BOUNDARY VALUE PROBLEM FOR A LOADED EQUATION ELLIPTIC-HYPERBOLIC TYPE IN A DOUBLY CONNECTED DOMAIN
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O.Kh. Abdullaev
2014-06-01
Full Text Available We study the existence and uniqueness of the solution of one boundary value problem for the loaded elliptic-hyperbolic equation of the second order with two lines of change of type in double-connected domain. Similar results have been received by D.M.Kuryhazov, when investigated domain is one-connected.
Triple solutions for multi-point boundary-value problem with p-Laplace operator
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Yansheng Liu
2009-11-01
Full Text Available Using a fixed point theorem due to Avery and Peterson, this article shows the existence of solutions for multi-point boundary-value problem with p-Laplace operator and parameters. Also, we present an example to illustrate the results obtained.
Positive solutions of second-order singular boundary value problem with a Laplace-like operator
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Ge Weigao
2005-01-01
Full Text Available By use of the concavity of solution for an associate boundary value problem, existence criteria of positive solutions are given for the Dirichlet BVP , , , where is odd and continuous with , , and may change sign and be singular along a curve in .
Existence of a positive solution to a right focal boundary value problem
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R. I. Avery
2010-01-01
Full Text Available In this paper we apply the recent extension of the Leggett-Williams Fixed Point Theorem which requires neither of the functional boundaries to be invariant to the second order right focal boundary value problem. We demonstrate a technique that can be used to deal with a singularity and provide a non-trivial example.
L1-Solutions of Boundary Value Problems for Implicit Fractional Order Differential Equations
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Mouffak Benchohra
2015-12-01
Full Text Available The aim of this paper is to present new results on the existence of solutions for a class of boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem.
Boundary-value problems for first and second order functional differential inclusions
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Shihuang Hong
2003-03-01
Full Text Available This paper presents sufficient conditions for the existence of solutions to boundary-value problems of first and second order multi-valued differential equations in Banach spaces. Our results obtained using fixed point theorems, and lead to new existence principles.
Huiqin Lu
2012-01-01
By constructing a special cone in ${C}^{1}[0,2\\pi ]$ and the fixed point theorem, this paper investigates second-order singular semipositone periodic boundary value problems with dependence on the first-order derivative and obtains the existence of multiple positive solutions. Further, an example is given to demonstrate the applications of our main results.
Solvability of boundary value problem at resonance for third-order ...
Indian Academy of Sciences (India)
This paper is devoted to the study of boundary value problem of third- order functional differential equations. We obtain some existence results for the prob- lem at resonance under the condition that the nonlinear terms is bounded or generally unbounded. In this paper we mainly use the topological degree theory. Keywords ...
Positive Solutions of a Nonlinear Fourth-order Integral Boundary Value Problem
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Benaicha Slimane
2016-07-01
Full Text Available In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.
A free-boundary value problem related to auto ignition of ...
African Journals Online (AJOL)
We examine a free boundary value problem related to auto ignition of combustible fluid in insulation materials. The criteria for the existence of similarity solution of the model equations are established. The conditions for the existence of unique solution are also stated. The numerical results which show the influence of ...
Approximate solution to a singular perturbed boundary value problem of thermal shielding
Latypov, I. I.
2017-11-01
The paper aims to investigate the problem of distribution of a non-regular, non-steady-state thermal field in the porous thermal shield material irradiated by a high flow of energy. A mathematical model of the original problem is stated in the form of a singular perturbed boundary value problem of a thermal conductivity equation with the nonlinear boundary conditions on moving boundaries. Its solution is obtained as asymptotic Poincare-type expansions in powers of small parameters.
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Paul W. Eloe
2016-11-01
Full Text Available We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green's functions are then constructed as convolutions of lower order Green's functions. Comparison theorems are known for the Green's functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question for a family of Green's functions that do not apparently have convolution representations.
Numerical solution of system of boundary value problems using B-spline with free parameter
Gupta, Yogesh
2017-01-01
This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation.
Initial-boundary value problem with a nonlocal condition for a viscosity equation
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Abdelfatah Bouziani
2002-01-01
Full Text Available This paper deals with the proof of the existence, uniqueness, and continuous dependence of a strong solution upon the data, for an initial-boundary value problem which combine Neumann and integral conditions for a viscosity equation. The proof is based on an energy inequality and on the density of the range of the linear operator corresponding to the abstract formulation of the studied problem.
Positive solutions for singular three-point boundary-value problems
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Zengqin Zhao
2007-11-01
Full Text Available In this paper, we present the Green's functions for a second-order linear differential equation with three-point boundary conditions. We give exact expressions of the solutions for the linear three-point boundary problems by the Green's functions. As applications, we study uniqueness and iteration of the solutions for a nonlinear singular second-order three-point boundary value problem.
Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach
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Panos K. Palamides
2012-01-01
Full Text Available We investigate the existence of positive or a negative solution of several classes of four-point boundary-value problems for fourth-order ordinary differential equations. Although these problems do not always admit a (positive Green's function, the obtained solution is still of definite sign. Furthermore, we prove the existence of an entire continuum of solutions. Our technique relies on the continuum property (connectedness and compactness of the solutions funnel (Kneser's Theorem, combined with the corresponding vector field.
Monotone and convex positive solutions for fourth-order multi-point boundary value problems
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Chunfang Shen
2011-01-01
Full Text Available Abstract The existence results of multiple monotone and convex positive solutions for some fourth-order multi-point boundary value problems are established. The nonlinearities in the problems studied depend on all order derivatives. The analysis relies on a fixed point theorem in a cone. The explicit expressions and properties of associated Green's functions are also given. MSC: 34B10; 34B15.
A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations
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A. L. Marhoune
2009-01-01
Full Text Available We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.
Coercive solvability of the nonlocal boundary value problem for parabolic differential equations
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P. E. Sobolevskii
2001-06-01
Full Text Available The nonlocal boundary value problem, vÃ¢Â€Â²(t+Av(t=f(t(0Ã¢Â‰Â¤tÃ¢Â‰Â¤1,v(0=v(ÃŽÂ»+ÃŽÂ¼(0<ÃŽÂ»Ã¢Â‰Â¤1, in an arbitrary Banach space E with the strongly positive operator A, is considered. The coercive stability estimates in HÃƒÂ¶lder norms for the solution of this problem are proved. The exact Schauder's estimates in HÃƒÂ¶lder norms of solutions of the boundary value problem on the range {0Ã¢Â‰Â¤tÃ¢Â‰Â¤1,xÃ¢Â„ÂÃ¢Â€Â‰n} for 2m-order multidimensional parabolic equations are obtaine.
On explicit and numerical solvability of parabolic initial-boundary value problems
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Lepsky Olga
2006-01-01
Full Text Available A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain ( being a bounded domain, and being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution . In the paper, the result is obtained not just for the operator , but also for an arbitrary parabolic differential operator , where is an elliptic operator in of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation in is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables.
Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative
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Erkinjon Karimov
2016-10-01
Full Text Available In this work, we consider a number of boundary-value problems for time-fractional heat equation with the recently introduced Caputo-Fabrizio derivative. Using the method of separation of variables, we prove a unique solvability of the stated problems. Moreover, we have found an explicit solution to certain initial value problem for Caputo-Fabrizio fractional order differential equation by reducing the problem to a Volterra integral equation. Different forms of solution were presented depending on the values of the parameter appeared in the problem.
The initial boundary value problem for free-evolution formulations of general relativity
Hilditch, David; Ruiz, Milton
2018-01-01
We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate primarily on boundaries that are geometrically determined by the outermost normal observer to spacelike slices of the foliation. We present high-order-derivative boundary conditions for the gauge, constraint violating and gravitational wave degrees of freedom of the formulation. Second order derivative boundary conditions are presented in terms of the conformal variables used in numerical relativity simulations. Using Kreiss–Agranovich–Métivier theory we demonstrate, in the frozen coefficient approximation, that with sufficiently high order derivative boundary conditions the initial boundary value problem can be rendered boundary stable. The precise number of derivatives required depends on the gauge. For a choice of the gauge condition that renders the system strongly hyperbolic of constant multiplicity, well-posedness of the initial boundary value problem follows in this approximation. Taking into account the theory of pseudo-differential operators, it is expected that the nonlinear problem is also well-posed locally in time.
Successive Iteration of Positive Solutions for Fourth-Order Two-Point Boundary Value Problems
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Yongping Sun
2013-01-01
Full Text Available We are concerned with a fourth-order two-point boundary value problem. We prove the existence of positive solutions and establish iterative schemes for approximating the solutions. The interesting point of our method is that the nonlinear term is involved with all lower-order derivatives of unknown function, and the iterative scheme starts off with a known cubic function or the zero function. Finally we give two examples to verify the effectiveness of the main results.
Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method
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Mustafa Inc
2014-01-01
Full Text Available We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM, homotopy perturbation method (HPM, Adomian decomposition method (ADM, variation of parameters method (VPM, and homotopy analysis method (HAM. Obtained results show that our method is very effective.
Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach
Amenta, Alex; Auscher, Pascal
2016-01-01
We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Besov-Hardy-Sobolev (BHS) spaces. Our approach uses minimal assumptions on the coefficients, and in particular does not require De Giorgi-Nash-Moser estimates. Our results are completely new for the Hardy-Sobolev case, and in the Besov case they extend results recently obtai...
Infinitely many solutions for a fourth-order boundary-value problem
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Seyyed Mohsen Khalkhali
2012-09-01
Full Text Available In this article we consider the existence of infinitely many solutions to the fourth-order boundary-value problem $$displaylines{ u^{iv}+alpha u''+eta(x u=lambda f(x,u+h(u,quad xin]0,1[cr u(0=u(1=0,cr u''(0=u''(1=0,. }$$ Our approach is based on variational methods and critical point theory.
Positive solutions for a nonlinear periodic boundary-value problem with a parameter
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Jingliang Qiu
2012-08-01
Full Text Available Using topological degree theory with a partially ordered structure of space, sufficient conditions for the existence and multiplicity of positive solutions for a second-order nonlinear periodic boundary-value problem are established. Inspired by ideas in Guo and Lakshmikantham [6], we study the dependence of positive periodic solutions as a parameter approaches infinity, $$ lim_{lambdao +infty}|x_{lambda}|=+infty,quadhbox{or}quad lim_{lambdao+infty}|x_{lambda}|=0. $$
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Jinhua Wang
2010-01-01
Full Text Available We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.
On nonlinear boundary value problems with deviating arguments and discontinuous right hand side
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B. C. Dhage
1993-01-01
Full Text Available In this paper we shall study the existence of the extremal solutions of a nonlinear boundary value problem of a second order differential equation with general Dirichlet/Neumann form boundary conditions. The right hand side of the differential equation is assumed to contain a deviating argument, and it is allowed to possess discontinuities in all the variables. The proof is based on a generalized iteration method.
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Yang Xiao-Jun
2015-01-01
Full Text Available In the present paper we investigate the fractal boundary value problems for the Fredholm\\Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results. [Projekat Ministarstva nauke Republike Srbije, br. OI 174001, III41006 i br. TI 35006
Multi-point boundary value problems for linear functional-differential equations
Czech Academy of Sciences Publication Activity Database
Domoshnitsky, A.; Hakl, Robert; Půža, Bedřich
2017-01-01
Roč. 24, č. 2 (2017), s. 193-206 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : boundary value problems * linear functional-differential equations * functional-differential inequalities Subject RIV: BA - General Mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0076/gmj-2016-0076. xml
Monotone methods for solving a boundary value problem of second order discrete system
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Wang Yuan-Ming
1999-01-01
Full Text Available A new concept of a pair of upper and lower solutions is introduced for a boundary value problem of second order discrete system. A comparison result is given. An existence theorem for a solution is established in terms of upper and lower solutions. A monotone iterative scheme is proposed, and the monotone convergence rate of the iteration is compared and analyzed. The numerical results are given.
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Abdelkader Boucherif
2006-06-01
Full Text Available In this paper we investigate the existence of positive solutions of two-point boundary value problems for nonlinear second order differential equations of the form $(py^{\\prime}^{\\prime}(t+q(ty(t=f(t,y(t,y^{\\prime}(t$, where $f$ is a Carathéodory function, which may change sign, with respect to its second argument, infinitely many times.
An optimal existence theorem for positive solutions of a four-point boundary value problem
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Man Kam Kwong
2009-12-01
Full Text Available We are interested in the existence of positive solutions to a four-point boundary value problem of the differential equation $ y''(t + a(tf(y(t=0 $ on $ [0,1] $. The value of $y$ at $0$ and $1$ are each a multiple of $y(t$ at an interior point. Many known existence criteria are based on the limiting values of $ f(u/u $ as $u$ approaches $0$ and infinity.
Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator
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Ding Yonghong
2011-01-01
Full Text Available We consider the existence, multiplicity of positive solutions for the integral boundary value problem with -Laplacian , , , , where is an odd, increasing homeomorphism from onto . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term is involved with the first-order derivative explicitly.
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Domoshnitsky Alexander
2009-01-01
Full Text Available We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation
Bory Reyes, Juan; Abreu Blaya, Ricardo; Rodríguez Dagnino, Ramón Martin; Kats, Boris Aleksandrovich
2018-01-01
The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we present an appropriate hyperholomorphic approach to the RBVP associated to the two dimensional Helmholtz equation in R^2 . Our analysis is based on a suitable operator calculus.
Fractional boundary value problems with multiple orders of fractional derivatives and integrals
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Sotiris K. Ntouyas
2017-04-01
Full Text Available In this article we study a new class of boundary value problems for fractional differential equations and inclusions with multiple orders of fractional derivatives and integrals, in both fractional differential equation and boundary conditions. The Sadovski's fixed point theorem is applied in the single-valued case while, in multi-valued case, the nonlinear alternative for contractive maps is used. Some illustrative examples are also included.
Deniz, Sinan; Bildik, Necdet
2016-06-01
In this paper, we use Adomian Decomposition Method (ADM) to solve the singularly perturbed fourth order boundary value problem. In order to make the calculation process easier, first the given problem is transformed into a system of two second order ODEs, with suitable boundary conditions. Numerical illustrations are given to prove the effectiveness and applicability of this method in solving these kinds of problems. Obtained results shows that this technique provides a sequence of functions which converges rapidly to the accurate solution of the problems.
Student Solutions Manual to Boundary Value Problems and Partial Differential Equations
Powers, David L
2005-01-01
This student solutions manual accompanies the text, Boundary Value Problems and Partial Differential Equations, 5e. The SSM is available in print via PDF or electronically, and provides the student with the detailed solutions of the odd-numbered problems contained throughout the book.Provides students with exercises that skillfully illustrate the techniques used in the text to solve science and engineering problemsNearly 900 exercises ranging in difficulty from basic drills to advanced problem-solving exercisesMany exercises based on current engineering applications
Questions on solvability of exterior boundary value problems with fractional boundary conditions
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Berikbol Torebek
2016-05-01
Full Text Available In this paper we study questions on solvability of some boundary value problems for the Laplace equation with boundary integro-differential operators in the exterior of a unit ball. We study properties of the given integral - differential operators of fractional order in a class of functions which are harmonic outside a ball. We prove theorems about existence and uniqueness of a solution of the problem. We construct explicit form of the solution of the problem in integral form, by solving the Dirichlet problem.
Use of Green's functions in the numerical solution of two-point boundary value problems
Gallaher, L. J.; Perlin, I. E.
1974-01-01
This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.
New fixed point approach for a fully nonlinear fourth order boundary value problem
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Dang Quang A
2018-10-01
Full Text Available In this paper we propose a method for investigating the solvability and iterative solution of a nonlinear fully fourth order boundary value problem. Namely, by the reduction of the problem to an operator equation for the right-hand side function we establish the existence and uniqueness of a solution and the convergence of an iterative process. Our method completely differs from the methods of other authors and does not require the condition of boundedness or linear growth of the right-hand side function on infinity. Many examples, where exact solutions of the problems are known or not, demonstrate the effectiveness of the obtained theoretical results.
Choudhury, Anupam Pal; Crippa, Gianluca; Spinolo, Laura V.
2017-12-01
We establish existence and uniqueness results for initial-boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions.
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Zhiyong Wang
2008-09-01
Full Text Available In this paper, we study the existence of positive solutions for the nonlinear nth-order with m-point singular boundary-value problem. By using the fixed point index theory and a new fixed point theorem in cones, the existence of countably many positive solutions for a nonlinear singular boundary value problem are obtained.
Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations
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Olivier Sarbach
2012-08-01
Full Text Available Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
A New technique of Initial Boundary Value Problems Using Homotopy Analysis Method
Wang, D. M.; Zhang, W.; Yao, M. H.; Liu, Y. L.
2017-10-01
In this paper, a new homotopy analysis technique which is applying to solve initial boundary value problems of partial differential equations by admitted both the initial and boundary conditions in the recursive relation to obtain a good approximate solution for the problem is proposed. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Furthermore, we can easily control and adjust the convergence domain and rate of series solutions by the convergence control parameter. The effectiveness of the approach is verified by several examples.
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Ying Wang
2015-03-01
Full Text Available In this article, we study the existence of multiple positive solutions for singular semipositone boundary-value problem (BVP with integral boundary conditions on infinite intervals. By using the properties of the Green's function and the Guo-Krasnosel'skii fixed point theorem, we obtain the existence of multiple positive solutions under conditions concerning the nonlinear functions. The method in this article can be used for a large number of problems. We illustrate the validity of our results with an example in the last section.
OpenMP for 3D potential boundary value problems solved by PIES
KuŻelewski, Andrzej; Zieniuk, Eugeniusz
2016-06-01
The main purpose of this paper is examination of an application of modern parallel computing technique OpenMP to speed up the calculation in the numerical solution of parametric integral equations systems (PIES). The authors noticed, that solving more complex boundary problems by PIES sometimes requires large computing time. This paper presents the use of OpenMP and fast C++ linear algebra library Armadillo for boundary value problems modelled by 3D Laplace's equation and solved using PIES. The testing example shows that the use of mentioned technologies significantly increases speed of calculations in PIES.
Regular boundary value problems for the heat equation with scalar parameters
Kalmenov, Tynysbek Sh.; Besbaev, Gani; Medetbekova, Ryskul
2017-09-01
This paper belongs to the general theory of well-posed initial-boundary value problems for parabolic equations. The classical construction of a boundary value problem is as follows: an equation and a boundary condition are given. It is necessary to investigate the solvability of this problem and properties of the solution if it exists (in the sense of belonging to some space). Beginning with the papers of J. von Neumann and M.I. Vishik (1951), there exists another more general approach: an equation and a space are given, right-hand parts of the equation and boundary conditions, and a solution must belong to this space. It is necessary to describe all the boundary conditions, for which the problem is correctly solvable in this space. Further development of this theory was given by M. Otelbaev, who constructed a complete theory for ordinary differential operators and for symmetric semibounded operators in a Banach space. In this paper we find regular solution of the regular boundary problem for the heat equation with scalar parameter.
Boundary-value problems for wave equations with data on the whole boundary
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Makhmud A. Sadybekov
2016-10-01
Full Text Available In this article we propose a new formulation of boundary-value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of boundary-value problem in the classical and generalized senses. To substantiate the well-posedness of this problem it is necessary to have an effective representation of the general solution of the problem. In this direction we obtain a convenient representation of the general solution for the wave equation in a rectangular domain based on d'Alembert classical formula. The constructed general solution automatically satisfies the boundary conditions by a spatial variable. Further, by setting different boundary conditions according to temporary variable, we get some functional or functional-differential equations. Thus, the proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations.
A simple finite element method for boundary value problems with a Riemann–Liouville derivative
Jin, Bangti
2016-02-01
© 2015 Elsevier B.V. All rights reserved. We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα-^{1} in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and ^{L2}(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.
Fixed set theorems for discrete dynamics and nonlinear boundary-value problems
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Robert Brooks
2011-05-01
Full Text Available We consider self-mappings of Hausdorff topological spaces which map compact sets to compact sets and establish the existence of invariant (fixed sets. The fixed set results are used to provide fixed set analogues of well-known fixed point theorems. An algorithm is employed to compute the existence of fixed sets which are self-similar in a generalized sense. Some numerical examples are given. The utility of the abstract result is further illustrated via the study of a boundary value problem for a system of differential equations
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Guotao Wang
2012-02-01
D^\\alpha_{0^+} u(t +a(tf(u(\\theta(t=0,&03\\,\\, (n\\in\\mathbb{N},~D^\\alpha_{0^+}$ is the standard Riemann-Liouville fractional derivative of order $\\alpha,$ $f: [0,\\infty\\to [0,\\infty,$ $a: [0,1]\\to (0,\\infty$ and $\\theta: (0,1\\to (0,1]$ are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established.
Solving eighth-order boundary value problems using differential transformation method
Hussin, Che Haziqah Che; Mandangan, Arif
2014-12-01
In this study, we solved linear and nonlinear eighth-order boundary value problems using Differential Transformation Method. Then we calculate the error of DTM and compare the results with other methods such as modified application of the variational iteration method (MVAM), homotopy perturbation method (HPM) and modified Adomian decomposition method (MADM). We compared the errors of each method with exact solutions. We provided several numerical examples in order to show the accuracy and efficiency of present method. The results showed that the DTM is more accurate in comparison with those obtained by other methods.
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Marwan Abukhaled
2013-01-01
Full Text Available The variational iteration method is applied to solve a class of nonlinear singular boundary value problems that arise in physiology. The process of the method, which produces solutions in terms of convergent series, is explained. The Lagrange multipliers needed to construct the correctional functional are found in terms of the exponential integral and Whittaker functions. The method easily overcomes the obstacle of singularities. Examples will be presented to test the method and compare it to other existing methods in order to confirm fast convergence and significant accuracy.
Amenta, Alex; Auscher, Pascal
2017-01-01
International audience; In this monograph our main goal is to study the well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems div A∇u = 0 on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Hardy–Sobolev and Besov spaces. Our approach uses minimal assumptions on the coefficients A, and in particular does not require De Giorgi–Nash–Moser estimates. Our results are completely new for the H...
Iterative Method for Solving the Second Boundary Value Problem for Biharmonic-Type Equation
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Dang Quang A.
2012-01-01
Full Text Available Solving boundary value problems (BVPs for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation, which describes the deflection of a plate resting on a biparametric elastic foundation. The convergence rate of the method is established. The optimal value of the iterative parameter is found. Several numerical examples confirm the efficiency of the proposed method.
Investigation of solutions of state-dependent multi-impulsive boundary value problems
Czech Academy of Sciences Publication Activity Database
Rontó, András; Rachůnková, I.; Rontó, M.; Rachůnek, L.
2017-01-01
Roč. 24, č. 2 (2017), s. 287-312 ISSN 1072-947X R&D Projects: GA ČR(CZ) GA14-06958S Institutional support: RVO:67985840 Keywords : state-dependent multi-impulsive systems * non-linear boundary value problem * parametrization technique Subject RIV: BA - General Mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2016-0084/gmj-2016-0084. xml
A combined analytic-numeric approach for some boundary-value problems
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Mustafa Turkyilmazoglu
2016-02-01
Full Text Available A combined analytic-numeric approach is undertaken in the present work for the solution of boundary-value problems in the finite or semi-infinite domains. Equations to be treated arise specifically from the boundary layer analysis of some two and three-dimensional flows in fluid mechanics. The purpose is to find quick but accurate enough solutions. Taylor expansions at either boundary conditions are computed which are next matched to the other asymptotic or exact boundary conditions. The technique is applied to the well-known Blasius as well as Karman flows. Solutions obtained in terms of series compare favorably with the existing ones in the literature.
Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations
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Xianghu Liu
2014-01-01
Full Text Available This paper is concerned with the sufficient conditions for the existence of solutions for a class of generalized antiperiodic boundary value problem for nonlinear fractional impulsive differential equations involving the Riemann-Liouville fractional derivative. Firstly, we introduce the fractional calculus and give the generalized R-L fractional integral formula of R-L fractional derivative involving impulsive. Secondly, the sufficient condition for the existence and uniqueness of solutions is presented. Finally, we give some examples to illustrate our main results.
Fixed point results for G-α-contractive maps with application to boundary value problems.
Hussain, Nawab; Parvaneh, Vahid; Roshan, Jamal Rezaei
2014-01-01
We unify the concepts of G-metric, metric-like, and b-metric to define new notion of generalized b-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes of G-α -admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.
Roul, Pradip; Warbhe, Ujwal
2017-08-01
The classical homotopy perturbation method proposed by J. H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999) is useful for obtaining the approximate solutions for a wide class of nonlinear problems in terms of series with easily calculable components. However, in some cases, it has been found that this method results in slowly convergent series. To overcome the shortcoming, we present a new reliable algorithm called the domain decomposition homotopy perturbation method (DDHPM) to solve a class of singular two-point boundary value problems with Neumann and Robin-type boundary conditions arising in various physical models. Five numerical examples are presented to demonstrate the accuracy and applicability of our method, including thermal explosion, oxygen-diffusion in a spherical cell and heat conduction through a solid with heat generation. A comparison is made between the proposed technique and other existing seminumerical or numerical techniques. Numerical results reveal that only two or three iterations lead to high accuracy of the solution and this newly improved technique introduces a powerful improvement for solving nonlinear singular boundary value problems (SBVPs).
Holota, P.
The purpose of this paper is to discuss the relation between the classical methods in the solution of the geodetic boundary value problems and the parameterization of the disturbing potential within the modern concepts. Therefore, a tie is investigated between the integral representation of the disturbing potential and the Hilbert space approach to solution of the respective boundary value problem. Problems are consid- ered that include the use of the global reference field, terrestrial, satellite and airborne data. Integral kernels of various kind and properties, band-limited and non-band lim- ited functions are used in quality of a trial system and a function basis. The concept is interpreted in terms of variational methods with their natural relation to the Dirich- let principle or alternatively to the Lax-Milgram theorem. Subsequently an attention is paid to the linear system for the numerical coefficients in the representation of the disturbing potential and finally an iteration process is constructed to treat the effect of the topography.
Conjugate Gradient Method with Ritz Method for the Solution of Boundary Value Problems
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Victor Onomza WAZIRI
2007-01-01
Full Text Available In this paper, we wish to determine the optimal control of a one-variable boundary value problem using the Ritz algorithm. The posed optimal control problem was inadequate to achieve our goal using the Conjugate Gradient Method version developed by (Harsdoff, 1976. It is anticipated that other operators from some given different problems may sustain the application of the algorithm if the approximate solutions terms are properly chosen quadratic functionals. The graphical solution given at the end of section five of the paper, however, shows that our problem can not have an optimal minimum value since the minimum output is not unique. The optimal value obtained using Mathcad program codes may constitute a conjugate gradient approximate numerical value. As observed from the graphical output, Ritz algorithm could give credence for wider horizon in the engineering computational methods for vibrations of mechanical components and simulates.
Existence of solutions to fractional boundary-value problems with a parameter
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Ya-Ning Li
2013-06-01
Full Text Available This article concerns the existence of solutions to the fractional boundary-value problem $$displaylines{ -frac{d}{dt} ig(frac{1}{2} {}_0D_t^{-eta}+ frac{1}{2}{}_tD_{T}^{-eta}igu'(t=lambda u(t+abla F(t,u(t,quad hbox{a.e. } tin[0,T], cr u(0=0,quad u(T=0. }$$ First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u, we show the existence of weak solutions of the problem when $lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.
Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth-Order Boundary Value Problems
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A. H. Bhrawy
2011-01-01
Full Text Available We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs. An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
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B. Juarez Campos
2016-01-01
Full Text Available We study the mixed initial-boundary value problem for the capillary wave equation: iut+u2u=∂x3/2u, t>0, x>0; u(x,0=u0(x, x>0; u(0,t+βux(0,t=h(t, t>0, where ∂x3/2u=(1/2π∫0∞signx-y/x-yuyy(y dy. We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
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Hoi Ying Wong
2013-01-01
Full Text Available Turbo warrants are liquidly traded financial derivative securities in over-the-counter and exchange markets in Asia and Europe. The structure of turbo warrants is similar to barrier options, but a lookback rebate will be paid if the barrier is crossed by the underlying asset price. Therefore, the turbo warrant price satisfies a partial differential equation (PDE with a boundary condition that depends on another boundary-value problem (BVP of PDE. Due to the highly complicated structure of turbo warrants, their valuation presents a challenging problem in the field of financial mathematics. This paper applies the homotopy analysis method to construct an analytic pricing formula for turbo warrants under stochastic volatility in a PDE framework.
Quasisolutions of Inverse Boundary-Value Problem of Aerodynamics for Dense Airfoil Grids
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A.M. Elizarov
2016-12-01
Full Text Available In the process of turbomachinery development, it is of great importance to accurately design impellers and select their blade shape. One of the promising approaches to solving this problem is based on the theory of inverse boundary-value problems in aerodynamics. It helps to develop methods for profiling airfoil grids with predetermined properties in the same way as it is done for isolated airfoils. In this paper, methods have been worked out to find quasisolutions of the inverse boundary-value problem in aerodynamics for a plane airfoil grid. Two methods of quasisolution have been described. The first “`formal” method is similar, in its essence, to the method used for construction of quasisolution for an isolated airfoil. It has been shown that such quasisolutions provide satisfactory results for grids having a sufficiently large relative airfoil pitch. If pitch values are low, this method is unacceptable, because “modified” velocity distribution in some areas is significantly different from the original one in this case. For this reason, areas with significant changes in the angle of the tangent line appear in the airfoil contour and the flow region becomes multivalent. To satisfy the conditions of solvability in the case of grids having a small airfoil pitch, a new quasisolution construction method taking into account the specifics of the problem has been suggested. The desired effect has been achieved due to changes in the weighting function of the minimized functional. The comparison of the results of construction of the new quasisolution with the results obtained by the “formal” method has demonstrated that the constructed airfoils are very similar when the pitch is large. In the case of dense grids, it is clear that preference should be given to the second method, as it brings less distortion to the initial velocity distribution and, thus, allows to physically find an actual airfoil contour.
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Bozor Islomov
2015-08-01
Full Text Available We prove the unique solvability of a boundary-value problems for a third-order loaded integro-differential equation with variable coefficients, by reducing the equation to a Volterra integral equation.
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Mabrouk Briki
2016-05-01
Full Text Available In this paper, a fourth-order boundary value problem on the half-line is considered and existence of solutions is proved using a minimization principle and the mountain pass theorem.
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Muhammad Tahir
2012-11-01
Full Text Available This article shows the uniqueness of a solution to a Bitsadze system of equations, with a boundary-value problem that has four additional single point conditions. It also shows how to construct the solution.
On explicit and numerical solvability of parabolic initial-boundary value problems
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Olga Lepsky
2006-05-01
Full Text Available A homogeneous boundary condition is constructed for the parabolic equation (Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”u=f in an arbitrary cylindrical domain ÃŽÂ©ÃƒÂ—Ã¢Â„Â (ÃŽÂ©Ã¢ÂŠÂ‚Ã¢Â„Ân being a bounded domain, I and ÃŽÂ” being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”, but also for an arbitrary parabolic differential operator Ã¢ÂˆÂ‚t+A, where A is an elliptic operator in Ã¢Â„Ân of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation (Ã¢ÂˆÂ‚t+IÃ¢ÂˆÂ’ÃŽÂ”u=0 in ÃŽÂ©ÃƒÂ—Ã¢Â„Â is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables.
METHOD OF GREEN FUNCTIONS IN MATHEMATICAL MODELLING FOR TWO-POINT BOUNDARY-VALUE PROBLEMS
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E. V. Dikareva
2015-01-01
Full Text Available Summary. In many applied problems of control, optimization, system theory, theoretical and construction mechanics, for problems with strings and nods structures, oscillation theory, theory of elasticity and plasticity, mechanical problems connected with fracture dynamics and shock waves, the main instrument for study these problems is a theory of high order ordinary differential equations. This methodology is also applied for studying mathematical models in graph theory with different partitioning based on differential equations. Such equations are used for theoretical foundation of mathematical models but also for constructing numerical methods and computer algorithms. These models are studied with use of Green function method. In the paper first necessary theoretical information is included on Green function method for multi point boundary-value problems. The main equation is discussed, notions of multi-point boundary conditions, boundary functionals, degenerate and non-degenerate problems, fundamental matrix of solutions are introduced. In the main part the problem to study is formulated in terms of shocks and deformations in boundary conditions. After that the main results are formulated. In theorem 1 conditions for existence and uniqueness of solutions are proved. In theorem 2 conditions are proved for strict positivity and equal measureness for a pair of solutions. In theorem 3 existence and estimates are proved for the least eigenvalue, spectral properties and positivity of eigenfunctions. In theorem 4 the weighted positivity is proved for the Green function. Some possible applications are considered for a signal theory and transmutation operators.
Approximate series solution of nonlinear singular boundary value problems arising in physiology.
Singh, Randhir; Kumar, Jitendra; Nelakanti, Gnaneshwar
2014-01-01
We introduce an efficient recursive scheme based on Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems. This approach is based on a modification of the ADM; here we use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components. In fact, we develop the recursive scheme without any undetermined coefficients while computing the solution components. Unlike the classical ADM, the proposed method avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. The uniqueness of the solution is discussed. The convergence and error analysis of the proposed method are also established. The accuracy and reliability of the proposed method are examined by four numerical examples.
An efficient numerical technique for the solution of nonlinear singular boundary value problems
Singh, Randhir; Kumar, Jitendra
2014-04-01
In this work, a new technique based on Green's function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green's function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.
Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory
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Yang Wang
2017-01-01
Full Text Available This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: DT-αaxD0+αux=fx,ux, x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=ess infx∈0,Tax>0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.
Solution matching for a three-point boundary-value problem on atime scale
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Martin Eggensperger
2004-07-01
Full Text Available Let $mathbb{T}$ be a time scale such that $t_1, t_2, t_3 in mathbb{T}$. We show the existence of a unique solution for the three-point boundary value problem $$displaylines{ y^{DeltaDeltaDelta}(t = f(t, y(t, y^Delta(t, y^{DeltaDelta}(t, quad t in [t_1, t_3] cap mathbb{T},cr y(t_1 = y_1, quad y(t_2 = y_2, quad y(t_3 = y_3,. }$$ We do this by matching a solution to the first equation satisfying a two-point boundary conditions on $[t_1, t_2] cap mathbb{T}$ with a solution satisfying a two-point boundary conditions on $[t_2, t_3] cap mathbb{T}$.
Method for solving moving boundary value problems for linear evolution equations
Fokas; Pelloni
2000-05-22
We introduce a method of solving initial boundary value problems for linear evolution equations in a time-dependent domain, and we apply it to an equation with dispersion relation omega(k), in the domain l(t)integral representation in the complex k plane, involving either an integral of exp[ikx-iomega(k)t]rho(k) along a time-dependent contour, or an integral of exp[ikx-iomega(k)t]rho(k, &kmacr;) over a fixed two-dimensional domain. The functions rho(k) and rho(k,&kmacr;) can be computed through the solution of a system of Volterra linear integral equations. This method can be generalized to nonlinear integrable partial differential equations.
A symmetric solution of a multipoint boundary value problem at resonance
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2006-01-01
Full Text Available We apply a coincidence degree theorem of Mawhin to show the existence of at least one symmetric solution of the nonlinear second-order multipoint boundary value problem u ″ ( t = f ( t , u ( t , | u ′ ( t | , t ∈ ( 0 , 1 , u ( 0 = ∑ i = 1 n μ i u ( ξ i , u ( 1 − t = u ( t , t ∈ ( 0 , 1 ] , where 0 < ξ 1 < ξ 2 < … ≤ ξ n 1 / 2 , ∑ i = 1 n μ i = 1 , f : [ 0 , 1 ] × ℝ 2 → ℝ with f ( t , x , y = f ( 1 − t , x , y , ( t , x , y ∈ [ 0 , 1 ] × ℝ 2 , satisfying the Carathéodory conditions.
Numerical continuation methods for dynamical systems path following and boundary value problems
Krauskopf, Bernd; Galan-Vioque, Jorge
2007-01-01
Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel''s 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects ...
Modelling of hydrogen thermal desorption spectrum in nonlinear dynamical boundary-value problem
Kostikova, E. K.; Zaika, Yu V.
2016-11-01
One of the technological challenges for hydrogen materials science (including the ITER project) is the currently active search for structural materials with various potential applications that will have predetermined limits of hydrogen permeability. One of the experimental methods is thermal desorption spectrometry (TDS). A hydrogen-saturated sample is degassed under vacuum and monotone heating. The desorption flux is measured by mass spectrometer to determine the character of interactions of hydrogen isotopes with the solid. We are interested in such transfer parameters as the coefficients of diffusion, dissolution, desorption. The paper presents a distributed boundary-value problem of thermal desorption and a numerical method for TDS spectrum simulation, where only integration of a nonlinear system of low order (compared with, e.g., the method of lines) ordinary differential equations (ODE) is required. This work is supported by the Russian Foundation for Basic Research (project 15-01-00744).
Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions
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Ciprian G. Gal
2017-01-01
Full Text Available Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞, we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α1, α2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L∞-setting.
Valent, Tullio
1988-01-01
In this book I present, in a systematic form, some local theorems on existence, uniqueness, and analytic dependence on the load, which I have recently obtained for some types of boundary value problems of finite elasticity. Actually, these results concern an n-dimensional (n ~ 1) formal generalization of three-dimensional elasticity. Such a generalization, be sides being quite spontaneous, allows us to consider a great many inter esting mathematical situations, and sometimes allows us to clarify certain aspects of the three-dimensional case. Part of the matter presented is unpublished; other arguments have been only partially published and in lesser generality. Note that I concentrate on simultaneous local existence and uniqueness; thus, I do not deal with the more general theory of exis tence. Moreover, I restrict my discussion to compressible elastic bodies and I do not treat unilateral problems. The clever use of the inverse function theorem in finite elasticity made by STOPPELLI [1954, 1957a, 1957b]...
Kiselev, Yu. N.; Orlov, M. V.; Orlov, S. M.
2015-11-01
An infinite-horizon two-sector economy model with a Cobb-Douglas production function and a utility function that is an integral functional with discounting and a logarithmic integrand is investigated. The application of Pontryagin's maximum principle yields a boundary value problem with special conditions at infinity. The search for the solution of the maximum-principle boundary value problem is complicated by singular modes in its optimal solution. In the construction of the solution to the problem, they are described in analytical form. Additionally, a special version of the sweep method in continuous form is proposed, which is of interest from theoretical and computational points of view. An important result is the proof of the optimality of the extremal solution obtained by applying the maximum-principle boundary value problem.
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Dongyuan Liu
2015-01-01
Full Text Available We consider the following state dependent boundary-value problem D0+αy(t-pD0+βg(t,y(σ(t+f(t,y(τ(t=0, 0
Advances in the study of boundary value problems for nonlinear integrable PDEs
Pelloni, Beatrice
2015-02-01
In this review I summarize some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations (PDEs) in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the unified transform or Fokas transform, that provides a substantial generalization of the classical inverse scattering transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the inverse scattering transform follows the ‘separation of variables’ philosophy, albeit in a nonlinear setting, the unified transform is based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalization to certain nonlinear cases of particular significance.
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Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
The CFL condition for spectral approximations to hyperbolic initial-boundary value problems
Gottlieb, David; Tadmor, Eitan
1990-01-01
The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.
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Omer Kelesoglu
2014-01-01
Full Text Available Adomian decomposition method (ADM is applied to linear nonhomogeneous boundary value problem arising from the beam-column theory. The obtained results are expressed in tables and graphs. We obtain rapidly converging results to exact solution by using the ADM. This situation indicates that the method is appropriate and reliable for such problems.
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M. Mamajonov
2014-06-01
Full Text Available This paper studies the methods of investigation of some boundary value problems for a class of parabolic-hyperbolic equations of the third order in the hexagonal concave areas that take advantage of the study of problems of mathematical physics in the magistracy.
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D. Goos
2015-01-01
Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
Liu, Shuangqian; Yang, Xiongfeng
2017-01-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
Existence Results for a Coupled System of Nonlinear Fractional Boundary Value Problems at Resonance
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Ruijuan Liu
2013-01-01
Full Text Available Some new Banach spaces are established. Based on those new Banach spaces and by using the coincidence degree theory, we present the existence results for a coupled system of nonlinear fractional differential equations with multipoint boundary value conditions at resonance case.
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Gai Gongqi
2011-01-01
Full Text Available Abstract This article studies the boundary value problems for the third-order nonlinear singular difference equations Δ 3 u ( i - 2 + λ a ( i f ( i , u ( i = 0 , i ∈ [ 2 , T + 2 ] , satisfying five kinds of different boundary value conditions. This article shows the existence of positive solutions for positone and semi-positone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone. MSC [2008]: 34B15; 39A10.
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Neda Khodabakhshi
2013-12-01
Full Text Available This paper deals with the existence results for solutions of coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain. Also, we give an illustrative example in order to indicate the validity of our assumptions.
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Wang Gang
2011-01-01
Full Text Available Abstract A 2m-point boundary value problem for a coupled system of nonlinear fractional differential equations is considered in this article. An existence result is obtained with the use of the coincidence degree theory. MSC: 34B17; 34L09.
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Lingling Cheng
2014-02-01
Full Text Available In this article, we discuss the existence of solutions to boundary-value problems for a coupled system of fractional differential equations with p-Laplacian operator at resonance. We prove the existence of solutions when $\\dim \\ker L\\geq 2$, using the coincidence degree theory by Mawhin.
Even number of positive solutions for 3nth order three-point boundary value problem on time scales
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K. Prasad
2011-12-01
Full Text Available We establish the existence of at least two positive solutions for the 3nth order three-point boundary value problem on time scales by using Avery-Henderson fixed point theorem. We also establish the existence of at least 2m positive solutions for an arbitrary positive integer m.
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Mitsuhiro Nakao
2014-01-01
Full Text Available We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a 'loan' method and use a difference inequality on the energy.
Yu Zhang
2014-01-01
The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.
M. K. Hasan; Y. H. Ng; J. Sulaiman
2013-01-01
This paper present the implementation of a new ordering strategy on Successive Overrelaxation scheme on two dimensional boundary value problems. The strategy involve two directions alternatingly; from top and bottom of the solution domain. The method shows to significantly reduce the iteration number to converge. Four numerical experiments were carried out to examine the performance of the new strategy.
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Lv Xuezhe
2010-01-01
Full Text Available Abstract The existence and uniqueness of positive solution is obtained for the singular second-order -point boundary value problem for , , , where , , are constants, and can have singularities for and/or and for . The main tool is the perturbation technique and Schauder fixed point theorem.
Di Lizia, P.; Armellin, R.; Bernelli Zazzera, F; Berz, M.
2008-01-01
Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two- body problem. However, a certain level of approximation always affects the dynamical models adopted to design the nominal trajectory of a spacecraft. Dynamical perturbations usually act on the spacecraft in real scenarios, deviating i...
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Javed Ali
2012-01-01
Full Text Available We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM. The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM, variational iteration (VIM, homotopy perturbation (HPM, and variational iteration decomposition method (VIDM. The results show that the proposed method is more effective and reliable.
Nau, Tobias
2012-01-01
Tobias Nau addresses initial boundary value problems in cylindrical space domains with the aid of modern techniques from functional analysis and operator theory. In particular, the author uses concepts from Fourier analysis of functions with values in Banach spaces and the operator-valued functional calculus of sectorial operators. He applies abstract results to concrete problems in cylindrical space domains such as the heat equation subject to numerous boundary conditions and equations arising from fluid dynamics.
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W. M. Abd-Elhameed
2015-01-01
Full Text Available The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.
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Zulqurnain Sabir
2014-06-01
Full Text Available In this paper, computational intelligence technique are presented for solving multi-point nonlinear boundary value problems based on artificial neural networks, evolutionary computing approach, and active-set technique. The neural network is to provide convenient methods for obtaining useful model based on unsupervised error for the differential equations. The motivation for presenting this work comes actually from the aim of introducing a reliable framework that combines the powerful features of ANN optimized with soft computing frameworks to cope with such challenging system. The applicability and reliability of such methods have been monitored thoroughly for various boundary value problems arises in science, engineering and biotechnology as well. Comprehensive numerical experimentations have been performed to validate the accuracy, convergence, and robustness of the designed scheme. Comparative studies have also been made with available standard solution to analyze the correctness of the proposed scheme.
Maksimova, N.V.; Akhmetov, R. G.
2013-01-01
The work deals with a boundary value problem for a quasilinear partial elliptical equation. The equation describes a stationary process of convective diffusion near a cylinder and takes into account the value of a chemical reaction for large Peclet numbers and for large constant of chemical reaction. The quantity the rate constant of the chemical reaction and Peclet number is assumed to have a constant value. The leading term of the asymptotics of the solution is constructed in the boundary l...
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Zhang Peiguo
2011-01-01
Full Text Available Abstract By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator. MSC: 34B15; 34B16.
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Hua Luo
2012-01-01
Full Text Available Let be a time scale with 0,T∈. We give a global description of the branches of positive solutions to the nonlinear boundary value problem of second-order dynamic equation on a time scale , uΔΔ(t+f(t,uσ(t=0, t∈[0,T], u(0=u(σ2(T=0, which is not necessarily linearizable. Our approaches are based on topological degree theory and global bifurcation techniques.
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Cristian Enache
2006-06-01
Full Text Available For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of u(x and |Ã¢ÂˆÂ‡u|2, where u(x are the solutions of our problems. From these inequalities, we derive, using Hopf's maximum principles, some maximum principles for the appropriate combinations of u(x and |Ã¢ÂˆÂ‡u|2, and we list a few examples of problems to which these maximum principles may be applied.
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A. M. Shkhagapsoev
2017-11-01
Full Text Available We consider the boundary-value problem for a third-order equation of parabolic type with the fractional derivative of Caputo. By the method of energy inequalities an a priori estimate of the solution of the analogue of the second boundary value problem for an equation with multiple characteristics.
Pskhu, A. V.
2017-12-01
We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan– Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient condition for unique solubility is the condition of Hölder smoothness for the lateral boundary of the domain. The corresponding results for equations with Riemann– Liouville and Caputo derivatives are particular cases of results obtained here.
Ardalan, A.; Safari, A.; Grafarend, E.
2003-04-01
A new ellipsoidal gravimetric-satellite altimetry boundary value problem has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential (ii) gravity intensity (iii) deflection of vertical and (iv) satellite altimetry data. The developed boundary value problem is enjoying the ellipsoidal nature and as such can take advantage of high precision GPS observations in the set-up of the problem. The highlights of the solution are as follows: begin{itemize} Application of ellipsoidal harmonic expansion up to degree/order and ellipsoidal centrifugal field for the reduction of global gravity and isostasy effects from the gravity observable at the surface of the Earth. Application of ellipsoidal Newton integral on the equal area map projection surface for the reduction of residual mass effects within a radius of 55 km around the computational point. Ellipsoidal harmonic downward continuation of the residual observables from the surface of the earth down to the surface of reference ellipsoid using the ellipsoidal height of the observation points derived from GPS. Restore of the removed effects at the application points on the surface of reference ellipsoid. Conversion of the satellite altimetry derived heights of the water bodies into potential. Combination of the downward continued gravity information with the potential equivalent of the satellite altimetry derived heights of the water bodies. Application of ellipsoidal Bruns formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights (i.e. ellipsoidal heights of the geoid) with respect to the reference ellipsoid. Computation of the high-resolution geoid of Iran has successfully tested this new methodology!
Application of Two-Parameter Extrapolation for Solution of Boundary-Value Problem on Semi-Axis
Zhidkov, E P
2000-01-01
A method for refining approximate eigenvalues and eigenfunctions for a boundary-value problem on a half-axis is suggested. To solve the problem numerically, one has to solve a problem on a finite segment [0,R] instead of the original problem on the interval [0,\\infty). This replacement leads to eigenvalues' and eigenfunctions' errors. To choose R beforehand for obtaining their required accuracy is often impossible. Thus, one has to resolve the problem on [0,R] with larger R. If there are two eigenvalues or two eigenfunctions that correspond to different segments, the suggested method allows one to improve the accuracy of the eigenvalue and the eigenfunction for the original problem by means of extrapolation along the segment. This approach is similar to Richardson's method. Moreover, a two-parameter extrapolation is described. It is combination of the extrapolation along the segment and Richardson's extrapolation along a discretization step.
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Sekson Sirisubtawee
2017-01-01
Full Text Available We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn or the error remainder functions (ERn(x of each problem are calculated.
Yang, Q.; Stainier, L.; Ortiz, M.
2006-02-01
A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids is presented. The coupled thermo-mechanical boundary-value problem under consideration consists of the equilibrium problem for a deformable, inelastic and dissipative solid with the heat conduction problem appended in addition. The variational formulation allows for general dissipative solids, including finite elastic and plastic deformations, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rules, as well as heat conduction. We show that a joint potential function exists such that both the conservation of energy and the balance of linear momentum equations follow as Euler-Lagrange equations. The identification of the joint potential requires a careful distinction between equilibrium and external temperatures, which are equal at equilibrium. The variational framework predicts the fraction of dissipated energy that is converted to heat. A comparison of this prediction and experimental data suggests that α-titanium and Al2024-T conform to the variational framework.
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Chuanzhi Bai
2008-12-01
Full Text Available In this paper, we investigate the existence of multiple positive solutions of the fourth-order four-point boundary-value problems $$displaylines{ y^{(4}(t = h(t g(y(t, y''(t, quad 0 < t < 1, cr y(0 = y(1 = 0, cr a y''(xi_1-b y'''(xi_1 = 0, quad c y''(xi_2+d y'''(xi_2 = 0, }$$ where $0 < xi_1 < xi_2 < 1$. We show the existence of three positive solutions by applying the Avery and Peterson fixed point theorem in a cone, here $h(t$ may change sign on $[0, 1]$.
Positive solutions to a generalized second-order three-point boundary-value problem on time scales
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Hua Luo
2005-02-01
Full Text Available Let $mathbb{T}$ be a time scale with $0,T in mathbb{T}$. We investigate the existence and multiplicity of positive solutions to the nonlinear second-order three-point boundary-value problem $$displaylines{ u^{Delta abla}(t+a(tf(u(t=0,quad tin[0, T]subset mathbb{T},cr u(0=eta u(eta,quad u(T=alpha u(eta }$$ on time scales $mathbb{T}$, where 0, 0less than $alpha$ less than $frac{T}{eta}$, 0 less than $eta$ less than $frac{T-alphaeta}{T-eta}$ are given constants.
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Baoqiang Yan
2008-10-01
Full Text Available Using the fixed point theorem in cones, this paper shows the existence of multiple positive solutions for the singular $m$-point boundary-value problem $$displaylines{ x''(t+a(tf(t,x(t,x'(t=0,quad 0
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Meiqiang Feng
2014-01-01
Full Text Available The author considers an impulsive boundary value problem involving the one-dimensional p-Laplacian -(φp (u′′=λωtft,u, 00 and μ>0 are two parameters. Using fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of λ>0 and μ>0. The exact upper and lower bounds for these positive solutions are also given. Moreover, the approach to deal with the impulsive term is different from earlier approaches. In this paper, our results cover equations without impulsive effects and are compared with some recent results by Ding and Wang.
Feleqi, Ermal
2016-02-01
Estimates in suitable Lebesgue or Sobolev norms for the deviation of solutions and eigenfunctions of second-order uniformly elliptic Dirichlet boundary value problems subject to domain perturbation in terms of natural distances between the domains are given. The main estimates are formulated via certain natural and easily computable ;atlas; distances for domains with Lipschitz continuous boundaries. As a corollary, similar estimates in terms of more ;classical; distances such as the Hausdorff distance or the Lebesgue measure of the symmetric difference of domains are derived. Sharper estimates are also proved to hold in smoother classes of domains.
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George N. Galanis
2005-10-01
Full Text Available In this paper we prove the existence of positive solutions for the three-point singular boundary-value problem$$ -[phi _{p}(u']'=q(tf(t,u(t,quad 0
Solvability of fractional multi-point boundary-value problems with p-Laplacian operator at resonance
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Tengfei Shen
2014-02-01
Full Text Available In this article, we consider the multi-point boundary-value problem for nonlinear fractional differential equations with $p$-Laplacian operator: $$\\displaylines{ D_{0^+}^\\beta \\varphi_p (D_{0^+}^\\alpha u(t = f(t,u(t,D_{0^+}^{\\alpha - 2} u(t,D_{0^+}^{\\alpha - 1} u(t, D_{0^+}^\\alpha u(t,\\quad t \\in (0,1, \\cr u(0 = u'(0=D_{0^+}^\\alpha u(0 = 0,\\quad D_{0^+}^{\\alpha - 1} u(1 = \\sum_{i = 1}^m {\\sigma_i D_{0^+}^{\\alpha - 1} u(\\eta_i } , }$$ where $2 < \\alpha \\le 3$, $0 < \\beta \\le 1$, $3 < \\alpha + \\beta \\le 4$, $\\sum_{i = 1}^m {\\sigma_i } = 1$, $D_{0^+}^\\alpha$ is the standard Riemann-Liouville fractional derivative. $\\varphi_{p}(s=|s|^{p-2}s$ is p-Laplacians operator. The existence of solutions for above fractional boundary value problem is obtained by using the extension of Mawhin's continuation theorem due to Ge, which enrich konwn results. An example is given to illustrate the main result.
Multipoint Singular Boundary-Value Problem for Systems of Nonlinear Differential Equations
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Zdeněk Šmarda
2009-01-01
Full Text Available A singular Cauchy-Nicoletti problem for a system of nonlinear ordinary differential equations is considered. With the aid of combination of Ważewski's topological method and Schauder's principle, the theorem concerning the existence of a solution of this problem (having the graph in a prescribed domain is proved.
Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation
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A. S. Berdyshev
2011-01-01
Full Text Available Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.
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Hammad Khalil
2016-01-01
Full Text Available The paper is devoted to the study of operational matrix method for approximating solution for nonlinear coupled system fractional differential equations. The main aim of this paper is to approximate solution for the problem under two different types of boundary conditions, m^-point nonlocal boundary conditions and mixed derivative boundary conditions. We develop some new operational matrices. These matrices are used along with some previously derived results to convert the problem under consideration into a system of easily solvable matrix equations. The convergence of the developed scheme is studied analytically and is conformed by solving some test problems.
K/S two-point-boundary-value problems. [for orbital trajectory optimization
Jezewski, D. J.
1976-01-01
A method for developing the missing general K/S (Kustaanheimo/Stiefel) boundary conditions is presented, with use of the formalism of optimal control theory. As an illustrative example, the method is applied to the K/S Lambert problem to derive the missing terminal condition. The necessary equations are developed for a solution to this problem with the generalized eccentric anomaly, E, as the independent variable. This formulation, requiring the solution of only one nonlinear, well-behaved equation in one unknown, E, results in considerable simplification of the problem.
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Vrabel Robert
2011-01-01
Full Text Available Abstract This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.
A cut finite element method for the Bernoulli free boundary value problem
National Research Council Canada - National Science Library
Burman, Erik; Elfverson, Daniel; Hansbo, Peter; Larson, Mats G; Larsson, Karl
2017-01-01
We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion...
Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems
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Sertan Alkan
2015-01-01
Full Text Available We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sinc-Galerkin method is a very efficient tool in solving space-fractional partial differential equations.
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
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Mesloub Said
2008-01-01
Full Text Available This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.
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A. Anguraj
2014-02-01
Full Text Available We study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. We establish the Sufficient conditions and unique solution for given problem. An Example is also explained to the main results.
Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem
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Aixia Qian
2005-11-01
Full Text Available We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions. The study of such a problem is based on the variational methods and critical point theory. We prove the conclusion by using the symmetric mountain-pass theorem under the Cerami condition.
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations
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H. Batarfi
2015-01-01
Full Text Available We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.
Kot, V. A.
2017-11-01
The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.
High regularity of the solution of a nonlinear parabolic boundary-value problem
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Luminita Barbu
2002-05-01
Full Text Available The aim of this paper is to report some results concerning high regularity of the solution of a nonlinear parabolic problem with a linear parabolic differential equation in one spatial dimension and nonlinear boundary conditions. We show that any regularity can be reached provided that appropriate smoothness of the data and compatibility assumptions are required.
Favini, Angelo; Rocca, Elisabetta; Schimperna, Giulio; Sprekels, Jürgen
2017-01-01
This volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; well-posedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a self-contained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs.
Xie, Lie-Jun; Zhou, Cai-Lian; Xu, Song
2016-01-01
In this work, an effective numerical method is developed to solve a class of singular boundary value problems arising in various physical models by using the improved differential transform method (IDTM). The IDTM applies the Adomian polynomials to handle the differential transforms of the nonlinearities arising in the given differential equation. The relation between the Adomian polynomials of those nonlinear functions and the coefficients of unknown truncated series solution is given by a simple formula, through which one can easily deduce the approximate solution which takes the form of a convergent series. An upper bound for the estimation of approximate error is presented. Several physical problems are discussed as illustrative examples to testify the validity and applicability of the proposed method. Comparisons are made between the present method and the other existing methods.
Fayolle, Guy; Malyshev, Vadim
2017-01-01
This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous applications and are of interest in several areas of mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum Physics. This second edition consists of two parts. Part I is a revised upgrade of the first edition (1999), with additional recent results on the group of a random walk. The theoretical approach given therein has been developed by the authors since the early 1970s. By using Complex Function Theory, Boundary Value Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed for solving functional equations of two complex variables, which can also be applied to characterize the Transient Behavior of the walks, as well as to find explicit solutions to the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable Systems. Part II borrows spec...
A Simulation Study of the Overdetermined Geodetic Boundary Value Problem Using Collocation
1989-03-01
VALUE PROBLEM 2.1 Fundamental principles of collocation The method of collocation was introduced in geodesy through Moritz ’ work of gravity anomaly...set of given linear functionals ( Moritz , 1980, p. 85). This definition is consistent with two aspects of collocation : the prediction aspect where...is given by Moritz (1980). For the sake of completeness the general least- squares collocation model with parameters is briefly described here in
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Alexander N. Kvitko
2017-01-01
Full Text Available An algorithm for constructing a control function that transfers a wide class of stationary nonlinear systems of ordinary differential equations from an initial state to a final state under certain control restrictions is proposed. The algorithm is designed to be convenient for numerical implementation. A constructive criterion of the desired transfer possibility is presented. The problem of an interorbital flight is considered as a test example and it is simulated numerically with the presented method.
Convergence of a continuous BGK model for initial boundary-value problems for conservation laws
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Driss Seghir
2001-11-01
Full Text Available We consider a scalar conservation law in the quarter plane. This equation is approximated in a continuous kinetic Bhatnagar-Gross-Krook (BGK model. The convergence of the model towards the unique entropy solution is established in the space of functions of bounded variation, using kinetic entropy inequalities, without special restriction on the flux nor on the equilibrium problem's data. As an application, we establish the hydrodynamic limit for a $2imes2$ relaxation system with general data. Also we construct a new family of convergent continuous BGK models with simple maxwellians different from the $chi$ models.
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Aziz Khan
2017-01-01
Full Text Available We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov’s fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.
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Patrick Winkert
2010-01-01
Full Text Available Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the p-Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fuc̆ik spectrum of the p-Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.
Alessandrini, Giovanni; de Hoop, Maarten V.; Gaburro, Romina
2017-12-01
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body Ω\\subset{R}n when the so-called Neumann-to-Dirichlet map is locally given on a non-empty curved portion Σ of the boundary \\partialΩ . We prove that anisotropic conductivities that are a priori known to be piecewise constant matrices on a given partition of Ω with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.
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Chengjun Yuan
2011-02-01
where $\\lambda$ is a parameter, $\\alpha\\in(n-1, n]$ is a real number and $n\\geq 3$, and $\\mathbf{D}_{0+}^\\alpha$ is the Riemann-Liouville's fractional derivative, and $f, g$ are continuous and semipositone. We give properties of Green's function of the boundary value problem, and derive an interval on $\\lambda$ such that for any $\\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions.
Periodic boundary-value problems and the Dancer-Fucik spectrum under conditions of resonance
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David A. Bliss
2011-08-01
Full Text Available We prove the existence of solutions to the nonlinear $2 pi$-periodic problem $$displaylines{ u''(x+mu u^+(x-u u^-(x+g(x,u(x=f(x,,quad xin (0,2pi,,cr u(0-u(2pi =0 ,, quad u'(0 - u'(2pi=0, }$$ where the point $(mu,u$ lies in the Dancer-Fucik spectrum, with $$ 0< frac{4}{9}mu leqslant u
Multiple solutions to a boundary value problem for an n-th order nonlinear difference equation
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Susan D. Lauer
1998-11-01
Full Text Available We seek multiple solutions to the n-th order nonlinear difference equation $$Delta^n x(t= (-1^{n-k} f(t,x(t,quad t in [0,T]$$ satisfying the boundary conditions $$x(0 = x(1 = cdots = x(k - 1 = x(T + k + 1 = cdots = x(T+ n = 0,.$$ Guo's fixed point theorem is applied multiple times to an operator defined on annular regions in a cone. In addition, the hypotheses invoked to obtain multiple solutions to this problem involves the condition (A $f:[0,T] imes {mathbb R}^+ o {mathbb R}^+$ is continuous in $x$, as well as one of the following: (B $f$ is sublinear at $0$ and superlinear at $infty$, or (C $f$ is superlinear at $0$ and sublinear at $infty$.
Extremal points for a higher-order fractional boundary-value problem
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Aijun Yang
2015-06-01
Full Text Available The Krein-Rutman theorem is applied to establish the extremal point, $b_0$, for a higher-order Riemann-Liouville fractional equation, $D_{0+}^{\\alpha}y+p(ty = 0$, $0
Boundary-value problems for second-order differential operators with nonlocal boundary conditions
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Mohamed Denche
2007-04-01
Full Text Available In this paper, we study a second-order differential operator combining weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions, called regular and non regular cases, we prove that the resolvent decreases with respect to the spectral parameter in $L^{p}(0,1$, but there is no maximal decrease at infinity for $p>1$. Furthermore, the studied operator generates in $L^{p}(0,1 $, an analytic semi group for $p=1$ in the regular case, and an analytic semi group with singularities for $p>1$, in both cases, and for $p=1$, in the non regular case only. The obtained results are then used to show the correct solvability of a mixed problem for parabolic partial differential equation with non regular boundary conditions.
The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates
DEFF Research Database (Denmark)
Grubb, Gerd
2011-01-01
For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ+ of the boundary and a Dirichlet condition on the other part Σ−. We show a Kreĭn resolvent formula, where the difference between its...... resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely sjj2/(n−1)→C0,+2/(n−1), where C0,+ is proportional...
Boundary value problem for one-dimensional fractional differential advection-dispersion equation
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Khasambiev Mokhammad Vakhaevich
2014-07-01
Full Text Available An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.
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I. J. Cabrera
2012-01-01
Full Text Available We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: D0+αu(t+f(t,u(t=0, 0
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Hytham. A. Alkresheh
2016-02-01
Full Text Available In this paper, an algorithm based on a new modification, developed by Duan and Rach, for the Adomian decomposition method (ADM is generalized to find positive solutions for boundary value problems involving nonlinear fractional ordinary differential equations. In the proposed algorithm the boundary conditions are used to convert the nonlinear fractional differential equations to an equivalent integral equation and then a recursion scheme is used to obtain the analytical solution components without the use of undetermined coefficients. Hence, there is no requirement to solve a nonlinear equation or a system of nonlinear equations of undetermined coefficients at each stage of approximation solution as per in the standard ADM. The fractional derivative is described in the Caputo sense. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm.
Duan, Jun-Sheng; Rach, Randolph; Wazwaz, Abdul-Majid
2014-11-01
In this paper, we present a reliable algorithm to calculate positive solutions of homogeneous nonlinear boundary value problems (BVPs). The algorithm converts the nonlinear BVP to an equivalent nonlinear Fredholm- Volterra integral equation.We employ the multistage Adomian decomposition method for BVPs on two or more subintervals of the domain of validity, and then solve the matching equation for the flux at the interior point, or interior points, to determine the solution. Several numerical examples are used to highlight the effectiveness of the proposed scheme to interpolate the interior values of the solution between boundary points. Furthermore we demonstrate two novel techniques to accelerate the rate of convergence of our decomposition series solutions by increasing the number of subintervals and adjusting the lengths of subintervals in the multistage Adomian decomposition method for BVPs.
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Liaqat Ali
2016-09-01
Full Text Available In this research work a new version of Optimal Homotopy Asymptotic Method is applied to solve nonlinear boundary value problems (BVPs in finite and infinite intervals. It comprises of initial guess, auxiliary functions (containing unknown convergence controlling parameters and a homotopy. The said method is applied to solve nonlinear Riccati equations and nonlinear BVP of order two for thin film flow of a third grade fluid on a moving belt. It is also used to solve nonlinear BVP of order three achieved by Mostafa et al. for Hydro-magnetic boundary layer and micro-polar fluid flow over a stretching surface embedded in a non-Darcian porous medium with radiation. The obtained results are compared with the existing results of Runge-Kutta (RK-4 and Optimal Homotopy Asymptotic Method (OHAM-1. The outcomes achieved by this method are in excellent concurrence with the exact solution and hence it is proved that this method is easy and effective.
Ion Cârstea, Cătălin; Wang, Jenn-Nan
2017-12-01
In the inverse boundary value problems of isotropic elasticity and complex conductivity, we derive estimates for the volume fraction of an inclusion whose physical parameters satisfy suitable gap conditions. For both the inclusion and the background medium we assume that the material coefficients are constant. In the elasticity case we require one measurement for the lower bound and another for the upper one. In the complex conductivity case we need three measurements for the lower bound and three for the upper. We accomplish this with the help of the ‘translation method’ which consists of perturbing the minimum principle associated with the equation by either a null-Lagrangian or a quasi-convex quadratic form.
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Bernard K. Bonzi
2012-01-01
Full Text Available In this article we study the nonlinear homogeneous Neumann boundary-value problem $$displaylines{ b(u-hbox{div} a(x,abla u=fquad hbox{in } Omegacr a(x,abla u.eta=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded open domain in $mathbb{R}^{N}$, $N geq 3$ and $eta$ the outer unit normal vector on $partialOmega$. We prove the existence and uniqueness of a weak solution for $f in L^{infty}(Omega$ and the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.
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Smail Djebali
2011-02-01
Full Text Available This work is devoted to the existence of nontrivial positive solutions for a class of second-order nonlinear multi-point boundary-value problems on the positive half-line. The novelty of this work is that the nonlinearity may exhibit a singularity at the origin simultaneously with respect to the solution and its derivative; moreover it satisfies quite general growth conditions far from the origin, including polynomial growth. New existence results of single, twin and triple solutions are proved using the fixed point index theory on appropriate cones in weighted Banach spaces together with two-functional and three-functional fixed point theorems. The singularity is treated by means of approximation and compactness arguments. The proofs of the existence results rely heavily on several sharp estimates and useful properties of the corresponding Green's function.
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Dankowicz, Harry; Schilder, Frank
This paper presents a collocation toolbox for multi-point, boundary-value problems. This toolbox has been recently developed by the authors to support general-purpose parameter continuation of sets of constrained orbit segments, such as i) segmented trajectories in hybrid dynamical systems......, for example, mechanical systems with impacts, friction, and switching control, ii) homoclinic orbits represented by an equilibrium point and a finite-time trajectory that starts and ends near this equilibrium point, and iii) collections of trajectories that represent quasi-periodic invariant tori...... the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations....
Dujardin, G. M.
2009-08-12
This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas\\' transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over. © 2009 The Royal Society.
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Kun Wang
2012-08-01
Full Text Available In this article we study the existence and multiplicity of positive solutions for the system of second-order boundary value problems involving first order derivatives $$displaylines{ -u''=f(t, u, u', v, v',cr -v''=g(t, u, u', v, v',cr u(0=u'(1=0,quad v(0=v'(1=0. }$$ Here $f,gin C([0,1]imes mathbb{R}_+^{4}, mathbb{R}_+(mathbb{R}_+:=[0,infty$. We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing Jensen's integral inequality for concave functions and $mathbb{R}_+^2$-monotone matrices.
Holota, Petr; Nesvadba, Otakar
2017-04-01
The aim of this paper is to discuss the solution of the linearized gravimetric boundary value problem by means of the method of successive approximations. We start with the relation between the geometry of the solution domain and the structure of Laplace's operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. Laplace's operator has a relatively simple structure in terms of ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from an oblate ellipsoid of revolution, even if it is optimally fitted. Therefore, an alternative is discussed. A system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces is used. Clearly, the structure of Laplace's operator is more complicated in this case. It was deduced by means of tensor calculus and in a sense it represents the topography of the physical surface of the Earth. Nevertheless, the construction of the respective Green's function is more simple, if the solution domain is transformed. This enables the use of the classical Green's function method together with the method of successive approximations for the solution of the linear gravimetric boundary value problem expressed in terms of new coordinates. The structure of iteration steps is analyzed and where useful also modified by means of the integration by parts. Comparison with other methods is discussed.
Elliptic boundary value problems
Maz'ya, V G; Plamenevskii, B A; Stupyali, L; Plamenevskii, B A
1984-01-01
The papers in this volume have been selected, translated, and edited from publications not otherwise translated into English under the auspices of the AMS-ASL-IMS Committee on Translations from Russian and Other Foreign Languages.
Porz, Lucas; Grombein, Thomas; Seitz, Kurt; Heck, Bernhard; Wenzel, Friedemann
2017-04-01
Regional height reference systems are generally related to individual vertical datums defined by specific tide gauges. The discrepancies of these vertical datums with respect to a unified global datum cause height system biases that range in an order of 1-2 m at a global scale. One approach for unification of height systems relates to the solution of a Geodetic Boundary Value Problem (GBVP). In particular, the fixed GBVP, using gravity disturbances as boundary values, is solved at GNSS/leveling benchmarks, whereupon height datum offsets can be estimated by least squares adjustment. In spherical approximation, the solution of the fixed GBVP is obtained by Hotine's spherical integral formula. However, this method relies on the global availability of gravity data. In practice, gravity data of the necessary resolution and accuracy is not accessible globally. Thus, the integration is restricted to an area within the vicinity of the computation points. The resulting truncation error can reach several meters in height, making height system unification without further consideration of this effect unfeasible. This study analyzes methods for reducing the truncation error by combining terrestrial gravity data with satellite-based global geopotential models and by modifying the integral kernel in order to accelerate the convergence of the resulting potential. For this purpose, EGM2008-derived gravity functionals are used as pseudo-observations to be integrated numerically. Geopotential models of different spectral degrees are implemented using a remove-restore-scheme. Three types of modification are applied to the Hotine-kernel and the convergence of the resulting potential is analyzed. In a further step, the impact of these operations on the estimation of height datum offsets is investigated within a closed loop simulation. A minimum integration radius in combination with a specific modification of the Hotine-kernel is suggested in order to achieve sub-cm accuracy for the
Ibdah, H.; Khuri, S. A.; Sayfy, A.
2014-09-01
The ultimate purpose of this article is to introduce and describe a combined approach, based on asymptotic boundary conditions (ABCs) and a fourth order cubic B-spline collocation, for the numerical solution of a general class of two-point linear boundary-value problems (BVPs) over a semi-infinite interval that arises in various engineering applications. The scheme will be extended and then implemented to handle a system of BVPs. The idea of the proposed strategy is to first reduce the condition at infinity to an asymptotic boundary condition that approaches the specified value at infinity over a large finite interval. Then, the problem complimented with the resulting ABC is solved using a fourth-order spline collocation approach constructed over uniform meshes. The scheme is numerically verified to have a fourth order rate of convergence. The work is illustrated by considering a number of test examples that confirm the accuracy, efficient treatment of the boundary condition at infinity, and applicability of the approach. The computational results show that the scheme is reliable, converges fast, and compares very well with the existing analytic solutions.
On a nonlocal boundary value problem for the two-term time-fractional diffusion-wave equation
Bazhlekova, E.
2013-10-01
We study a nonlocal boundary value problem for the spatially one-dimensional diffusion-wave equation with two fractional Caputo time-derivatives of different orders α and β, where 1 telegraph equation. In the limiting case of one time-derivative of order 2 (wave equation) the oscillation amplitude of the system increases infinitely with time in the absence of forcing terms in the formulation of the problem. We first give an explanation for this resonant behaviour. Then we study the damping effect due to decreasing of the order of differentiation from 2 to αɛ (1,2) and/or adding a second fractional term. Our considerations are based on the generalized eigenfunction expansion of the solution. The time-dependent components in this expansion are studied in detail. Their properties are derived from the representation as a Laplace inverse integral. It appears that the time-dependent components exhibit oscillations with decreasing amplitude for sufficiently large t, and algebraic decay for t → ∞, except in the case α = 2,β = 1 (telegraph equation), when this decay is exponential. To illustrate the analytical formulas, results of numerical calculations and plots are presented.
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K. R. Prasad
2014-01-01
Full Text Available This paper establishes the existence of at least three positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems, D0+β1(ϕp(D0+α1u(t=f1(t,u(t,v(t, t∈(0,1, D0+β2(ϕp(D0+α2v(t=f2(t,u(t,v(t, t∈(0,1, u(0=D0+q1u(0=0, γu(1+δD0+q2u(1=0, D0+α1u(0=D0+α1u(1=0, v(0=D0+q1v(0=0, γv(1+δD0+q2v(1=0, D0+α2v(0=D0+α2v(1=0, by applying five functionals fixed point theorem.
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Zhenlai Han
2012-11-01
Full Text Available In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t+a(tf(u=0, quad 0
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Nguyen Manh Hung
2008-03-01
Full Text Available In this paper, we consider the second initial boundary value problem for strongly general Schrodinger systems in both the finite and the infinite cylinders $Q_T, 0
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Jing Niu
2013-01-01
reproducing kernel on infinite interval is obtained concisely in polynomial form for the first time. Furthermore, as a particular effective application of this method, we give an explicit representation formula for calculation of reproducing kernel in reproducing kernel space with boundary value conditions.
Trifonov, E. V.
2017-07-01
We propose a procedure for multiplying solutions of linear and nonlinear one-dimensional wave equations, where the speed of sound can be an arbitrary function of one variable. We obtain exact solutions. We show that the functional series comprising these solutions can be used to solve initial boundary value problems. For this, we introduce a special scalar product.
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Yanmei Sun
2012-01-01
Full Text Available By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.
Frederickson, P. O.; Wessel, W. R.
1979-01-01
Certain physical processes are modeled by partial differential equations which are parabolic over part of the domain and elliptic over the remainder. A family of semi-implicit algorithms which are well suited to initial-boundary value problems of this mixed type is discussed. One important feature of these algorithms is the use of an approximate inverse for the solution of the implicit linear system. A strong error analysis results in an estimate of the total error as a function of approximate inverse error e and time step h.
Tian, Shou-Fu
2017-09-01
In this paper, we implement the Fokas method in order to study initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the half-line, with Lax pairs involving 3× 3 matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution \\{ p(x, t), q(x, t)\\} can be written in terms of the solution of a 3× 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k) , which are respectively determined by the initial values and boundary values at x=0 . Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.
Yan, Zhenya
2017-05-01
We extend the idea of the Fokas unified transform to investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair on the half-line. The solution of this system can be expressed in terms of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x = 0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation. These results may be of the potential significance in both spinor Bose-Einstein condensates and the theory of multi-component integrable systems.
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Gang Wu
2013-01-01
Full Text Available We study a system of second-order dynamic equations on time scales (p1u1∇Δ(t-q1(tu1(t+λf1(t,u1(t,u2(t=0,t∈(t1,tn,(p2u2∇Δ(t-q2(tu2(t+λf2(t,u1(t, u2(t=0, satisfying four kinds of different multipoint boundary value conditions, fi is continuous and semipositone. We derive an interval of λ such that any λ lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.
A Third-Order p-Laplacian Boundary Value Problem Solved by an SL(3,ℝ Lie-Group Shooting Method
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Chein-Shan Liu
2013-01-01
Full Text Available The boundary layer problem for power-law fluid can be recast to a third-order p-Laplacian boundary value problem (BVP. In this paper, we transform the third-order p-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of r∈[0,1]. The present SL(3,ℝ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order p-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4 method to obtain a quite accurate numerical solution of the p-Laplacian.
Taha, Mohamed
2014-06-01
In the present work, the recursive differentiation method (RDM) is introduced and implemented to obtain analytical solutions for differential equations governing different types of boundary value prob- lems (BVP). Then, the method is applied to investigate the static behaviour of a beam-column resting on a two parameter foundation subjected to different types of lateral loading. The analytical solutions obtained using RDM and Adomian decomposition method (ADM) are found similar but the RDM requires less mathematical effort. It is indicated that the RDM is reliable, straightforward and efficient for investigation of BVP in finite domains. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. However, it is indicated that, in the case of beam-columns resting on foundations, the beam-column may be buckled in a higher mode rather than a lower one, then the critical load in that case is that accompanies the higher mode. This result is very important to avoid static instability as it is widely common that the buckling load of the first buckling mode is always the smaller one, which is true only in the case of the beam-columns without foundations.
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Jihui Zhang
2008-06-01
Full Text Available This paper deals with the existence and iteration of positive solutions for the following one-dimensional p-Laplacian boundary value problems: (ÃÂ•p(uÃ¢Â€Â²(tÃ¢Â€Â²+a(tf(t,u(t,uÃ¢Â€Â²(t=0, tÃ¢ÂˆÂˆ(0,1, subject to some boundary conditions. By making use of monotone iterative technique, not only we obtain the existence of positive solutions for the problems, but also we establish iterative schemes for approximating the solutions.
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Xuemei Zhang
2014-01-01
Full Text Available This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument -x′′t-atx′t+btxt=ωtft, xαt, t∈0, 1; x′0=0, x1-∫01htxtdt=0, where a∈0, 1, 0, +∞, b∈C0, 1, 0, +∞ and, ω may be singular at t=0 or/and at t=1. Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.
A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
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Denche M
2006-01-01
Full Text Available We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.
A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
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S. Djezzar
2006-02-01
Full Text Available We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.
Zhu, Qiao-Zhen; Fan, En-Gui; Xu, Jian
2017-10-01
The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov–Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation. Supported by grants from the National Science Foundation of China under Grant No. 11671095, National Science Foundation of China under Grant No. 11501365, Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No 15YF1408100, and the Hujiang Foundation of China (B14005)
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Chi-Chang Wang
2013-09-01
Full Text Available This paper seeks to use the proposed residual correction method in coordination with the monotone iterative technique to obtain upper and lower approximate solutions of singularly perturbed non-linear boundary value problems. First, the monotonicity of a non-linear differential equation is reinforced using the monotone iterative technique, then the cubic-spline method is applied to discretize and convert the differential equation into the mathematical programming problems of an inequation, and finally based on the residual correction concept, complex constraint solution problems are transformed into simpler questions of equational iteration. As verified by the four examples given in this paper, the method proposed hereof can be utilized to fast obtain the upper and lower solutions of questions of this kind, and to easily identify the error range between mean approximate solutions and exact solutions.
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Khaleghi Moghadam Mohsen
2017-08-01
Full Text Available Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.
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Yuji Liu
2016-11-01
Full Text Available Firstly we prove existence and uniqueness of solutions of Cauchy problems of linear fractional differential equations (LFDEs with two variable coefficients involving Caputo fractional derivative, Riemann-Liouville derivative, Caputo type Hadamard derivative and Riemann-Liouville type Hadamard fractional derivatives with order q in [n-1,n by using the iterative method.
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Alessia E. Kogoj
2016-12-01
Full Text Available For every bounded open set Ω in RN+1, we study the first boundary problem for a wide class of hypoelliptic evolution operators. The operators are assumed to be endowed with a well behaved global fundamental solution that allows us to construct a generalized solution in the sense of Perron-Wiener of the Dirichlet problem. Then, we give a criterion of regularity for boundary points in terms of the behavior, close to the point, of the fundamental solution of the involved operator. We deduce exterior conetype criteria for operators of Kolmogorov-Fokker-Planck-type, for the heat operators and more general evolution invariant operators on Lie groups. Our criteria extend and generalize the classical parabolic-cone condition for the classical heat operator due to Effros and Kazdan. The results presented are contained in [K16].
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Erdoğan Şen
2013-01-01
Full Text Available We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t+a(tf(u=0, 01, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ0 are parameters, a:(0,1→[0,+∞, and f:[0,+∞→[0,+∞ are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.
Liao, Kai-Pin; Matalon, Moshe; Pantano, Carlos
2011-11-01
We present a new numerical method to determine the edge flame velocity in a counterflow as an eigenvalue of the two-dimensional boundary-value problem for the variable density equations in the zero Mach number limit. The method utilizes a collocated arrangement of all variables in space and relies on discrete mass conservation using centered second-order accurate finite-differences. The finite element method approach, weak form, is adopted to determine the discretization near boundary and ensure well-posedness of the equations. Pressure and velocities are coupled and solved iteratively, while energy and species equations are segregated and solved sequentially. The method is coupled with pseudo-arc length continuation to explore the full parametric dependence of the solution. The edge-flame velocity and structure under the combined effect of strain and heat release will be presented.
Feehan, Paul M. N.
2017-09-01
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton [9] in their study of the porous medium equation or the degeneracy of the Heston operator [21] in mathematical finance. Existence of a solution to the partial Dirichlet problem on a half-ball, where the operator becomes degenerate on the flat boundary and a Dirichlet condition is only imposed on the spherical boundary, provides the key additional ingredient required for our Perron method. Surprisingly, proving existence of a solution to this partial Dirichlet problem with ;mixed; boundary conditions on a half-ball is more challenging than one might expect. Due to the difficulty in developing a global Schauder estimate and due to compatibility conditions arising where the ;degenerate; and ;non-degenerate boundaries; touch, one cannot directly apply the continuity or approximate solution methods. However, in dimension two, there is a holomorphic map from the half-disk onto the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half-ball onto the infinite ;slab;. The solution to the partial Dirichlet problem on the half-ball can thus be converted to a partial Dirichlet problem on the slab, albeit for an operator which now has exponentially growing coefficients. The required Schauder regularity theory and existence of a solution to the partial Dirichlet problem on the slab can nevertheless be obtained using previous work of the author and C. Pop [16]. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of
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Ruslan V. Zhalnin
2017-12-01
Full Text Available Introduction: In this paper, we present a priori error analysis of the solution of a homogeneous boundary value problem for a second-order differential equation by the discontinuous Galerkin method on staggered grids. Materials and Methods: This study is based on the unified hp-version error analysis of local discontinuous Galerkin method proposed by Castillo et al. [Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, 2002]. The purpose of this paper is to present a new approach to the error analysis of the solution of parabolic equations by the discontinuous Galerkin method on staggered grids. Results: We suggest that approximation errors depend on the characteristic size of the cells and the degree of polynomials used in the basis functions. The necessary lemmas are formulated for the problem solution. The complete proof of the lemmas formulated is carried out. We formulated and proved a theorem, in which a priori error estimates are given for solving parabolic equations using the discontinuous Galerkin method on staggered grids Discussion and Conclusions: The obtained results are consistent with similar studies of other authors and complement them. Further work on this topic involves the study of diffusion-type equations of order higher than the first and the production of a posteriori error estimates.
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Muhammad Iqbal
2017-01-01
Full Text Available We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: D cαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],D cαyt-ft,yt=gt,xt,Iαxt, t∈J=0,1, D cpx0=ψxη1, x′0=0,…,xn-20=0, D cpx1=ψxη2, D cpy0=ψyη1, y′0=0,…,yn-20=0, D cpy1=ψyη2, where p,η1,η2∈0,1, ψ is linear, D cα is Caputo fractional derivative of order α, with n-1<α≤n, n∈N, and Iα is fractional integral of order α. The nonlinear functions f, g are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.
1983-07-01
type of (1.11) was successfully studied by Murat-Simon (1977), Chesnais (1975), Pironneau (.1976) and Dervieux (1981). The characteristic of our method...1959). Chesnais , D., On the existence of a solution in a domain identification problem, J. of Math. Anal. and Appl., Vol. 52, No. 2 (1975). Ciarlet, Ph
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O. P. Kupenko
2016-05-01
Full Text Available We study a Dirichlet optimal control problem for a nonlinear elliptic anisotropic p-Laplace equation with control and state constraints. The matrix-valued coecients we take as controls and in the linear part of dierential operator we consider coecients to be unbounded skew-symmetric matrix. We show that, in spite of unboundedness of the non-linear dierential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable.
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Rutkauskas Stasys
2011-01-01
Full Text Available Abstract A system of elliptic equations which are irregularly degenerate at an inner point is considered in this article. The equations are weakly coupled by a matrix that has multiple zero eigenvalue and corresponding to it adjoint vectors. Two statements of a well-posed Dirichlet type problem in the class of smooth functions are given and sufficient conditions on the existence and uniqueness of the solutions are obtained.
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Ghasem Alizadeh Afrouzi
2006-10-01
Full Text Available In this paper, we establish an equivalent statement to minimax inequality for a special class of functionals. As an application, we prove the existence of three solutions to the Dirichlet problem $$displaylines{ -u''(x+m(xu(x =lambda f(x,u(x,quad xin (a,b,cr u(a=u(b=0, }$$ where $lambda>0$, $f:[a,b]imes mathbb{R}o mathbb{R}$ is a continuous function which changes sign on $[a,b]imes mathbb{R}$ and $m(xin C([a,b]$ is a positive function.
Recursive recovery of Markov transition probabilities from boundary value data
Energy Technology Data Exchange (ETDEWEB)
Patch, Sarah Kathyrn [Univ. of California, Berkeley, CA (United States)
1994-04-01
In an effort to mathematically describe the anisotropic diffusion of infrared radiation in biological tissue Gruenbaum posed an anisotropic diffusion boundary value problem in 1989. In order to accommodate anisotropy, he discretized the temporal as well as the spatial domain. The probabilistic interpretation of the diffusion equation is retained; radiation is assumed to travel according to a random walk (of sorts). In this random walk the probabilities with which photons change direction depend upon their previous as well as present location. The forward problem gives boundary value data as a function of the Markov transition probabilities. The inverse problem requires finding the transition probabilities from boundary value data. Problems in the plane are studied carefully in this thesis. Consistency conditions amongst the data are derived. These conditions have two effects: they prohibit inversion of the forward map but permit smoothing of noisy data. Next, a recursive algorithm which yields a family of solutions to the inverse problem is detailed. This algorithm takes advantage of all independent data and generates a system of highly nonlinear algebraic equations. Pluecker-Grassmann relations are instrumental in simplifying the equations. The algorithm is used to solve the 4 x 4 problem. Finally, the smallest nontrivial problem in three dimensions, the 2 x 2 x 2 problem, is solved.
Xu Fuyi; Meng Zhaowei
2009-01-01
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
Generalized Green's functions for higher order boundary value matrix differential systems
Directory of Open Access Journals (Sweden)
R. J. Villanueva
1992-01-01
Full Text Available In this paper, a Green's matrix function for higher order two point boundary value differential matrix problems is constructed. By using the concept of rectangular co-solution of certain algebraic matrix equation associated to the problem, an existence condition as well as an explicit closed form expression for the solution of possibly not well-posed boundary value problems is given avoiding the increase of the problem dimension.
Global Behavior of the Components for the Second Order
An Yulian; Ma Ruyun
2008-01-01
Abstract We consider the nonlinear eigenvalue problems , , , , where , , and for , with ; ; . There exist two constants such that and , . Using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of the above problems.
Energy Technology Data Exchange (ETDEWEB)
Pereira, Luis Carlos Martins
1998-06-15
New Petrov-Galerkin formulations on the finite element methods for convection-diffusion problems with boundary layers are presented. Such formulations are based on a consistent new theory on discontinuous finite element methods. Existence and uniqueness of solutions for these problems in the new finite element spaces are demonstrated. Some numerical experiments shows how the new formulation operate and also their efficacy. (author)
Analysis of Blasius Equation for Flat-Plate Flow with Infinite Boundary Value
DEFF Research Database (Denmark)
Miansari, M. O.; Miansari, M. E.; Barari, Amin
2010-01-01
and write the nonlinear differential equation in the state space format, and then solve the initial value problem instead of boundary value problem. The significance of linear part is a key factor in convergence. A first seen linear part may lead to an unstable solution, therefore an extra term is added......This paper applies the homotopy perturbation method (HPM) to determine the well-known Blasius equation with infinite boundary value for Flat-plate Flow. We study here the possibility of reducing the momentum and continuity equations to ordinary differential equations by a similarity transformation...
Boundary value problemfor multidimensional fractional advection-dispersion equation
Directory of Open Access Journals (Sweden)
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
Multiplicity of solutions for elliptic boundary value problems
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Yiwei Ye
2014-06-01
Full Text Available In this article, we study the existence of infinitely many solutions for the semilinear elliptic equation $-\\Delta u+a(xu=f(x,u$ in a bounded domain of $\\mathbb{R}^N$ $(N\\geq 3$ with the Dirichlet boundary conditions, where the primitive of the nonlinearity $f$ is either superquadratic at infinity or subquadratic at zero.
Periodic and boundary value problems for second order differential ...
Indian Academy of Sciences (India)
Liouville and periodic boundary conditions. The vector field (, , ) is Caratheodory and in some instances the continuity condition on or is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well ...
RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems
Farrell, Patricio
2013-01-01
In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multilevel fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations. © 2013 Society for Industrial and Applied Mathematics.
Periodic and boundary value problems for second order differential ...
Indian Academy of Sciences (India)
Abstract. In this paper we study second order scalar differential equations with. Sturm–Liouville and periodic boundary conditions. The vector field fًt; x; yق is. Caratheodory and in some instances the continuity condition on x or y is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as ...
On numerical-analytic techniques for boundary value problems
Czech Academy of Sciences Publication Activity Database
Rontó, András; Rontó, M.; Shchobak, N.
2012-01-01
Roč. 12, č. 3 (2012), s. 5-10 ISSN 1335-8243 Institutional support: RVO:67985840 Keywords : numerical-analytic method * periodic successive approximations * Lyapunov-Schmidt method Subject RIV: BA - General Mathematics http://www.degruyter.com/view/j/aeei.2012.12.issue-3/v10198-012-0035-1/v10198-012-0035-1. xml ?format=INT
Existence results for non-autonomous elliptic boundary value problems
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V. Anuradha
1994-07-01
Full Text Available $$-Delta u(x = lambda f(x, u;quad x in Omega$$ $$u(x + alpha(x frac{partial u(x}{partial n} = 0;quad x in partial Omega$$ where $lambda > 0$, $Omega$ is a bounded region in $Bbb{R}^N$; $N geq 1$ with smooth boundary $partial Omega$, $alpha(x geq 0$, $n$ is the outward unit normal, and $f$ is a smooth function such that it has either sublinear or restricted linear growth in $u$ at infinity, uniformly in $x$. We also consider $f$ such that $f(x, u u leq 0$ uniformly in $x$, when $|u|$ is large. Without requiring any sign condition on $f(x, 0$, thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given $lambda in (lambda_{n}, lambda_{n + 1}$ where $lambda_{k}$ is the $k$-th eigenvalue of $-Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $lambda$ near $lambda_{1}$, and for $lambda$ large when $f$ is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $lambda$ small.
Numerical solution of fuzzy boundary value problems using Galerkin ...
Indian Academy of Sciences (India)
, China; Department of Mathematics, Kalinga Institute of Industrial Technology, Bhubaneswar, Odisha 751 024, India; Department of Mathematics, National Institute of Technology, Rourkela, Odisha 769 008, India; Department of Mathematics, ...
Nonlinear second-order multivalued boundary value problems
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
Abstract. In this paper we study nonlinear second-order differential inclusions involv- ing the ordinary vector p-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities ...
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.
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Yuriy Povstenko
2016-01-01
Full Text Available The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.
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Valery Romanovski
2008-12-01
Full Text Available We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
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Romanovski Valery
2008-01-01
Full Text Available We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.
Existence of Triple Positive Solutions for Second-Order Discrete Boundary Value Problems
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Yanping Guo
2007-01-01
Full Text Available By using a new fixed-point theorem introduced by Avery and Peterson (2001, we obtain sufficient conditions for the existence of at least three positive solutions for the equation Δ2x(k−1+q(kf(k,x(k,Δx(k=0, for k∈{1,2,…,n−1}, subject to the following two boundary conditions: x(0=x(n=0 or x(0=Δx(n−1=0, where n≥3.
A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems
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Mohammed AL-Smadi
2014-01-01
Full Text Available The objective of this paper is to present a numerical iterative method for solving systems of first-order ordinary differential equations subject to periodic boundary conditions. This iterative technique is based on the use of the reproducing kernel Hilbert space method in which every function satisfies the periodic boundary conditions. The present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case. Furthermore, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration. Indeed, three numerical examples are provided to illustrate the effectiveness of the present method. Results obtained show that the numerical scheme is very effective and convenient for solving systems of first-order ordinary differential equations with periodic boundary conditions.
Brito, Irene; Mena, Filipe C
2017-08-01
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
2009-02-01
Full Text Available We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.
Existence of positive solutions for a system of semipositone fractional boundary value problems
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Johnny Henderson
2016-05-01
Full Text Available We investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions.
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Johnny Henderson
2016-01-01
Full Text Available We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.
A Neumann problem for a system depending on the unknown boundary values of the solution
Directory of Open Access Journals (Sweden)
Pablo Amster
2013-01-01
Full Text Available A semilinear system of second order ODEs under Neumann conditions is studied. The system has the particularity that its nonlinear term depends on the (unknown Dirichlet values $y(0$ and $y(1$ of the solution. Asymptotic and non-asymptotic sufficient conditions of Landesman-Lazer type for existence of solutions are given. We generalize our previous results for a scalar equation, and a well known result by Nirenberg for a nonlinearity independent of $y(0$ and $y(1$.
Existence of Positive Solutions of Nonlinear Second-Order Periodic Boundary Value Problems
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Ma Ruyun
2010-01-01
Full Text Available Abstract This paper is devoted to study the existence of periodic solutions of the second-order equation , where is a Carathéodory function, by combining a new expression of Green's function together with Dancer's global bifurcation theorem. Our main results are sharp and improve the main results by Torres (2003.
Existence of positive solutions for multi-point boundary value problems
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B. Karna
2007-11-01
where $a_j,b_j\\in[0,\\infty, \\ j=1, 2, \\ldots, m,$ with $0<\\sum_{j=1}^{m}a_j<1, \\ 0<\\sum_{j=1}^{m}b_j<1,$ and $ \\eta_j \\in(0,1$ with $0<\\eta_1<\\eta_2<\\ldots <\\eta_m<1,$ under certain conditions on $f$ and $p$ using the Krasnosel'skii fixed point theorem for certain values of $\\lambda$. We use the positivity of the Green's function and cone theory to prove our results.
Even-order self-adjoint boundary value problems for proportional derivatives
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Douglas R. Anderson
2017-09-01
Full Text Available In this study, even order self-adjoint differential equations incorporating recently introduced proportional derivatives, and their associated self-adjoint boundary conditions, are discussed. Using quasi derivatives, a Lagrange bracket and bilinear functional are used to obtain a Lagrange identity and Green's formula; this also leads to the classification of self-adjoint boundary conditions. Next we connect the self-adjoint differential equations with the theory of Hamiltonian systems and (n,n-disconjugacy. Specific formulas of Green's functions for two and four iterated proportional derivatives are also derived.
Existence of One-Signed Solutions of Discrete Second-Order Periodic Boundary Value Problems
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Ruyun Ma
2012-01-01
Full Text Available We prove the existence of one-signed periodic solutions of second-order nonlinear difference equation on a finite discrete segment with periodic boundary conditions by combining some properties of Green's function with the fixed-point theorem in cones.
Domoshnitsky, Alexander
2014-01-01
The impulsive delay differential equation is considered (Lx)(t) = x′(t) + ∑i=1 m p i(t)x(t − τ i(t)) = f(t), t ∈ [a, b], x(t j) = β j x(t j − 0), j = 1,…, k, a = t 0 Green's functions for this equation are obtained. PMID:24719584
Positive non-symmetric solutions of a non-linear boundary value problem
Directory of Open Access Journals (Sweden)
Samuel Peres
2013-11-01
Full Text Available This paper deals with a non-linear second order ordinary differential equation with symmetric non-linear boundary conditions, where both of the non-linearities are of power type. It provides results concerning the existence and multiplicity of positive non-symmetric solutions for values of parameters not considered before. The main tool is the shooting method.
On solutions of some fractional $m$-point boundary value problems at resonance
Directory of Open Access Journals (Sweden)
Zhanbing Bai
2010-06-01
is considered, where $1< \\alpha \\leq 2,$ is a real number, $D_{0+}^\\alpha$ and $I_{0+}^{\\alpha}$ are the standard Riemann-Liouville differentiation and integration, and $f:[0,1]\\times R^2 \\to R$ is continuous and $e \\in L^1[0,1]$, and $\\eta_i \\in (0, 1, \\beta_i \\in R, i=1,2, \\cdots, m-2$, are given constants such that $\\sum_{i=1}^{m-2}\\beta_i=1$. By using the coincidence degree theory, some existence results of solutions are established.
Brito, Irene; Mena, Filipe C.
2017-08-01
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
Domoshnitsky, Alexander; Volinsky, Irina
2014-01-01
The impulsive delay differential equation is considered (Lx)(t) = x'(t) + ∑(i=1)(m) p(i)(t)x(t - τ(i) (t)) = f(t), t ∈ [a, b], x(t j) = β(j)x(t(j - 0)), j = 1,…, k, a = t0 Green's functions for this equation are obtained.
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Alexander Domoshnitsky
2014-01-01
Full Text Available The impulsive delay differential equation is considered (Lx(t=x′(t+∑i=1mpi(tx(t-τi(t=f(t, t∈[a,b], x(tj=βjx(tj-0, j=1,…,k, a=t0
Introduction to partial differential equations from Fourier series to boundary-value problems
Broman, Arne
2010-01-01
This well-written, advanced-level text introduces students to Fourier analysis and some of its applications. The self-contained treatment covers Fourier series, orthogonal systems, Fourier and Laplace transforms, Bessel functions, and partial differential equations of the first and second orders. Over 260 exercises with solutions reinforce students' grasp of the material. 1970 edition.
Positive solutions for systems of nth order three-point nonlocal boundary value problems
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Johnny Henderson
2007-09-01
Full Text Available Intervals of the parameter $\\lambda$ are determined for which there exist positive solutions for the system of nonlinear differential equations, $u^{(n} + \\lambda a(t f(v = 0, \\ v^{(n} +\\lambda b(t g(u = 0, $ for $0 < t <1$, and satisfying three-point nonlocal boundary conditions, $u(0 = 0, u'(0 = 0, \\ldots, u^{(n-2}(0 = 0, \\ u(1=\\alpha u(\\eta, v(0 = 0, v'(0 = 0, \\ldots, v^{(n-2}(0 = 0, \\ v(1=\\alpha v(\\eta$. A Guo-Krasnosel'skii fixed point theorem is applied.
On a system of higher-order multi-point boundary value problems
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Johnny Henderson
2012-07-01
Full Text Available We investigate the existence and nonexistence of positive solutions for a system of nonlinear higher-order ordinary differential equations subject to some multi-point boundary conditions.
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Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
Solvability of boundary value problem at resonance for third-order ...
Indian Academy of Sciences (India)
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei Province 05003, People's Republic of China; School of Mathematical Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, People's Republic of China; Department of Mathematics, Beijing Institute of Technology, ...
High Order Difference Methods for Quasilinear Elliptic Boundary Value Problems on General Regions.
1980-02-01
8217 Mx i jz2 2q Ifn q .2 2jZ ~x) (5.2) h2 2 z-x)) f(xz(x),V(x) L + O(h2 (q+l)+Q) for x E h . In irregular mesh points an ,summing the derivatives of z...conditions 1, Comm. Pure Appi. Math. 12, 623-727 (1959). [2] Ames, W. F.-. Nonlinear partial differential eouiations in engineering , Academic Press, New York
Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
DEFF Research Database (Denmark)
Abels, Helmut; Grubb, Gerd; Wood, Ian Geoffrey
2014-01-01
. In the present work we show that pseudodifferential methods can be used to obtain a full characterization, including Kreĭn resolvent formulas, of the realizations of nonselfadjoint second-order operators on C32+ε domains; more precisely, we treat domains with Bp,232-smoothness and operators with Hq1...
Application of Splines to the Numerical Solution of Two-Point Boundary- Value Problems
1978-12-01
ARE THE NEWLY COMPUTED S O L U T I O N . NGTE THEY OCCUPY THE SAME MEMORY AS D (SEE CALL PROCEO IN THE MAIN PGM) CALL PAGES D3 5 0 J C A S E = t...ther assumpt ions are made to fully specify the six unknowns , then the quint ic would be de termined . The fol lowing procedure is p roposed
Problems in Hydrodynamics and Partial Differential Equations.
CALCULUS OF VARIATIONS, RESEARCH MANAGEMENT ), NONLINEAR DIFFERENTIAL EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS, EQUATIONS, INEQUALITIES, MEASURE THEORY , INTEGRALS, ABSTRACTS
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TIAN Jialei
2015-11-01
Full Text Available By using the ground as the boundary, Molodensky problem usually gets the solution in form of series. Higher order terms reflect the correction between a smooth surface and the ground boundary. Application difficulties arise from not only computational complexity and stability maintenance, but also data-intensiveness. Therefore, in this paper, starting from the application of external gravity disturbance, Green formula is used on digital terrain surface. In the case of ignoring the influence of horizontal component of the integral, the expression formula of external disturbance potential determined by boundary value consisted of ground gravity anomalies and height anomaly difference are obtained, whose kernel function is reciprocal of distance and Poisson core respectively. With this method, there is no need of continuation of ground data. And kernel function is concise, and suitable for the stochastic computation of external disturbing gravity field.
On Perturbation Solutions for Axisymmetric Bending Boundary Values of a Deep Thin Spherical Shell
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Rong Xiao
2014-01-01
Full Text Available On the basis of the general theory of elastic thin shells and the Kirchhoff-Love hypothesis, a fundamental equation for a thin shell under the moment theory is established. In this study, the author derives Reissner’s equation with a transverse shear force Q1 and the displacement component w. These basic unknown quantities are derived considering the axisymmetry of the deep, thin spherical shell and manage to constitute a boundary value question of axisymmetric bending of the deep thin spherical shell under boundary conditions. The asymptotic solution is obtained by the composite expansion method. At the end of this paper, to prove the correctness and accuracy of the derivation, an example is given to compare the numerical solution by ANSYS and the perturbation solution. Meanwhile, the effects of material and geometric parameters on the nonlinear response of axisymmetric deep thin spherical shell under uniform external pressure are also analyzed in this paper.
Analysis of vibroacoustic properties of dimpled beams using a boundary value model
Myers, Kyle R.
Attention has been given recently to the use of dimples as a means of passively altering the vibroacoustic properties of structures. Because of their geometric complexity, previous studies have modeled dimpled structures using the finite element method. However, the dynamics of dimpled structures are not completely understood. The goal of this study is to provide a better understanding of these structures through the development of a boundary value model (BVM) using Hamilton's Variational Principle. The focus of this study is on dimpled beams, which represent the simplest form of a dimpled structure. A general model of a beam with N dimples in free vibration is developed. Since dimples formed via a stamping process do not change the mass of the beam, the dimple thickness is less than that of the straight segments. Differential equations of motion that describe the normal and axial motion of the dimpled beams are derived. Their numerical solution yields the natural frequencies and analytical mode shapes of a dimpled beam. The accuracy of this model is checked against those obtained using the finite element method, as well as the analytical studies on the vibrations of arches, and shown to be accurate. The effect of dimple placement, dimple angle, its chord length, its thickness, as well as beam boundary conditions on beam natural frequencies and mode shapes are investigated. For beams with axially restrictive boundary conditions, the results.
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Liu Yang
2014-01-01
Full Text Available We consider the existence of infinitely many classical solutions to a class of impulsive differential equations with Dirichlet boundary value condition. Our main tools are based on variant fountain theorems and variational method. We study the case in which the nonlinearity is sublinear. Some recent results are extended and improved.
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Gryshchuk Serhii V.
2017-04-01
Full Text Available We consider a commutative algebra over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e222=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2}^{2}=0, e_{1}^{2}+e_{2}^{2}\
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Dhany Rajendran
2016-03-01
$$\\displaylines{ -z'' = h(t \\frac{f(z}{z^\\beta} \\quad \\text{in } (0,1 ,\\cr z(t> 0 \\quad \\text{in } (0,1,\\cr z(0= z(1=0 , }$$ where $ 0 0$ and $0<\\alpha<1-\\beta$. When there exist two pairs of sub-supersolutions $(\\psi_1,\\phi_1$ and $(\\psi_2,\\phi_2$ where $\\psi_1\\leq \\psi_2\\leq \\phi_1, \\psi_1\\leq \\phi_2\\leq \\phi_1 $ with $\\psi_2 \
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Abdelhalim Ebaid
2014-01-01
Full Text Available The exact solution for any physical model is of great importance in the applied science. Such exact solution leads to the correct physical interpretation and it is also useful in validating the approximate analytical or numerical methods. The exact solution for the peristaltic transport of a Jeffrey fluid with variable viscosity through a porous medium in an asymmetric channel has been achieved. The main advantage of such exact solution is the avoidance of any kind of restrictions on the viscosity parameter α, unlike the previous study in which the restriction α ≪ 1 has been put to achieve the requirements of the regular perturbation method. Hence, various plots have been introduced for the exact effects of the viscosity parameter, Daray’s number, porosity, amplitude ratio, Jeffrey fluid parameter, and the amplitudes of the waves on the pressure rise and the axial velocity. These exact effects have been discussed and further compared with those approximately obtained in the literature by using the regular perturbation method. The comparisons reveal that remarkable differences have been detected between the current exact results and those approximately obtained in the literature for the axial velocity profile and the pressure rise.
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Bila Adolphe Kyelem
2017-04-01
Full Text Available In this article, we prove the existence of solutions for some discrete nonlinear difference equations subjected to a potential boundary type condition. We use a variational technique that relies on Szulkin's critical point theory, which ensures the existence of solutions by ground state and mountain pass methods.
Mao, Feiyue; Wang, Wei; Min, Qilong; Gong, Wei
2015-06-01
Fernald method is regarded as the standard method for retrieving lidar data, but the retrieval can be performed only when a boundary value is given. Generally, we can select clear atmosphere above the tropopause as a reference to determine the boundary value, but we need to use the slope method to fit the boundary value when the detecting range is lower than the tropopause. The slope method involves significant uncertainty because this algorithm is based on two hypotheses: one is that aerosol vertical distribution is homogeneous, and the other is that either molecule or aerosol components exist in the atmosphere. To reduce the uncertainty, we proposed a new approach, which segments a signal into "uniform" sub-signals to avoid the first hypothesis, and then uses nonlinear two-component fitting to avoid the second one. Compared with the approach based on the slope method, the new approach obtained more accurate boundary values and retrieving results for both of the simulated and real signals. Thus the automatic segmentation algorithm and the two-component fitting method are useful for determining the reference bin and boundary values when the effective detecting range of lidar is lower than the tropopause.
Peng, Yaguang; Li, Wei; Wang, Yang; Bo, Jian; Chen, Hui
2015-01-01
To explore a scientific boundary of WHtR to evaluate central obesity and CVD risk factors in a Chinese adult population. The data are from the Prospective Urban Rural Epidemiology (PURE) China study that was conducted from 2005-2007. The final study sample consisted of 43 841 participants (18 019 men and 25 822 women) aged 35-70 years. According to the group of CVD risk factors proposed by Joint National Committee 7 version and the clustering of risk factors, some diagnosis parameters, such as sensitivity, specificity and receiver operating characteristic (ROC) curve least distance were calculated for hypertension, diabetes, high serum triglyceride (TG), high serum low density lipoprotein cholesterol (LDL-C), low serum high density lipoprotein cholesterol (HDL-C) and clustering of risk factors (number≥2) to evaluate the efficacy at each value of the WHtR cut-off point. The upper boundary value for severity was fixed on the point where the specificity was above 90%. The lower boundary value, which indicated above underweight, was determined by the percentile distribution of WHtR, specifically the 5th percentile (P5) for both males and females population. Then, based on convenience and practical use, the optimal boundary values of WHtR for underweight and obvious central obesity were determined. For the whole study population, the optimal WHtR cut-off point for the CVD risk factor cluster was 0.50. The cut-off points for severe central obesity were 0.57 in the whole population. The upper boundary values of WHtR to detect the risk factor cluster with specificity above 90% were 0.55 and 0.58 for men and women, respectively. Additionally, the cut-off points of WHtR for each of four cardiovascular risk factors with specificity above 90% in males ranged from 0.55 to 0.56, whereas in females, it ranged from 0.57 to 0.58. The P5 of WHtR, which represents the lower boundary values of WHtR that indicates above underweight, was 0.40 in the whole population. WHtR 0.50 was an
Bifurcation of solutions of nonlinear Sturm–Liouville problems
Directory of Open Access Journals (Sweden)
Gulgowski Jacek
2001-01-01
Full Text Available A global bifurcation theorem for the following nonlinear Sturm–Liouville problem is given Moreover we give various versions of existence theorems for boundary value problems The main idea of these proofs is studying properties of an unbounded connected subset of the set of all nontrivial solutions of the nonlinear spectral problem , associated with the boundary value problem , in such a way that .
On solution of the integral equations for the potential problems of two circular-strips
Directory of Open Access Journals (Sweden)
C. Sampath
1988-01-01
Dirichlet and Newmann boundary value problems of two equal infinite coaxial circular strips in various branches of potential theory. For illustration, these solutions are applied to solve some boundary value problems in electrostatics, hydrodynamics, and expressions for the physical quantities of interest are derived.
Torre Sangrà, David de la; Fantino, Elena
2015-01-01
Lambert’s problem is the orbital boundary-value problem constrained by two points and elapsed time. It is one of the most extensively studied problems in celestial mechanics and astrodynamics, and, as such, it has always attracted the interest of mathematicians and engineers. Its solution lies at the base of algorithms for, e.g., orbit determination, orbit design (mission planning), space rendezvous and interception, space debris correlation, missile and spacecraft targeting. There is abundan...
Izzo, Dario
2014-01-01
The orbital boundary value problem, also known as Lambert Problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an...
Directory of Open Access Journals (Sweden)
Fuyi Xu
2010-04-01
\\end{array}\\right.$$ where $1\\leq k\\leq s\\leq m-2, a_i, b_i\\in(0,+\\infty$ with $0<\\sum_{i=1}^{k}b_{i}-\\sum_{i=k+1}^{s}b_{i}<1, 0<\\sum_{i=1}^{m-2}a_{i}<1, 0<\\xi_1<\\xi_2<\\cdots<\\xi_{m-2}<\\rho(T$, $f\\in C( [0,+\\infty,[0,+\\infty$, $a(t$ may be singular at $t=0$. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results.
1974-09-01
directed oppositely to the force of gravity in both cwo and three dimensions. The undisturbed water surface will be taken as the (x,z) plane. As stated...SNAME I RL. Waid A . I1 1 Aerojet-General 1 R. Lacy W.C. Beckwith I Ro*-,rt Perkins 1 Bethlehem Steel Sparrows 1 Ray Kramer - A.D. Haff, Tech Mgr I
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Yuji Liu
2003-12-01
Full Text Available In this article, we study the differential equation $$ (-1^{n-p} x^{(n}(t=f(t,x(t,x'(t,dots,x^{(n-1}(t, $$ subject to the multi-point boundary conditions $$displaylines{ x^{(i}(0=0 quad hbox{for }i=0,1,dots,p-1,cr x^{(i}(1=0 quad hbox{for }i=p+1,dots,n-1,cr sum_{i=1}^malpha_ix^{(p}(xi_i=0, }$$ where $1le ple n-1$. We establish sufficient conditions for the existence of at least one solution at resonance and another at non-resonance. The emphasis in this paper is that $f$ depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.
Higher order nonlinear degenerate elliptic problems with weak monotonicity
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Youssef Akdim
2006-09-01
Full Text Available We prove the existence of solutions for nonlinear degenerate elliptic boundary-value problems of higher order. Solutions are obtained using pseudo-monotonicity theory in a suitable weighted Sobolev space.
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Vadim L. Khaikov
2018-01-01
Full Text Available The estimating of a projectile initial velocity is formulated as a two-point boundary value problem. To solve it, the data of a Doppler Radar or the results of solving the Cauchy problem can be used. The projectile initial velocity v0 estimation process is based on the numerical solution of a system of ordinary differential equations and the bisection method. The iterative calculating process is interrupted when a predetermined accuracy of a projectile initial velocity and a predetermined value of the width of velocity's search interval is reached. In the article, the block diagram of the algorithm for the projectile initial velocity process is developed. The Mathcad program code for mathematical modeling and a computer simulation of the projectile initial velocity estimation process for a 57mm armor-piercing projectile of ZIS-2 anti-tank gun 1943 model is given. / Задача оценки начальной скорости снаряда сформулирована в виде двухточечной граничной задачи. Для её решения могут быть использованы данные доплеровского измерителя скорости или результаты решения задачи Коши. Приведен алгоритм оценки v0, базирующийся на совокупности численного решения системы дифференциальных уравнений (СДУ полёта снаряда и метода бисекции. Итерационный процесс оценки начальной скорости прерывается при достижении заранее назначенной величины погрешности и заблаговременно установленного значения ширины интервала поиска. В статье представлена блок-схема алгоритма
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Jie Liu
2014-01-01
discusses the nonconforming rotated Q1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.
Inverse boundary spectral problems
Kachalov, Alexander; Lassas, Matti
2001-01-01
Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems.Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: ""Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary value
Domain decomposition methods for hyperbolic problems
Indian Academy of Sciences (India)
In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the 2 norms of the residuals and a term which is the ...
Numerical methods for hyperbolic differential functional problems
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Roman Ciarski
2008-01-01
Full Text Available The paper deals with the initial boundary value problem for quasilinear first order partial differential functional systems. A general class of difference methods for the problem is constructed. Theorems on the error estimate of approximate solutions for difference functional systems are presented. The convergence results are proved by means of consistency and stability arguments. A numerical example is given.
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Bakhtiyor Kadirkulov
2014-02-01
Full Text Available In this article, we study a boundary value problem for a parabolic-hyperbolic equation with Caputo fractional derivative. Under certain conditions, we prove its unique solvability using methods of integral equations and Green's functions.
On Some Trends in Elliptic Problem Solvers.
1981-02-01
solution to boundary value problems. Mathematics At Comutation 31:333-390, 1977. [2] R. Chandra. ConiuireGradient Methods fr Partial Di Eaustion. PhD...include serious consideration of algorithms, architecture, and t o . .• applied mathematics , each traditionally the subject of an entire Uhanghl.yd
Higher order Nevanlinna functions and the inverse three spectra problem
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Olga Boyko
2016-01-01
Full Text Available The three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on \\([0,a]\\, the Dirichlet-Dirichlet problem on \\([0,a/2]\\ and the Neumann-Dirichlet problem on \\([a/2,a]\\ is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found.
Neighboring extremals of dynamic optimization problems with path equality constraints
Lee, A. Y.
1988-01-01
Neighboring extremals of dynamic optimization problems with path equality constraints and with an unknown parameter vector are considered in this paper. With some simplifications, the problem is reduced to solving a linear, time-varying two-point boundary-value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Two example problems are solved to illustrate the validity and usefulness of the solution technique.
The fencing problem and Coleochaete cell division.
Wang, Yuandi; Dou, Mingya; Zhou, Zhigang
2015-03-01
The findings in this study suggest that the solution of a boundary value problem for differential equation system can be used to discuss the fencing problem in mathematics and Coleochaete, a green algae, cell division. This differential equation model in parametric expression is used to simulate the two kinds of cell division process, one is for the usual case and the case with a "dead" daughter cell.
Low-cost control problems on perforated and non-perforated domains
Indian Academy of Sciences (India)
We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order ...
Energy Technology Data Exchange (ETDEWEB)
Schaefer, K.; Graus, H.; Remmel, R. [Deutsche Steinkohle AG, Sulzbach (Germany). Werkstaetten Saar/Technische Dienste
2000-07-01
Adaptation of a heat exchanger hot section and its associated burner installation of a hot gas engine to be newly developed to the utilisation of regenerative energy sources. The task definition of the Deutsche Steinkohle AG was to adapt a heat exchanger hot section of a Stirling-engine with the associated burner installation to the use of alternative energy carriers. In order to do this, it was necessary to produce a suitable basic engine. The engine was to work initially on the basis of 20 kW electrical power. The various influences to which the Stirling engine is subjected in industrial use were also to be examined and described. Calculation methods are presented as a result of the work, which make it possible to design the highly stressed components (heat exchanger, crank mechanism, and housing) and to assess these in terms of safety. Also shown are a large number of problems in the machine environment and solutions are addressed. It appears that with the presented burner installation, it is also possible to process the most varying regenerative fuel gases for further utilisation on the Stirling-engine. This was verified by the commissioning of an industrial plant for utilisation of low-methane-containing marsh gas. (orig.) [German] Die Deutsche Steinkohle AG sollte einen Heissteilwaermetauscher eines Stirling-Motors mit der dazugehoerigen Brenneranlage and die Verwendung alternativer Energietraeger anpassen. Hierzu war es notwendig, einen geeigneten Grundmotor zu erstellen. Der Grundmotor sollte zunaechst auf einer Basis von 20 kW elektrischer Leistung arbeiten. Des weiteren sollten die verschiedenen Einfluesse denen der Stirling-Motor im industriellen Einsatz unterliegt, untersucht und beschrieben werden. Als ein Ergebnis der Arbeiten werden Berechnungsverfahren vorgestellt, die es ermoeglichen, die hochbelasteten Komponenten (Waermetauscher, Kurbeltrieb, Gehaeuse) konstruktiv auszulegen und sicherheitstechnisch zu bewerten. Des weiteren werden zahlreiche
DEFF Research Database (Denmark)
Barari, Amin; Ganjavi, B.; Jeloudar, M. Ghanbari
2010-01-01
Purpose – In the last two decades with the rapid development of nonlinear science, there has appeared ever-increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general...... boundary value problems, but also are used as mathematical models in viscoelastic and inelastic flows. The purpose of this paper is to present the application of the homotopy-perturbation method (HPM) and variational iteration method (VIM) to solve some boundary value problems in structural engineering...... and fluid mechanics. Design/methodology/approach – Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics. Findings – Analytical solutions often fit under classical perturbation methods. However...
hp Spectral element methods for three dimensional elliptic problems ...
Indian Academy of Sciences (India)
Abstract. This is the first of a series of papers devoted to the study of h-p spec- tral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the ...
Solving the Stokes problem on a massively parallel computer
DEFF Research Database (Denmark)
Axelsson, Owe; Barker, Vincent A.; Neytcheva, Maya
2001-01-01
boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work...
Approximate solutions of some problems of scattering of surface ...
Indian Academy of Sciences (India)
A class of mixed boundary value problems (bvps), occurring in the study of scattering of surface water waves by thin vertical rigid barriers placed in water of finite depth, is examined for their approximate solutions. Two different placings of vertical barriers are analyzed, namely, (i) a partially immersed barrier and(ii) a bottom ...
Approximate solutions of some problems of scattering of surface ...
Indian Academy of Sciences (India)
A Choudhary
Abstract. A class of mixed boundary value problems (bvps), occurring in the study of scattering of surface water waves by thin vertical rigid barriers placed in water of finite depth, is examined for their approximate solutions. Two different placings of vertical barriers are analyzed, namely, (i) a partially immersed barrier and.
Izzo, Dario
2015-01-01
The orbital boundary value problem, also known as Lambert problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an efficient initial guess to an Householder iterative method that is then able to converge, for the single revolution case, in only two iterations. The resulting algorithm is compared, for single and multiple revolutions, to Gooding's procedure revealing to be numerically as accurate, while having a significantly smaller computational complexity.
Vision-based stereo ranging as an optimal control problem
Menon, P. K. A.; Sridhar, B.; Chatterji, G. B.
1992-01-01
The recent interest in the use of machine vision for flight vehicle guidance is motivated by the need to automate the nap-of-the-earth flight regime of helicopters. Vision-based stereo ranging problem is cast as an optimal control problem in this paper. A quadratic performance index consisting of the integral of the error between observed image irradiances and those predicted by a Pade approximation of the correspondence hypothesis is then used to define an optimization problem. The necessary conditions for optimality yield a set of linear two-point boundary-value problems. These two-point boundary-value problems are solved in feedback form using a version of the backward sweep method. Application of the ranging algorithm is illustrated using a laboratory image pair.
POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
Directory of Open Access Journals (Sweden)
FAOUZI HADDOUCHI
2015-11-01
Full Text Available In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP for the following second-order differential equation u''(t + \\lambda a(tf(u(t = 0; 0 0 is a parameter, 0 <\\eta < 1, 0 <\\alpha < 1/{\\eta}. . By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.
An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology
Beretta, Elena; Cavaterra, Cecilia; Cerutti, M. Cristina; Manzoni, Andrea; Ratti, Luca
2017-10-01
In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity \
An inverse problem for Maxwell’s equations with Lipschitz parameters
Pichler, Monika
2018-02-01
We consider an inverse boundary value problem for Maxwell’s equations, which aims to recover the electromagnetic material properties of a body from measurements on the boundary. We show that a Lipschitz continuous conductivity, electric permittivity, and magnetic permeability are uniquely determined by knowledge of all tangential electric and magnetic fields on the boundary of the body at a fixed frequency.
On the solvability of Dirichlet problem for the weighted p-Laplacian
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Ewa Szlachtowska
2012-01-01
Full Text Available The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted \\(p\\-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.
Sahli, B.; Bencheikh, L.
2010-11-01
The question of non-uniqueness in boundary integral equation formulations of exterior Neumann boundary-value problem in elasticity can be resolved by seeking the solution in the form of a single-layer potential. We present an analysis of the appropriate choice of the multipole coefficients which is optimal in the sense of minimizing the condition number of the boundary integral operator.
High-order time-accurate schemes for parabolic singular perturbation problems with convection
P.W. Hemker (Piet); G.I. Shishkin (Gregori); L.P. Shishkina
2001-01-01
textabstractWe consider the first boundary value problem for a singularly perturbed para-bo-lic PDE with convection on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh, condensing in the boundary layer,
Multiplicity of solutions for discrete problems with double-well potentials
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Josef Otta
2013-08-01
Full Text Available This article presents some multiplicity results for a general class of nonlinear discrete problems with double-well potentials. Variational techniques are used to obtain the existence of saddle-point type critical points. In addition to simple discrete boundary-value problems, partial difference equations as well as problems involving discrete $p$-Laplacian are considered. Also the boundedness of solutions is studied and possible applications, e.g. in image processing, are discussed.
A finite difference method for free boundary problems
Fornberg, Bengt
2010-04-01
Fornberg and Meyer-Spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a Cartesian grid. We show here how this procedure can be combined with a minimax-based optimization procedure to rapidly solve a wide range of elliptic-type free boundary value problems. © 2009 Elsevier B.V. All rights reserved.
Asymptotic analysis of the narrow escape problem in dendritic spine shaped domain: three dimensions
Li, Xiaofei; Lee, Hyundae; Wang, Yuliang
2017-08-01
This paper deals with the three-dimensional narrow escape problem in a dendritic spine shaped domain, which is composed of a relatively big head and a thin neck. The narrow escape problem is to compute the mean first passage time of Brownian particles traveling from inside the head to the end of the neck. The original model is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper we seek to transfer the original problem to a mixed Robin-Neumann boundary value problem by dropping the thin neck part, and rigorously derive the asymptotic expansion of the mean first passage time with high order terms. This study is a nontrivial three-dimensional generalization of the work in Li (2014 J. Phys. A: Math. Theor. 47 505202), where a two-dimensional analogue domain is considered.
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Obidjon Kh. Abdullaev
2016-06-01
Full Text Available In this work, we study the existence and uniqueness of solutions to non-local boundary value problems with integral gluing condition. Mixed type equations (parabolic-hyperbolic involving the Caputo fractional derivative have loaded parts in Riemann-Liouville integrals. Thus we use the method of integral energy to prove uniqueness, and the method of integral equations to prove existence.
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V. Rukavishnikov
2014-01-01
Full Text Available The existence and uniqueness of the Rv-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.
SEACAS Theory Manuals: Part 1. Problem Formulation in Nonlinear Solid Mechancis
Energy Technology Data Exchange (ETDEWEB)
Attaway, S.W.; Laursen, T.A.; Zadoks, R.I.
1998-08-01
This report gives an introduction to the basic concepts and principles involved in the formulation of nonlinear problems in solid mechanics. By way of motivation, the discussion begins with a survey of some of the important sources of nonlinearity in solid mechanics applications, using wherever possible simple one dimensional idealizations to demonstrate the physical concepts. This discussion is then generalized by presenting generic statements of initial/boundary value problems in solid mechanics, using linear elasticity as a template and encompassing such ideas as strong and weak forms of boundary value problems, boundary and initial conditions, and dynamic and quasistatic idealizations. The notational framework used for the linearized problem is then extended to account for finite deformation of possibly inelastic solids, providing the context for the descriptions of nonlinear continuum mechanics, constitutive modeling, and finite element technology given in three companion reports.
On the inverse problem for a heat-like equation
Directory of Open Access Journals (Sweden)
Igor Malyshev
1987-01-01
Full Text Available Using the integral representation of the solution of the boundary value problem for the equation with one time-dependent coefficient at the highest space-derivative three inverse problems are solved. Depending on the property of the coefficient we consider cases when the equation is of the parabolic type and two special cases of the degenerate/mixed type. In the parabolic case the corresponding inverse problem is reduced to the nonlinear Volterra integral equation for which the uniqueness of the solution is proved. For the special cases explicit formulae are derived. Both Ã‚Â“minimalÃ‚Â” and overspecified boundary data are considered.
Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates
Kitahara, M
1985-01-01
The boundary integral equation (BIE) method has been used more and more in the last 20 years for solving various engineering problems. It has important advantages over other techniques for numerical treatment of a wide class of boundary value problems and is now regarded as an indispensable tool for potential problems, electromagnetism problems, heat transfer, fluid flow, elastostatics, stress concentration and fracture problems, geomechanical problems, and steady-state and transient electrodynamics.In this book, the author gives a complete, thorough and detailed survey of the method. It pro
Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators
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Allaberen Ashyralyev
2014-01-01
Full Text Available The nonlocal boundary value problem for the parabolic differential equation v'(t+A(tv(t=f(t (0≤t≤T, v(0=v(λ+φ, 0<λ≤T in an arbitrary Banach space E with the dependent linear positive operator A(t is investigated. The well-posedness of this problem is established in Banach spaces C0β,γ(Eα-β of all Eα-β-valued continuous functions φ(t on [0,T] satisfying a Hölder condition with a weight (t+τγ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.
The mixed problem for the Laplacian in Lipschitz domains
Ott, Katharine A.; Brown, Russell M.
2009-01-01
We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under thes...
Regularity of spectral fractional Dirichlet and Neumann problems
DEFF Research Database (Denmark)
Grubb, Gerd
2016-01-01
Consider the fractional powers and of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley...... in the 1970's, we demonstrate how they imply regularity properties in full scales of -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on -calculus. We also include an overview...... of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian....
Approximated Solutions of Linear Quadratic Fractional Optimal Control Problems
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Soradi Zeid S.
2016-12-01
Full Text Available In this study we apply the Adomian decomposition method (ADM to approximate the solution of fractional optimal control problems (FOCPs where the dynamic of system is a linear control system with constant coefficient and the cost functional is defined in a quadratic form. First we stated the necessary optimality conditions in a form of fractional two point boundary value problem (TPBVP, then the ADM is used to solve the resulting fractional differential equations (FDEs. Some examples are provided to demonstrate the validity and applicability of the proposed method.
Uniqueness for a boundary identification problem in thermal imaging
Directory of Open Access Journals (Sweden)
Kurt Bryan
1998-11-01
Full Text Available An inverse problem for an initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in ${mathbb R}^n$ from measurements of Cauchy data on a known portion of the boundary. The dynamics in the interior of the region are governed by a differential operator of parabolic type. Utilizing a unique continuation result for evolution operators, along with the method of eigenfunction expansions, it is shown that uniqueness holds for a large and physically reasonable class of Cauchy data pairs.
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
Directory of Open Access Journals (Sweden)
Rai Nath Kabindra Rajeev
2009-01-01
Full Text Available In this paper, the solution of the one dimensional moving boundary problem with periodic boundary conditions is obtained with the help of variational iterational method. By using initial and boundary values, the explicit solutions of the equations have been derived, which accelerate the rapid convergence of the series solution. The method performs extremely well in terms of efficiency and simplicity. The temperature distribution and the position of moving boundary are evaluated and numerical results are presented graphically.
Two-Sided A Posteriori Error Bounds for Electro-Magneto Static Problems
Pauly, Dirk; Repin, Sergey
2011-01-01
This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between the exact and the approximate solution of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such ...
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Richard I. Avery
2000-05-01
Full Text Available We study the existence of solutions to the fourth order Lidstone boundary value problem $$displaylines{ y^{(4}(t = f(y(t,-y''(t,cr y(0=y''(0=y''(1=y(1=0,. }$$ By imposing growth conditions on $f$ and using a generalization of the multiple fixed point theorem by Leggett and Williams, we show the existence of at least three symmetric positive solutions. We also prove analogous results for difference equations.
Three-point third-order problems with a sign-changing nonlinear term
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Johnny Henderson
2014-08-01
Full Text Available In this article we study a well-known boundary value problem $$\\displaylines{ u'''(t = f(t, u(t, \\quad 0 1/2$. It is well-known that the standard method successfully applied to the semi-positone problem with $\\eta > 1/2$ does not work for $\\eta =1/2$ in the same setting. We treat the latter as a problem with a sign-changing term rather than a semi-positone problem. We apply Krasnosel'skii's fixed point theorem [4] to obtain positive solutions.
Application of Green's operator to quadratic variational problems
Directory of Open Access Journals (Sweden)
Nikolay V. Azbelev
2006-01-01
Full Text Available We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space \\(L_2\\ of quadratically summable functions. We get the necessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in \\(L_2\\ with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1 a variational problem with deviating argument, and 2 the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis.
A mixed semilinear parabolic problem from combustion theory
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Claudia Lederman
2001-01-01
Full Text Available We prove existence, uniqueness, and regularity of the solution to a mixed initial boundary-value problem. The equation is semilinear uniformly parabolic with principal part in divergence form, in a non-cylindrical space-time domain. Here we extend our results in cite{LVWmix} to a more general domain. As in cite{LVWmix}, we assume only mild regularity on the coefficients, on the non-cylindrical part of the lateral boundary (where the Dirichlet data are given, and on the Dirichlet data.
Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
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Golovaty Yuriy
2017-04-01
Full Text Available We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.
Boundary eigencurve problems involving the p-Laplacian operator
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Mohammed Ouanan
2008-05-01
Full Text Available In this paper, we show that for each $lambda in mathbb{R}$, there is an increasing sequence of eigenvalues for the nonlinear boundary-value problem $$displaylines{ Delta_pu=|u|^{p-2}u quad hbox{in } Omegacr | abla u|^{p-2}frac{partial u}{partial u}=lambda ho(x|u|^{p-2}u+mu|u|^{p-2}u quad hbox{on } partial Omega,; }$$ also we show that the first eigenvalue is simple and isolated. Some results about their variation, density, and continuous dependence on the parameter $lambda$ are obtained.
... Women Hair Loss Hand/Wrist/Arm Problems Headaches Hearing Problems Hip Problems Knee Problems Leg Problems Lower Back ... have ear pain or redness but is having problems hearing?YesNo Back to Questions Step 3 Possible Causes ...
Uniqueness theorems for variational problems by the method of transformation groups
Reichel, Wolfgang
2004-01-01
A classical problem in the calculus of variations is the investigation of critical points of functionals {\\cal L} on normed spaces V. The present work addresses the question: Under what conditions on the functional {\\cal L} and the underlying space V does {\\cal L} have at most one critical point? A sufficient condition for uniqueness is given: the presence of a "variational sub-symmetry", i.e., a one-parameter group G of transformations of V, which strictly reduces the values of {\\cal L}. The "method of transformation groups" is applied to second-order elliptic boundary value problems on Riemannian manifolds. Further applications include problems of geometric analysis and elasticity.
A note on the singular Sturm-Liouville problem with infinitely many solutions
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Nickolai Kosmatov
2002-09-01
Full Text Available We consider the Sturm-Liouville nonlinear boundary-value problem $$ displaylines{ -u''(t = a(tf(u(t, quad 0 0$ and $a(t$ is in a class of singular functions. Using a fixed point theorem we show that under certain growth conditions imposed on $f(u$ the problem admits infinitely many solutions. Submitted November 13, 2001. Published September 27, 2002. Math Subject Classifications: 34B16, 34B18. Key Words: Sturm-Liouville problem; Green's function; fixed point theorem; Holder's inequality; multiple solutions.
On a shock problem involving a nonlinear viscoelastic bar
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Tran Ngoc Diem
2005-11-01
Full Text Available We treat an initial boundary value problem for a nonlinear wave equation uttÃ¢ÂˆÂ’uxx+K|u|ÃŽÂ±u+ÃŽÂ»|ut|ÃŽÂ²ut=f(x,t in the domain 0
Exchange orbits in the planar 1 + 4 body problem
Bengochea, A.; Galán, J.; Pérez-Chavela, E.
2015-05-01
We study some doubly-symmetric orbits in the planar 1 + 2 n-body problem, that is the mass of the central body is significantly bigger than the other 2 n equal masses. The necessary and sufficient conditions for periodicity of the orbits are discussed. We also study numerically these kinds of orbits for the case n = 2. The system under study corresponds to one conformed by a planet and four satellites of equal mass. We determine a 1-parameter family of time-reversible invariant tori, related with the reversing symmetries of the equations of motion. The initial conditions of the orbits were determined by means of solving a boundary value problem with one free parameter. The numerical solution of the boundary value problem was obtained using the software AUTO. For the numerical analysis we have used the value of 3.5 × 10-4 as mass ratio of some satellite and the planet. In the computed solutions the satellites are in mean motion resonance 1:1 and they librate around a relative equilibria, that is a solution where the distances between the bodies remain constant for all time.
1988-07-08
initial-boundary value problems". 20. Speaker, The Gatlinburg IX Conference on Numerical Algebra, Univesity of Waterloo, Waterloo, Canada, July 1984...31, 1927 Education : Attended public schools in California Military Service: United States Navy, 1944-1946, honorable discharge Married: Rebecca...Microcomputer Laboratory at the University of California at Santa Barbara, under grants from: the Fund for the Improvement of Postsecondary Education
problem for the damped Boussinesq equation
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Vladimir V. Varlamov
1997-01-01
Full Text Available For the damped Boussinesq equation utt−2butxx=−αuxxxx+uxx+β(u2xx,x∈(0,π,t>0;α,b=const>0,β=const∈R1, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limit b→+0 in the constructed solution is investigated.
Varying domains in a general class of sublinear elliptic problems
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Santiago Cano-Casanova
2004-05-01
Full Text Available In this paper we use the linear theory developed in [8] and [9] to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the underlying domain. Our main theorem completes the results of Daners and Dancer [12] -and the references there in-, where the classical Robin problem was dealt with. Besides the fact that we are working with mixed non-classical boundary conditions, it must be mentioned that this paper is considering problems where bifurcation from infinity occurs; now a days, analyzing these general problems, where the coefficients are allowed to vary and eventually vanishing or changing sign, is focusing a great deal of attention -as they give rise to metasolutions (e.g., [20]-.
DEFF Research Database (Denmark)
Almegaard, Henrik
2014-01-01
The majority of literature on engineering design methods is focused on the processes of fulfilling the design goals as efficiently as possible. This paper will focus on - and discuss - the processes of determining the design goals: the specifications. The purpose is to draw attention to the inher...... to the inherent problems, dilemmas and possibilities in these processes bearing in mind that that the most important decisions in a design project are taken in the beginning of the project....
Sakhnovich, Lev A; Roitberg, Inna Ya
2013-01-01
This monograph fits theclearlyneed for books with a rigorous treatment of theinverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authorsdevelop a unified treatment of explicit and global solutions via the transfer matrix function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained andstarts with preliminaries and examples, and hencealso serves as an introduction for advanced graduate students in the field.
Regularity of solutions of the Neumann problem for the Laplace equation
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Dagmar Medkova
2006-11-01
Full Text Available Let u be a solution of the Neumann problem for the Laplace equation in G with the boundary condition g. It is shown that u ∈ L q (∂ G (equivalently, u ∈ Bq,21/q (G for 1 , u ∈ Lq 1/q (G for 2 ≤ q if and only if the single layer potential corresponding to the boundary condition g is in L q (∂ G . As a consequence we give a regularity result for some nonlinear boundary value problem.
Beck, Lisa; Bulíček, Miroslav; Málek, Josef; Süli, Endre
2017-08-01
We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic radial structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously to the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.
Cengizci, Süleyman; Atay, Mehmet Tarık; Eryılmaz, Aytekin
2016-01-01
This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.
... it could be a sign of a balance problem. Balance problems can make you feel unsteady. You may also ... injuries, such as a hip fracture. Some balance problems are due to problems in the inner ear. ...
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...
On rational approximation methods for inverse source problems
Rundell, William
2011-02-01
The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Such is the ubiquity of these problems, the underlying model can lead to a partial differential equation of any of the major types, but here we focus on the case of steady-state electrostatic or thermal imaging and consider boundary value problems for Laplace\\'s equation. Our inclusions are interior forces with compact support and our data consists of a single measurement of (say) voltage/current or temperature/heat flux on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler "equivalent point source" problem, and which uses a Newton scheme to improve the corresponding initial approximation. © 2011 American Institute of Mathematical Sciences.
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Garcia, F., Jr.
1974-01-01
A study of the solution problem of a complex entry optimization was studied. The problem was transformed into a two-point boundary value problem by using classical calculus of variation methods. Two perturbation methods were devised. These methods attempted to desensitize the contingency of the solution of this type of problem on the required initial co-state estimates. Also numerical results are presented for the optimal solution resulting from a number of different initial co-states estimates. The perturbation methods were compared. It is found that they are an improvement over existing methods.
Fuel-optimal trajectories for aeroassisted coplanar orbital transfer problem
Naidu, Desineni Subbaramaiah; Hibey, Joseph L.; Charalambous, Charalambos D.
1990-01-01
The optimal control problem arising in coplanar orbital transfer employing aeroassist technology is addressed. The maneuver involves the transfer from high to low earth orbit via the atmosphere, with the object of minimizing the total fuel consumption. Simulations are carried out to obtain the fuel-optimal trajectories for flying the spacecraft through the atmosphere. A highlight is the application of an efficient multiple-shooting method for treating the nonlinear two-point boundary value problem resulting from the optimizaion procedure. The strategy for the atmospheric portion of the minimum-fuel transfer is to fly at the maximum lift-to-drag ratio L/D initially in order to recover from the downward plunge, and then to fly at a negative L/D to level off the flight so that the vehicle skips out of the atmosphere with a flight path angle near zero degrees.
Inverse Sturm-Liouville problem with discontinuity conditions
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Mohammad Shahriari
2014-02-01
Full Text Available This paper deals with the boundary value problem involving the dierential equation ly := -y''+qy = λy, subject to the standard boundary conditions along with the following discontinuity conditions at a point a ε(0,π y(a + 0 = a1y(a - 0, y'(a + 0 = a1-1y'(a - 0 + a2y(a - 0, where q(x, a1,a2 are real, q ε L2(0,π and λ is a parameter independent of x. We develop the Hochestadt's result based on the transformation operator for inverse Sturm-Liouville problem when there are discontinuous conditions. Furthermore, we establish a formula for q(x - q~(x in the nite interval where q(x and q~(x are analogous functions.
Lectures on nonlinear evolution equations initial value problems
Racke, Reinhard
2015-01-01
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value p...
The semi-infinite strip problem in the mechanics of composite materials
Kaw, Autar K.; Goree, James G.
1991-01-01
Many problems, ranging from interface-fiber fracture to determining the strength of fiber-matrix interfaces in composites, can be greatly assisted by a complete understanding of the boundary value problem of a semi-infinite strip. The specific boundary value problem of a semi-infinite strip with symmetric tractions on the transverse edges and normal loads on the end is investigated in the present study. The nature of the singular behavior in the slope of the end vertical displacement at the corner points is found. An asymptotic analysis of the solution shows that, depending on the nature of the applied stresses in the vicinity of the corner points, the singularity in the slope of the end vertical displacement may either be a power or logarithmic type, both, or not exist at all. Illustrative examples, for which exact or numerical results are known, are given. Numerical procedures are also given. The direct application of the results of the paper are illustrated for several problems in the mechanics of composite materials.
Asymptotic behavior of positive solutions of a semilinear Dirichlet problem outside the unit ball
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Habib Maagli
2013-04-01
Full Text Available In this article, we are concerned with the existence, uniqueness and asymptotic behavior of a positive classical solution to the semilinear boundary-value problem $$displaylines{ -Delta u=a(xu^{sigma }quadext{in }D, cr lim _{|x|o 1}u(x= lim_{|x|o infty}u(x =0. }$$ Here D is the complement of the closed unit ball of $mathbb{R} ^n$ ($ngeq 3$, $sigma<1$ and the function a is a nonnegative function in $C_{m loc}^{gamma}(D$, $0
A numerical method for finding sign-changing solutions of superlinear Dirichlet problems
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Neuberger, J.M.
1996-12-31
In a recent result it was shown via a variational argument that a class of superlinear elliptic boundary value problems has at least three nontrivial solutions, a pair of one sign and one which sign changes exactly once. These three and all other nontrivial solutions are saddle points of an action functional, and are characterized as local minima of that functional restricted to a codimension one submanifold of the Hilbert space H-0-1-2, or an appropriate higher codimension subset of that manifold. In this paper, we present a numerical Sobolev steepest descent algorithm for finding these three solutions.
Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
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Safa Dridi
2015-01-01
Full Text Available In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \\[-\\Delta u=q(xu^{\\sigma }\\;\\text{in}\\;\\Omega,\\quad u_{|\\partial\\Omega}=0.\\] Here \\(\\Omega\\ is an annulus in \\(\\mathbb{R}^{n}\\, \\(n\\geq 3\\, \\(\\sigma \\lt 1\\ and \\(q\\ is a positive function in \\(\\mathcal{C}_{loc}^{\\gamma }(\\Omega \\, \\(0\\lt\\gamma \\lt 1\\, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.
Kawashita, Mishio
2017-01-01
In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of this body, a single pair of the temperature and heat flux is given as an observation datum for the inverse problem. It is found the minimum length of broken paths connecting arbitrary fixed point in the outside, a point on the boundary of the cavities and a po...
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W. Sinkala
2012-01-01
Full Text Available We use Lie symmetry analysis to solve a boundary value problem that arises in chemical engineering, namely, mass transfer during the contact of a solid slab with an overhead flowing fluid. This problem was earlier tackled using Adomian decomposition method (Fatoorehchi and Abolghasemi 2011, leading to the Adomian series form of solution. It turns out that the application of Lie group analysis yields an elegant form of the solution. After introducing the governing mathematical model and some preliminaries of Lie symmetry analysis, we compute the Lie point symmetries admitted by the governing equation and use these to construct the desired solution as an invariant solution.
Explaining the Mind: Problems, Problems
Harnad, Stevan
2001-01-01
The mind/body problem is the feeling/function problem: How and why do feeling systems feel? The problem is not just "hard" but insoluble (unless one is ready to resort to telekinetic dualism). Fortunately, the "easy" problems of cognitive science (such as the how and why of categorization and language) are not insoluble. Five books (by Damasio, Edelman/Tononi...
New direct linearizations for KdV and solutions of the other Cauchy problem
Sabatier, Pierre C.
1999-06-01
The author showed previously that there are several equations that "directly linearize" the Korteweg de Vries equation. They may give different classes of solutions and they correspond either to generalized Gelfand Levitan or generalized Marchenko inversion equations. The idea is developed here with a precise goal: solving by linear methods the "other Cauchy problem" for KdV, i.e., the boundary value problem where the solution is known at fixed x, together with its two first derivatives. After several new direct linearization equations are given and analyzed, the one that solves the problem is eventually obtained. It also corresponds to a new inverse spectral problem, whose scalar equations are fourth order, and that is first studied in the Gelfand Levitan form, and then studied and completely solved in the Marchenko form. The methods given here can be extended probably to most nonlinear integrable equations, and suggest several new problems.
Effective Numerical Methods for Solving Elliptical Problems in Strengthened Sobolev Spaces
D'yakonov, Eugene G.
1996-01-01
Fourth-order elliptic boundary value problems in the plane can be reduced to operator equations in Hilbert spaces G that are certain subspaces of the Sobolev space W(sub 2)(exp 2)(Omega) is identical with G(sup (2)). Appearance of asymptotically optimal algorithms for Stokes type problems made it natural to focus on an approach that considers rot w is identical with (D(sub 2)w - D(sub 1)w) is identical with vector of u as a new unknown vector-function, which automatically satisfies the condition div vector of u = 0. In this work, we show that this approach can also be developed for an important class of problems from the theory of plates and shells with stiffeners. The main mathematical problem was to show that the well-known inf-sup condition (normal solvability of the divergence operator) holds for special Hilbert spaces. This result is also essential for certain hydrodynamics problems.
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Unsupervised neural networks for solving Troesch's problem
Muhammad, Asif Zahoor Raja
2014-01-01
In this study, stochastic computational intelligence techniques are presented for the solution of Troesch's boundary value problem. The proposed stochastic solvers use the competency of a feed-forward artificial neural network for mathematical modeling of the problem in an unsupervised manner, whereas the learning of unknown parameters is made with local and global optimization methods as well as their combinations. Genetic algorithm (GA) and pattern search (PS) techniques are used as the global search methods and the interior point method (IPM) is used for an efficient local search. The combination of techniques like GA hybridized with IPM (GA-IPM) and PS hybridized with IPM (PS-IPM) are also applied to solve different forms of the equation. A comparison of the proposed results obtained from GA, PS, IPM, PS-IPM and GA-IPM has been made with the standard solutions including well known analytic techniques of the Adomian decomposition method, the variational iterational method and the homotopy perturbation method. The reliability and effectiveness of the proposed schemes, in term of accuracy and convergence, are evaluated from the results of statistical analysis based on sufficiently large independent runs.
Numerical solution of Lord-Shulman thermopiezoelectricity dynamical problem
Stelmashchuk, Vitaliy; Shynkarenko, Heorhiy
2018-01-01
Using Lord-Shulman hypothesis we formulate the initial boundary value problem and its corresponding variational problem of a generalized linear thermopiezoelectricity in terms of the displacement, electrical potential, temperature increment and heat flux, which describes the dynamic behavior of the coupled mechanic, electric and heat waves in pyroelectric materials. We construct the corresponding energy balance equation and determine input data regularity for the variational problem, which guarantees the existence, uniqueness and stability of its solution in the problem energy norm. Based on these results, we propose a numerical scheme for solving this problem, which includes spatial finite element semi-discretization and one-step recurrent time integration procedures and generalizes the similar one for classic thermopiezoelectricity problem. We give the sufficient conditions on the values of the scheme parameters which guarantee properties of conservatism and unconditional stability of the scheme. The rest of the article is devoted to the analysis of performed numerical experiments with 1D model problem and their results are then compared with the ones obtained by the other researchers.
On one nonlocal problem for the Euler–Darboux equation
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Mikhail V. Dolgopolov
2016-06-01
Full Text Available The boundary value problem with displacement is determined for the generalized Euler–Darboux equation in the field representing the first quadrant. This problem, unlike previous productions, specifies two conditions, connect integrals and fractional derivatives from the values of the sought solution in the boundary points. On the line of singularity of the coefficients of the equations the matching conditions continuous with respect to the solution and its normal derivation are considered. The authors took for the basis of solving the earlier obtained by themselves the Cauchy problem solution of the special class due to the integral representations of one of the specified functions acquired simple form both for positive and for negative values of Euler–Darboux equation parameter. The nonlocal problem set by the authors is reduced to the system of Volterra integral equations with unpacked operators, the only solution which is given explicitly in the corresponding class of functions. From the above the uniqueness of the solution of nonlocal problem follows. The existence is proved by the direct verification. This reasoning allowed us to obtain the solution of nonlocal problem in the explicit form both for the positive and for the negative values of Euler–Darboux equation parameter.
Skovhus, Randi Boelskifte; Thomsen, Rie
2017-01-01
This article introduces a method to critical reviews and explores the ways in which problems have been formulated in knowledge production on career guidance in Denmark over a 10-year period from 2004 to 2014. The method draws upon the work of Bacchi focussing on the "What's the problem represented to be" (WPR) approach. Forty-nine…
... Read MoreDepression in Children and TeensRead MoreBMI Calculator Hearing ProblemsLoss in the ability to hear or discriminate ... This flow chart will help direct you if hearing loss is a problem for you or a ...
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Nahed S. Hussein
2014-01-01
Full Text Available A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.
Ruggeri, Fabrizio
2016-05-12
In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.
Solution of the Dirichlet Problem for the Poisson's Equation in a Multidimensional Infinite Layer
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O. D. Algazin
2015-01-01
Full Text Available The paper considers the multidimensional Poisson equation in the domain bounded by two parallel hyperplanes (in the multidimensional infinite layer. For an n-dimensional half-space method of solving boundary value problems for linear partial differential equations with constant coefficients is a Fourier transform to the variables in the boundary hyperplane. The same method can be used for an infinite layer, as is done in this paper in the case of the Dirichlet problem for the Poisson equation. For strip and infinite layer in three-dimensional space the solutions of this problem are known. And in the three-dimensional case Green's function is written as an infinite series. In this paper, the solution is obtained in the integral form and kernels of integrals are expressed in a finite form in terms of elementary functions and Bessel functions. A recurrence relation between the kernels of integrals for n-dimensional and (n + 2 -dimensional layers was obtained. In particular, is built the Green's function of the Laplace operator for the Dirichlet problem, through which the solution of the problem is recorded. Even in three-dimensional case we obtained new formula compared to the known. It is shown that the kernel of the integral representation of the solution of the Dirichlet problem for a homogeneous Poisson equation (Laplace equation is an approximate identity (δ-shaped system of functions. Therefore, if the boundary values are generalized functions of slow growth, the solution of the Dirichlet problem for the homogeneous equation (Laplace is written as a convolution of kernels with these functions.
Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms
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Myron K. Grammatikopoulos
2003-01-01
Full Text Available In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems on the plane. Protter studied these problems in a 3-D domain $Omega_0$, bounded by two characteristic cones $Sigma_1$ and $Sigma_{2,0}$, and by a plane region $Sigma_0$. It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995 discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on $Sigma_0$: the strong power-type singularity appears in the generalized solution on the characteristic cone $Sigma_{2,0}$. In the present paper we consider the case of third boundary-value problem on $Sigma_0$ and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifica ally, for Protter's problems in $mathbb{R}^{3}$ it is shown here that for any $nin N$ there exists a $C^{n}({Omega}_0$-function, for which the corresponding unique generalized solution belongs to $C^{n}({Omega}_0slash O$ and has a strong power type singularity at the point $O$. This singularity is isolated at the vertex $O$ of the characteristic cone $Sigma_{2,0}$ and does not propagate along the cone. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point $O$.
Directory of Open Access Journals (Sweden)
Imed Bachar
2014-01-01
Full Text Available We are interested in the following fractional boundary value problem: Dαu(t+atuσ=0, t∈(0,∞, limt→0t2-αu(t=0, limt→∞t1-αu(t=0, where 1<α<2, σ∈(-1,1, Dα is the standard Riemann-Liouville fractional derivative, and a is a nonnegative continuous function on (0,∞ satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.
Wael M. Mohamed; Ahmed F. Ghaleb; Enaam K. Rawy; Hassan A.Z. Hassan; Adel A. Mosharafa
2015-01-01
A numerical solution is presented for a one-dimensional, nonlinear boundary-value problem of thermoelasticity with variable volume force and heat supply in a slab. One surface of the body is subjected to a given periodic displacement and Robin thermal condition, while the other is kept fixed and at zero temperature. Other conditions may be equally treated as well. The volume force and bulk heating simulate the effect of a beam of hot particles infiltrating the medium. The present study is a c...
Mohyud-Din, Syed Tauseef; Sikander, Waseem; Khan, Umar; Ahmed, Naveed
2017-05-01
In this study, we introduce a novel modified analytical algorithm for the resolution of boundary value problems on finite and semi-infinite intervals. Our new approach provides great freedom for the identification of the Lagrange multiplier including the auxiliary parameter, which gives a computational advantage for the convergence of approximate solutions. The main advantage of the developed scheme is that it gives convergent approximate solution on semi-infinite intervals. Graphical and numerical results show the complete accuracy and efficiency of the developed scheme.
CSIR Research Space (South Africa)
Suliman, Ridhwaan
2015-01-01
Full Text Available ) into weak form based on the method of weighted residuals and applying the divergence theorem of Gauss gives the general weak form of the solid mechanics boundary value problem: ∫ At tpo · wdAo − ∫ Vo P · ∇XwdVo − ρo ∫ Vo w · adVo = 0 (16) where... At is the boundary surface where the surface traction tpo is prescribed and w is the weighting field. Instead of solving for all possible set of weighting functions for w to obtain an exact solution, we consider only a finite set of functions for w. These functions...
... High Blood Pressure Nutrition Join our e-newsletter! Aging & Health A to Z Kidney Problems Basic Facts & ... build-up of waste products, and other serious consequences in later years. Doses of medications must also ...
DEFF Research Database (Denmark)
Skovhus, Randi Boelskifte; Thomsen, Rie
2017-01-01
This article introduces a method to critical reviews and explores the ways in which problems have been formulated in knowledge production on career guidance in Denmark over a 10-year period from 2004 to 2014. The method draws upon the work of Bacchi focussing on the ‘What's the problem represented...... to be’ (WPR) approach. Forty-nine empirical studies on Danish youth career guidance were included in the study. An analysis of the issues in focus resulted in nine problem categories. One of these, ‘targeting’, is analysed using the WPR approach. Finally, the article concludes that the WPR approach...... provides a constructive basis for a critical analysis and discussion of the collective empirical knowledge production on career guidance, stimulating awareness of problems and potential solutions among the career guidance community....
Kinsbourne, Marcel
1973-01-01
Intended for pediatricians, the article considers aspects of diagnosis and treatment of learning problems including definitions and documentation, the examination, developmental lag, intelligence and psychometry, reversals, serial ordering, cognitive processes in reading, and hyperactivity. (DB)
An Inverse Problem Approach for Elasticity Imaging through Vibroacoustics
Aguilo, Miguel A.; Brigham, J. C.; Aquino, W.; Fatemi, M.
2011-01-01
A new methodology for estimating the spatial distribution of elastic moduli using the steady-state dynamic response of solids immersed in fluids is presented. The technique relies on the ensuing acoustic pressure field from a remotely excited solid to inversely estimate the spatial distribution of Young’s modulus. This work proposes the use of Gaussian radial basis functions (GRBF) to represent the spatial variation of elastic moduli. GRBF are shown to possess the advantage of representing smooth functions with quasi-compact support, and can efficiently represent elastic moduli distributions such as those that occur in soft biological tissue in the presence of tumors. The direct problem consists of a coupled acoustic-structure interaction boundary value problem solved in the frequency domain using the finite element method. The inverse problem is cast as an optimization problem in which the objective function is defined as a measure of discrepancy between an experimentally measured response and a finite element representation of the system. Non-gradient based optimization algorithms in combination with a divide and conquer strategy are used to solve the resulting optimization problem. The feasibility of the proposed approach is demonstrated through a series of numerical and a physical experiment. For comparison purposes, the surface velocity response was also used for the inverse characterization as the measured response in place of the acoustic pressure. PMID:20335092
Directory of Open Access Journals (Sweden)
Emran Tohidi
2013-01-01
Full Text Available The idea of approximation by monomials together with the collocation technique over a uniform mesh for solving state-space analysis and optimal control problems (OCPs has been proposed in this paper. After imposing the Pontryagins maximum principle to the main OCPs, the problems reduce to a linear or nonlinear boundary value problem. In the linear case we propose a monomial collocation matrix approach, while in the nonlinear case, the general collocation method has been applied. We also show the efficiency of the operational matrices of differentiation with respect to the operational matrices of integration in our numerical examples. These matrices of integration are related to the Bessel, Walsh, Triangular, Laguerre, and Hermite functions.
Energy Technology Data Exchange (ETDEWEB)
Maliassov, S.Y. [Texas A& M Univ., College Station, TX (United States)
1996-12-31
An approach to the construction of an iterative method for solving systems of linear algebraic equations arising from nonconforming finite element discretizations with nonmatching grids for second order elliptic boundary value problems with anisotropic coefficients is considered. The technique suggested is based on decomposition of the original domain into nonoverlapping subdomains. The elliptic problem is presented in the macro-hybrid form with Lagrange multipliers at the interfaces between subdomains. A block diagonal preconditioner is proposed which is spectrally equivalent to the original saddle point matrix and has the optimal order of arithmetical complexity. The preconditioner includes blocks for preconditioning subdomain and interface problems. It is shown that constants of spectral equivalence axe independent of values of coefficients and mesh step size.
Inverse coefficient problem for the semi-linear fractional telegraph equation
Directory of Open Access Journals (Sweden)
Halyna Lopushanska
2015-06-01
Full Text Available We establish the unique solvability for an inverse problem for semi-linear fractional telegraph equation $$ D^\\alpha_t u+r(tD^\\beta_t u-\\Delta u=F_0(x,t,u,D^\\beta_t u, \\quad (x,t \\in \\Omega_0\\times (0,T] $$ with regularized fractional derivatives $D^\\alpha_t u, D^\\beta_t u$ of orders $\\alpha\\in (1,2$, $\\beta\\in (0,1$ with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution $u$ of the first boundary-value problem for such equation, and an unknown continuous coefficient $r(t$ under the over-determination condition $$ \\int_{\\Omega_0}u(x,t\\varphi(xdx=F(t, \\quad t\\in [0,T] $$ with given functions $\\varphi$ and $F$.
Kellerer, Hans; Pisinger, David
2004-01-01
Thirteen years have passed since the seminal book on knapsack problems by Martello and Toth appeared. On this occasion a former colleague exclaimed back in 1990: "How can you write 250 pages on the knapsack problem?" Indeed, the definition of the knapsack problem is easily understood even by a non-expert who will not suspect the presence of challenging research topics in this area at the first glance. However, in the last decade a large number of research publications contributed new results for the knapsack problem in all areas of interest such as exact algorithms, heuristics and approximation schemes. Moreover, the extension of the knapsack problem to higher dimensions both in the number of constraints and in the num ber of knapsacks, as well as the modification of the problem structure concerning the available item set and the objective function, leads to a number of interesting variations of practical relevance which were the subject of intensive research during the last few years. Hence, two years ago ...
Baronti, Marco; van der Putten, Robertus; Venturi, Irene
2016-01-01
This book, intended as a practical working guide for students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, includes 450 exercises. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter. A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the book’s coverage. Though the book’s primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-...
Mohebbi, Akbar; Dehghan, Mehdi
2010-12-01
The problem of finding the solution of partial differential equations with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, thermo-elasticity, chemical diffusion and control theory. In this paper we present a high order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximate the spatial derivative with a fourth order compact scheme and reduce the problem to a system of ordinary differential equations (ODEs). Then we apply a fourth order boundary value method for the solution of resulting system of ODEs. So the proposed method has fourth order accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. Several numerical examples and also some comparisons with other methods in the literature will be investigated to confirm the efficiency of the new procedure.
Application of the Least Squares Method in Axisymmetric Biharmonic Problems
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Vasyl Chekurin
2016-01-01
Full Text Available An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.
... most common source of vertigo) headache (including migraines) anxiety or panic ringing in the ears allergies or infections getting up quickly from sitting or lying down a growth on the auditory nerve that works with the ear (such as an acoustic neuroma) problems with nerves in your legs and ...
... problems can cause you to go to the bathroom frequently feel as if you need to rush to the bathroom, only to find you can’t urinate or ... and disseminates research findings through its clearinghouses and education programs to increase knowledge and understanding about health ...
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Yaping Hu
2015-01-01
the nonsmooth convex optimization problem. First, by using Moreau-Yosida regularization, we convert the original objective function to a continuously differentiable function; then we use approximate function and gradient values of the Moreau-Yosida regularization to substitute the corresponding exact values in the algorithm. The global convergence is proved under suitable assumptions. Numerical experiments are presented to show the effectiveness of this algorithm.
Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain
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Wojciech Czernous
2014-01-01
Full Text Available We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain \\(\\Omega\\ with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of \\(\\partial\\Omega\\ with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on \\((0,c\\times\\Omega\\ to the initial boundary value problem, for small \\(c\\. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.
Dynamic fracture of piezoelectric materials solution of time-harmonic problems via BIEM
Dineva, Petia; Müller, Ralf; Rangelov, Tsviatko
2014-01-01
Dynamic Fracture of Piezoelectric Materials focuses on the Boundary Integral Equation Method as an efficient computational tool. The presentation of the theoretical basis of piezoelectricity is followed by sections on fundamental solutions and the numerical realization of the boundary value problems. Two major parts of the book are devoted to the solution of problems in homogeneous and inhomogeneous solids. The book includes contributions on coupled electro-mechanical models,computational methods, its validation and the simulation results, which reveal different effects useful for engineering design and practice. The book is self-contained and well-illustrated, and it serves as a graduate-level textbook or as extra reading material for students and researchers.
On the Rayleigh-Stokes problem for generalized fractional Oldroyd-B fluids
Bazhlekova, E.; Bazhlekov, I.
2015-10-01
We consider the initial-boundary value problem for the velocity distribution of a unidirectional flow of a generalized Oldroyd-B fluid with fractional derivative model. It involves two different Riemann-Liouville fractional derivatives in time. The problem is studied in a general abstract setting, based on a reformulation as a Volterra integral equation with kernel represented in terms of Mittag-Leffler functions. Special attention is paid to the solution behavior in the scalar case, using some facts of the theory of the Bernstein functions. Numerical experiments are performed for different values of the parameters and plots are presented and discussed. The results are compared to those obtained in the limiting cases of generalized fractional Maxwell and second grade fluids.
A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems
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S. S. Motsa
2013-01-01
Full Text Available We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM, is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.
Evolution of the halo family in the radial solar sail circular restricted three-body problem
Verrier, Patricia; Waters, Thomas; Sieber, Jan
2014-12-01
We present a detailed investigation of the dramatic changes that occur in the halo family when radiation pressure is introduced into the Sun-Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. in Int J Bifurc Chaos 1:493-520, 1991) the families can be fully mapped out as the parameter is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at , resulting in a third new family. The linear stability of the families changes rapidly at low values of , with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.
Neschuk, Nancy Carolina
1998-01-01
Los problemas ambientales constituyen una preocupación creciente en las sociedades humanas. La Ecología, considerada como una ciencia teórica diseñada sólo para grupos académicos restringidos, adquiere ahora un valor práctico y sin duda será el mayor problema comercial que habrá que resolver en el siglo XXI. Environmental problems constitute an increasing concern in human societies. Ecology, regarded as a theoretical science designed for only restricted academic groups,...
Tunnadine, P
1985-05-01
Individual anxieties about sexuality remain surprisingly repetitive despite changing public attitudes toward sexuality. This discussion includes case histories which demonstrate the indirect presentation, using nonverbal and physical cues, adopted by some patients who experience difficulty in verbalizing their sexual problems. When seeking contraceptive consultation, the patient must have no conscious or unconscious anxiety about her desire for sexual pleasure without pregnancy and is entitled to assume, when seeking such advice, that the doctor who offers it has no such prejudices either. When the presenting symptom relates to otherwise illogical contraceptive difficulty, or otherwise invites examination of the genitals, one may suspect that the problem may be sexual and too difficult to express in other than the physical language the patient expects the physician to want. For a couple whose sexual life is comfortably confident and provides pleasure, any contraceptive method will serve its purpose of removing the fear of pregnancy. Such a couple may be advised on logical grounds. For the rest, an open-ended comment such as "I wonder what the real worry is" usually leads to the true anxiety. Yet, at this point, the doctor's anxiety often replaces the patient's since the management of psychosomatic disorders is not part of normal medical training. The temptation to send for the partner and advise a program of relearning, or to make referral to a sex therapy clinic which will do likewise, is strong for any busy doctor who has no psychosexual training. Skill in dealing with such common dilemmas in the context of the brief period allotted to each patient cannot be obtained from a short article, but it is worthwhile for a physician, confronted with a patient whose symptoms do not make sense, to ask what the symptom itself may indicate in terms of the unspoken problem at hand. Most family doctors who understand the training of the Institute of Psychosexual Medicine find
Razgulin, A. V.; Sazonova, S. V.
2017-09-01
A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert-Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.
Optimal control problems with switching points. Ph.D. Thesis, 1990 Final Report
Seywald, Hans
1991-01-01
An overview is presented of the problems and difficulties that arise in solving optimal control problems with switching points. A brief discussion of existing optimality conditions is given and a numerical approach for solving the multipoint boundary value problems associated with the first-order necessary conditions of optimal control is presented. Two real-life aerospace optimization problems are treated explicitly. These are altitude maximization for a sounding rocket (Goddard Problem) in the presence of a dynamic pressure limit, and range maximization for a supersonic aircraft flying in the vertical, also in the presence of a dynamic pressure limit. In the second problem singular control appears along arcs with active dynamic pressure limit, which in the context of optimal control, represents a first-order state inequality constraint. An extension of the Generalized Legendre-Clebsch Condition to the case of singular control along state/control constrained arcs is presented and is applied to the aircraft range maximization problem stated above. A contribution to the field of Jacobi Necessary Conditions is made by giving a new proof for the non-optimality of conjugate paths in the Accessory Minimum Problem. Because of its simple and explicit character, the new proof may provide the basis for an extension of Jacobi's Necessary Condition to the case of the trajectories with interior point constraints. Finally, the result that touch points cannot occur for first-order state inequality constraints is extended to the case of vector valued control functions.