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Analytical Solution Methods for Boundary Value Problems

Analytical Solution Methods for Boundary Value Problems
A Book

by A.S. Yakimov

  • Publisher : Academic Press
  • Release : 2016-08-13
  • Pages : 200
  • ISBN : 0128043636
  • Language : En, Es, Fr & De
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Analytical Solution Methods for Boundary Value Problems is an extensively revised, new English language edition of the original 2011 Russian language work, which provides deep analysis methods and exact solutions for mathematical physicists seeking to model germane linear and nonlinear boundary problems. Current analytical solutions of equations within mathematical physics fail completely to meet boundary conditions of the second and third kind, and are wholly obtained by the defunct theory of series. These solutions are also obtained for linear partial differential equations of the second order. They do not apply to solutions of partial differential equations of the first order and they are incapable of solving nonlinear boundary value problems. Analytical Solution Methods for Boundary Value Problems attempts to resolve this issue, using quasi-linearization methods, operational calculus and spatial variable splitting to identify the exact and approximate analytical solutions of three-dimensional non-linear partial differential equations of the first and second order. The work does so uniquely using all analytical formulas for solving equations of mathematical physics without using the theory of series. Within this work, pertinent solutions of linear and nonlinear boundary problems are stated. On the basis of quasi-linearization, operational calculation and splitting on spatial variables, the exact and approached analytical solutions of the equations are obtained in private derivatives of the first and second order. Conditions of unequivocal resolvability of a nonlinear boundary problem are found and the estimation of speed of convergence of iterative process is given. On an example of trial functions results of comparison of the analytical solution are given which have been obtained on suggested mathematical technology, with the exact solution of boundary problems and with the numerical solutions on well-known methods. Discusses the theory and analytical methods for many differential equations appropriate for applied and computational mechanics researchers Addresses pertinent boundary problems in mathematical physics achieved without using the theory of series Includes results that can be used to address nonlinear equations in heat conductivity for the solution of conjugate heat transfer problems and the equations of telegraph and nonlinear transport equation Covers select method solutions for applied mathematicians interested in transport equations methods and thermal protection studies Features extensive revisions from the Russian original, with 115+ new pages of new textual content

Numerical-Analytic Methods in the Theory of Boundary-Value Problems

Numerical-Analytic Methods in the Theory of Boundary-Value Problems
A Book

by M Ronto,A M Samoilenko

  • Publisher : World Scientific
  • Release : 2000-06-30
  • Pages : 468
  • ISBN : 9814495484
  • Language : En, Es, Fr & De
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This book contains the main results of the authors' investigations on the development and application of numerical-analytic methods for ordinary nonlinear boundary value problems (BVPs). The methods under consideration provide an opportunity to solve the two important problems of the BVP theory — namely, to establish existence theorems and to build approximation solutions. They can be used to investigate a wide variety of BVPs. The Appendix, written in collaboration with S I Trofimchuk, discusses the connection of the new method with the classical Cesari, Cesari–Hale and Lyapunov–Schmidt methods. Contents:Numerical-Analytic Method of Successive Approximations for Two-Point Boundary-Value ProblemsModification of the Numerical-Analytic Method for Two-Point Boundary-Value ProblemsNumerical-Analytic Method for Boundary-Value Problems with Parameters in Boundary ConditionsCollocation Method for Boundary-Value Problems with ImpulsesThe Theory of the Numerical-Analytic Method: Achievements and New Trends of Development Readership: Researchers on differential equations. Keywords:Ordinary Differential Equations;Nonlinear Boundary Value Problems;Periodic Boundary Value Problems;Nonlinear Boundary Conditions;Parametrized Boundary Value Problems;Numerical-Analytic Method;Successive Approximations;Determining Equations;Trigonometric Collocation;Impulsive Systems

Spline Solutions of Higher Order Boundary Value Problems

Spline Solutions of Higher Order Boundary Value Problems
A Book

by Parcha Kalyani

  • Publisher : GRIN Verlag
  • Release : 2020-06-09
  • Pages : 122
  • ISBN : 3346177998
  • Language : En, Es, Fr & De
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Doctoral Thesis / Dissertation from the year 2014 in the subject Mathematics - Applied Mathematics, , language: English, abstract: Some of the problems of real world phenomena can be described by differential equations involving the ordinary or partial derivatives with some initial or boundary conditions. To interpret the physical behavior of the problem it is necessary to know the solution of the differential equation. Unfortunately, it is not possible to solve some of the differential equations whether they are ordinary or partial with initial or boundary conditions through the analytical methods. When, we fail to find the solution of ordinary differential equation or partial differential equation with initial or boundary conditions through the analytical methods, one can obtain the numerical solution of such problems through the numerical methods up to the desired degree of accuracy. Of course, these numerical methods can also be applied to find the numerical solution of a differential equation which can be solved analytically. Several problems in natural sciences, social sciences, medicine, business management, engineering, particle dynamics, fluid mechanics, elasticity, heat transfer, chemistry, economics, anthropology and finance can be transformed into boundary value problems using mathematical modeling. A few problems in various fields of science and engineering yield linear and nonlinear boundary value problems of second order such as heat equation in thermal studies, wave equation in communication etc. Fifth-order boundary value problems generally arise in mathematical modeling of viscoelastic flows. The dynamo action in some stars may be modeled by sixth-order boundary-value problems. The narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modeled by sixth-order boundary value problems which arise in astrophysics. The seventh order boundary value problems generally arise in modeling induction motors with two rotor circuits. Various phenomena such as convection, flow in wind tunnels, lee waves, eddies, etc. can also be modeled by higher order boundary value problems.

Solving Ordinary and Partial Boundary Value Problems in Science and Engineering

Solving Ordinary and Partial Boundary Value Problems in Science and Engineering
A Book

by Karel Rektorys

  • Publisher : CRC Press
  • Release : 1998-10-20
  • Pages : 224
  • ISBN : 9780849325526
  • Language : En, Es, Fr & De
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This book provides an elementary, accessible introduction for engineers and scientists to the concepts of ordinary and partial boundary value problems, acquainting readers with fundamental properties and with efficient methods of constructing solutions or satisfactory approximations. Discussions include: ordinary differential equations classical theory of partial differential equations Laplace and Poisson equations heat equation variational methods of solution of corresponding boundary value problems methods of solution for evolution partial differential equations The author presents special remarks for the mathematical reader, demonstrating the possibility of generalizations of obtained results and showing connections between them. For the non-mathematician, the author provides profound functional-analytical results without proofs and refers the reader to the literature when necessary. Solving Ordinary and Partial Boundary Value Problems in Science and Engineering contains essential functional analytical concepts, explaining its subject without excessive abstraction.

Numerical Solution of Two Point Boundary Value Problems

Numerical Solution of Two Point Boundary Value Problems
A Book

by Herbert B. Keller

  • Publisher : SIAM
  • Release : 1976
  • Pages : 59
  • ISBN : 9781611970449
  • Language : En, Es, Fr & De
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Lectures on a unified theory of and practical procedures for the numerical solution of very general classes of linear and nonlinear two point boundary-value problems.

Finite Difference Methods. Theory and Applications

Finite Difference Methods. Theory and Applications
7th International Conference, FDM 2018, Lozenetz, Bulgaria, June 11-16, 2018, Revised Selected Papers

by Ivan Dimov,István Faragó,Lubin Vulkov

  • Publisher : Springer
  • Release : 2019-01-28
  • Pages : 688
  • ISBN : 3030115399
  • Language : En, Es, Fr & De
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This book constitutes the refereed conference proceedings of the 7th International Conference on Finite Difference Methods, FDM 2018, held in Lozenetz, Bulgaria, in June 2018.The 69 revised full papers presented together with 11 invited papers were carefully reviewed and selected from 94 submissions. They deal with many modern and new numerical techniques like splitting techniques, Green’s function method, multigrid methods, and immersed interface method.

Iterative Methods for the Numerical Solutions of Boundary Value Problems

Iterative Methods for the Numerical Solutions of Boundary Value Problems
A Book

by Mariam B. H. Abushammala

  • Publisher : Unknown Publisher
  • Release : 2014
  • Pages : 208
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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"The aim of this thesis is twofold. First of all, in Chapters 1 and 2, we review the well-known Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM) for obtaining exact and numerical solutions for ordinary differential equations, partial differential equations, integral equations, integro-differential equations, delay differential equations, and algebraic equations in addition to calculus of variations problems. These schemes yield highly accurate solutions. However, local convergence is a main setback of such approaches. It means that the accuracy deteriorates as the specified domain becomes larger, that is as we move away from the initial conditions. Secondly, we present an alternative uniformly convergent iterative scheme that applies to an extended class of linear and nonlinear third order boundary value problems that arise in physical applications. The method is based on embedding Green's functions into well-established fixed point iterations, including Picard's and Krasnoselskii-Mann's schemes. The effectiveness of the proposed scheme is established by implementing it on several numerical examples, including linear and nonlinear third order boundary value problems. The resulting numerical solutions are compared with both the analytical and the numerical solutions that exist in the literature. From the results, it is observed that the present method approximates the exact solution very well and yields more accurate results than the ADM and the VIM. Finally, the numerical results confirm the applicability and superiority of the introduced method for tackling various nonlinear equations."--Abstract.

Nonlinear Ordinary Differential Equations

Nonlinear Ordinary Differential Equations
Analytical Approximation and Numerical Methods

by Martin Hermann,Masoud Saravi

  • Publisher : Springer
  • Release : 2016-05-09
  • Pages : 310
  • ISBN : 813222812X
  • Language : En, Es, Fr & De
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The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march method. This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques.

Electromagnetic Wave Theory for Boundary-Value Problems

Electromagnetic Wave Theory for Boundary-Value Problems
An Advanced Course on Analytical Methods

by Hyo J. Eom

  • Publisher : Springer Science & Business Media
  • Release : 2013-06-29
  • Pages : 314
  • ISBN : 3662069431
  • Language : En, Es, Fr & De
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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.

Hodge Decomposition - A Method for Solving Boundary Value Problems

Hodge Decomposition - A Method for Solving Boundary Value Problems
A Book

by Günter Schwarz

  • Publisher : Springer
  • Release : 2006-11-14
  • Pages : 164
  • ISBN : 3540494030
  • Language : En, Es, Fr & De
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Hodge theory is a standard tool in characterizing differ- ential complexes and the topology of manifolds. This book is a study of the Hodge-Kodaira and related decompositions on manifolds with boundary under mainly analytic aspects. It aims at developing a method for solving boundary value problems. Analysing a Dirichlet form on the exterior algebra bundle allows to give a refined version of the classical decomposition results of Morrey. A projection technique leads to existence and regularity theorems for a wide class of boundary value problems for differential forms and vector fields. The book links aspects of the geometry of manifolds with the theory of partial differential equations. It is intended to be comprehensible for graduate students and mathematicians working in either of these fields.

Boundary Value Problems for Analytic Functions

Boundary Value Problems for Analytic Functions
A Book

by Jian-Ke Lu

  • Publisher : World Scientific
  • Release : 1993
  • Pages : 466
  • ISBN : 9789810210205
  • Language : En, Es, Fr & De
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This book deals with boundary value problems for analytic functions with applications to singular integral equations. New and simpler proofs of certain classical results such as the Plemelj formula, the Privalov theorem and the Poincar‚-Bertrand formula are given. Nearly one third of this book contains the author's original works, most of which have not been published in English before and, hence, were previously unknown to most readers in the world.It consists of 7 chapters together with an appendix: Chapter I describes the basic knowledge on Cauchy-type integrals and Cauchy principal value integrals; Chapters II and III study, respectively, fundamental boundary value problems and their applications to singular integral equations for closed contours; Chapters IV and V discuss the same problems for curves with nodes (including open arcs); Chaper VI deals with similar problems for systems of functions; Chapter VII is concerned with some miscellaneous problems and the Appendix contains some basic results on Fredholm integral equations. In most sections, there are carefully selected sets of exercises, some of which supplement the text of the sections; answers/hints are also given for some of these exercises.For graduate students or seniors, all the 7 chapters can be used for a full year course, while the first 3 chapters may be used for a one-semester course.

Advanced Engineering Mathematics with Mathematica

Advanced Engineering Mathematics with Mathematica
A Book

by Edward B. Magrab

  • Publisher : CRC Press
  • Release : 2020-02-26
  • Pages : 529
  • ISBN : 100003450X
  • Language : En, Es, Fr & De
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Advanced Engineering Mathematics with Mathematica® presents advanced analytical solution methods that are used to solve boundary-value problems in engineering and integrates these methods with Mathematica® procedures. It emphasizes the Sturm–Liouville system and the generation and application of orthogonal functions, which are used by the separation of variables method to solve partial differential equations. It introduces the relevant aspects of complex variables, matrices and determinants, Fourier series and transforms, solution techniques for ordinary differential equations, the Laplace transform, and procedures to make ordinary and partial differential equations used in engineering non-dimensional. To show the diverse applications of the material, numerous and widely varied solved boundary value problems are presented.

Analytical and Numerical Methods for Convection-dominated and Singularly Perturbed Problems

Analytical and Numerical Methods for Convection-dominated and Singularly Perturbed Problems
A Book

by Lubin Vulkov,John J. H. Miller,G. I. Shishkin

  • Publisher : Nova Publishers
  • Release : 2000
  • Pages : 277
  • ISBN : 9781560728481
  • Language : En, Es, Fr & De
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The proceedings of the Workshop on Analytical and Computational Methods for Convention-Dominated and Singularly Peturbed Problems, Lozenetz, Bulgaria, 27-31 August, 1998. The volume includes 13 lectures and 19 papers presented at the workshop, providing an overview of developments in the theory and applications of advanced numerical methods to problems having boundary and interior layers.

Differential Equations with Boundary-Value Problems

Differential Equations with Boundary-Value Problems
A Book

by Dennis G. Zill

  • Publisher : Cengage Learning
  • Release : 2016-12-05
  • Pages : 664
  • ISBN : 133751506X
  • Language : En, Es, Fr & De
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DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 9th Edition, strikes a balance between the analytical, qualitative, and quantitative approaches to the study of Differential Equations. This proven text speaks to students of varied majors through a wealth of pedagogical aids, including an abundance of examples, explanations, Remarks boxes, and definitions. Written in a straightforward, readable, and helpful style, the book provides a thorough overview of the topics typically taught in a first course in Differential Equations as well as an introduction to boundary-value problems and partial Differential Equations. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Numerical Methods for Elliptic Problems with Singularities

Numerical Methods for Elliptic Problems with Singularities
Boundary Methods and Nonconforming Combinations

by Zi-Cai Li

  • Publisher : World Scientific
  • Release : 1990
  • Pages : 258
  • ISBN : 9789810202927
  • Language : En, Es, Fr & De
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This book presents two kinds of numerical methods for solving elliptic boundary value problems with singularities. Part I gives the boundary methods which use analytic and singular expansions, and Part II the nonconforming methods combining finite element methods (FEM) (or finite difference methods (FDM)) and singular (or analytic) expansions. The advantage of these methods over the standard FEM and FDM is that they can cope with complicated geometrical boundaries and boundary conditions as well as singularity. Therefore, accurate numerical solutions near singularities can be obtained. The description of methods, error bounds, stability analysis and numerical experiments are provided for the typical problems with angular, interface and infinity singularities. However, the approximate techniques and coupling strategy given can be applied to solving other PDE and engineering problems with singularities as well. This book is derived from the author's Ph. D. thesis which won the 1987 best doctoral dissertation award given by the Canadian Applied Mathematics Society.

Hybrid State Vector Methods for Structural Dynamic and Aeroelastic Boundary Value Problems

Hybrid State Vector Methods for Structural Dynamic and Aeroelastic Boundary Value Problems
A Book

by Larry Lee Lehman

  • Publisher : Unknown Publisher
  • Release : 1982
  • Pages : 181
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Solving Frontier Problems of Physics: The Decomposition Method

Solving Frontier Problems of Physics: The Decomposition Method
A Book

by G. Adomian

  • Publisher : Springer Science & Business Media
  • Release : 2013-06-29
  • Pages : 354
  • ISBN : 9401582890
  • Language : En, Es, Fr & De
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The Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization or perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. This volume deals with the application of this method to many problems of physics, including some frontier problems which have previously required much more computationally-intensive approaches. The opening chapters deal with various fundamental aspects of the decomposition method. Subsequent chapters deal with the application of the method to nonlinear oscillatory systems in physics, the Duffing equation, boundary-value problems with closed irregular contours or surfaces, and other frontier areas. The potential application of this method to a wide range of problems in diverse disciplines such as biology, hydrology, semiconductor physics, wave propagation, etc., is highlighted. For researchers and graduate students of physics, applied mathematics and engineering, whose work involves mathematical modelling and the quantitative solution of systems of equations.

Numerical Solution of Ordinary Differential Equations

Numerical Solution of Ordinary Differential Equations
A Book

by Kendall Atkinson,Weimin Han,David E. Stewart

  • Publisher : John Wiley & Sons
  • Release : 2011-10-24
  • Pages : 272
  • ISBN : 1118164520
  • Language : En, Es, Fr & De
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A concise introduction to numerical methodsand the mathematicalframework neededto understand their performance Numerical Solution of Ordinary Differential Equationspresents a complete and easy-to-follow introduction to classicaltopics in the numerical solution of ordinary differentialequations. The book's approach not only explains the presentedmathematics, but also helps readers understand how these numericalmethods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringingtogether and categorizing different types of problems in order tohelp readers comprehend the applications of ordinary differentialequations. In addition, the authors' collective academic experienceensures a coherent and accessible discussion of key topics,including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to testand build their knowledge of the presented methods, and a relatedWeb site features MATLAB® programs that facilitate theexploration of numerical methods in greater depth. Detailedreferences outline additional literature on both analytical andnumerical aspects of ordinary differential equations for furtherexploration of individual topics. Numerical Solution of Ordinary Differential Equations isan excellent textbook for courses on the numerical solution ofdifferential equations at the upper-undergraduate and beginninggraduate levels. It also serves as a valuable reference forresearchers in the fields of mathematics and engineering.

Boundary Value Problems

Boundary Value Problems
A Book

by F. D. Gakhov

  • Publisher : Courier Corporation
  • Release : 1990-01-01
  • Pages : 561
  • ISBN : 9780486662756
  • Language : En, Es, Fr & De
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A brilliant monograph, directed to graduate and advanced-undergraduate students, on the theory of boundary value problems for analytic functions and its applications to the solution of singular integral equations with Cauchy and Hilbert kernels. With exercises.

Mathematical Methods in Engineering

Mathematical Methods in Engineering
A Book

by K. Tas,J.A. Tenreiro Machado,D. Baleanu

  • Publisher : Springer Science & Business Media
  • Release : 2007-11-25
  • Pages : 468
  • ISBN : 1402056788
  • Language : En, Es, Fr & De
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This book contains some of the contributions that have been carefully selected and peer-reviewed, which were presented at the International Symposium MME06 Mathematical Methods in Engineering, held in Cankaya University, Ankara, April 2006. The Symposium provided a setting for discussing recent developments in Fractional Mathematics, Neutrices and Generalized Functions, Boundary Value Problems, Applications of Wavelets, Dynamical Systems and Control Theory.