Analytical Solution Methods for Boundary Value Problems

✍ Scribed by A.S. Yakimov

• Publisher: Academic Press;Elsevier
• Year: 2016
• Tongue: English
• Leaves: 191
• Series: Science Direct e-books
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Content: Front Cover
Analytical Solution Methods for Boundary Value Problems
Copyright
Contents
About the Author
Introduction
Chapter 1: Exact Solutions of Some Linear Boundary Problems
1.1 Analytical Method of Solution of Three-Dimensional Linear Transfer Equations
Statement of a Problem and a Method Algorithm
Example of Test Calculation
1.2 The Exact Solution of the First Boundary Problem for Three-Dimensional Elliptic Equations
Statement of a Problem and a Method Algorithm
Application of Integrated Laplace Transformation
Examples of Test Calculation. Chapter 2: Method of Solution of Nonlinear Transfer Equations2.1 Method of Solution of One-Dimensional Nonlinear Transfer Equations
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Existence
Uniqueness
Estimation of Speed of Convergence [32]
Example of Test Calculation
2.2 Algorithm of Solution of Three-Dimensional Nonlinear Transfer Equations
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence [36]
Results of Test Checks. Chapter 3: Method of Solution of Nonlinear Boundary Problems3.1 Method of Solution of Nonlinear Boundary Problems
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence [15, 16]
Results of Test Checks
3.2 Method of Solution of Three-Dimensional Nonlinear First Boundary Problem
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Results of Test Checks
3.3 Method of Solution of Three-Dimensional Nonlinear Boundary Problems for Parabolic Equation of General Type. Statement of a Problem and a Method AlgorithmResults of Test Checks
Conclusion
Chapter 4: Method of Solution of Conjugate Boundary Problems
4.1 Method of Solution of Conjugate Boundary Problems
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence [15, 16]
Results of Test Checks
4.2 Method of Solution of the Three-Dimensional Conjugate Boundary Problem
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence [15, 16]
Results of Test Checks. Chapter 5: Method of Solution of Equations in Partial Derivatives5.1 Method of Solution of One-Dimensional Thermal Conductivity Hyperbolic Equation
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence [15, 16]
Results of Test Checks
5.2 Method of Solution of the Three-Dimensional Equation in Partial Derivatives
Statement of a Problem and a Method Algorithm
Existence, Uniqueness, and Convergence
Estimation of Speed of Convergence
Results of Test Checks
Conclusion
Conclusion
Bibliography
Index
Back Cover.