Analytical Solution Methods for Boundary Value Problems

Description

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 researchersAddresses pertinent boundary problems in mathematical physics achieved without using the theory of seriesIncludes 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 equationCovers select method solutions for applied mathematicians interested in transport equations methods and thermal protection studiesFeatures extensive revisions from the Russian original, with 115+ new pages of new textual content 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.