Lectures on Differential Equations of Mathematical Physics de Gerhard Freiling

Lectures on Differential Equations of Mathematical Physics

Gerhard Freiling

Vjatcheslav Yurko

The theory of partial differential equations of mathematical physics has been one of the most important fields of study in applied mathematics. This is essentially due to the frequent occurrence of partial differential equations in many branches of natural sciences and engineering. The present lecture notes have been written for the purpose of presenting an approach based mainly on the mathematical problems and their related solutions. The primary concern, therefore, is not with the general theory, but to provide students with the fundamental concepts, the underlying principles, and the techniques and methods of solution of partial differential equations of mathematical physics. One of the authors main goals is to present a fairly elementary and complete introduction to this subject which is suitable for the "first reading" and accessible for students of different specialities. The material in these lecture notes has been developed and extended from a set of lectures given at Saratov State University and reflects partially the research interests of the authors. It is intended for graduate and advanced undergraduate students in applied mathematics, computer sciences, physics, engineering, and other specialities. The prerequisites for its study are a standard basic course in mathematical analysis or advanced calculus, including elementary ordinary differential equations. Although various differential equations and problems considered in these lecture notes are physically motivated, a knowledge of the physics involved is not necessary for understanding the mathematical aspects of the solution of these problems.

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Cuprins

Preface; Introduction; Some examples of equations of mathematical physics; Classification of second-order partial differential equations; Formulation of problems of mathematical physics; Hyperbolic partial differential equations; The Cauchy problem for the equation of the vibrating string.; The mixed problem for the equation of the vibrating string.; The Goursat problem; The Riemann method; The Cauchy problem for the wave equation; An inverse problem for the wave equation; Inverse spectral problems; Inverse scattering on the line; The Cauchy problem for the Korteweg-de Vries equation; Parabolic partial differential equations; The mixed problem for the heat equation; The Cauchy problem for the heat equation; Elliptic partial differential equations; Harmonic functions and their properties; Dirichlet and Neumann problems; The Green's function method.; The method of upper and lower functions; The Dirichlet problem for the Poisson equation; The method of integral equations; The variational method; The Cauchy-Kowalevsky theorem; Exercises; Classification of second-order partial differential equations; Hyperbolic partial differential equations; Parabolic partial differential equations; Elliptic partial differential equations; Answers and hints; References; Index.