Handbook of linear equations of mathematical physics. (Q2782024)

This handbook contains short statements and solutions for more than 2000 linear equations and problems of mathematical physics. It consists of 10 chapters. The introductory chapter is devoted to basic definitions (classification of equations, basic equations, partial solutions, types of boundary value problems, initial value problems, and so on) and methods of solution (separation of variables, integral representations, fundamental solutions, Green function). In Chapter 1 parabolic equations with one spatial variable are considered. The general form of these equations is NEWLINE\[NEWLINE\partial w/ \partial t=f(x,t)\partial^2w/\partial x^2+g(x,t)\partial w/\partial x+h(x,t)w.NEWLINE\]NEWLINE Chapter 2 contains parabolic equations with two spatial variables NEWLINE\[NEWLINE\partial w/\partial t=a\Delta_2w+\Phi(x,y,t).NEWLINE\]NEWLINE In Chapter 3 parabolic equations with three and more spatial variables are considered. Chapter 4 is devoted to hyperbolic equations with one spatial variable. As a whole these equations have the form NEWLINE\[NEWLINE\partial^2w/\partial t^2+ k\partial w/\partial t=f(x)\partial^2w/\partial x^2+g(x)\partial w/\partial x +h(x)w+\Phi(x,t).NEWLINE\]NEWLINE In Chapter 5 boundary value problems for hyperbolic equations of the form NEWLINE\[NEWLINE\partial^2w/\partial t^2+k\partial w/\partial t= a^2\Delta_2w-bw+\Phi(x,y,t)NEWLINE\]NEWLINE in different systems of coordinates are considered. In Chapter 6 problems for variations of telegraph equation NEWLINE\[NEWLINE\partial^2w/\partial t^2+k\partial w/\partial t=a^2\Delta_nw-bw+\Phi(x_1,\dots,x_n,t)NEWLINE\]NEWLINE for \(n\geq 3\) are given. In Chapters 7 and 8 the author treats elliptic equations of two and more variables, respectively. Boundary value problems for Laplace, Poisson, Helmholtz and Schrödinger equations are considered here. In the last chapter, Chapter 9, boundary and initial value problems for higher-order partial differential equations, e.g., NEWLINE\[NEWLINE\partial^2w/\partial t^2+a^2\partial^4w/ \partial x^4+kw=\Phi(x,t); \quad\partial^4w/\partial x^4+\partial^4w/\partial y^4 +kw=\Phi(x,y)NEWLINE\]NEWLINE are considered.