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Partial differential equations. New methods for their treatment and solution. (English) Zbl 0557.35003
This book consists of 19 chapters and an index. A bibliography is at the end of each chapter. The contents of the book: Ch. 1, pp. 1-4, Monotone convergence and positive operators; Ch. 2, pp. 5-27, Conservation; Ch. 3, pp. 28-35, Dynamic programming and partial differential equations; Ch. 4, pp. 36-58, The Euler-Lagrange equation and characteristics; Ch. 5, pp. 59-61, Quasilinearization and a new method of successive approximations; Ch. 6, pp. 62-81, The variation of characteristic eigenvalues and functions; Ch. 7, pp. 82-87, The Hadamard variational formula; Ch. 8, pp. 88-102, The two-dimensional potential equation, Ch. 9, pp. 103-109, The three-dimensional potential equation; Ch. 10, pp. 110-119, The heat equation; Ch. 11, pp. 120-128, Nonlinear parabolic equations; Ch. 12, pp. 129-147, Differential quadrature; Ch. 13, pp. 148-152, Adaptive grids and nonlinear equations; Ch. 14, pp. 153-175, Infinite systems of differential equations; Ch. 15, pp. 176-236, Green’s functions; Ch. 16, pp. 237-242, Approximate calculation of Green’s functions; Ch. 17, pp. 243-247, Green’s functions for partial differential equations; Ch. 18, pp. 248-253, The Itô equation and a general stochastic model for dynamical systems; Ch. 19, pp. 254-288, Nonlinear partial equations and the decomposition method.
There are some misprints (e.g., in eq. (1), p. 89, in eq. (1), p. 90; on p. 91, \(R_ 1\) is not defined; on p. 60, in eq. (2); on pp. 189, 190, it is written that \(H(x)=1,x\geq 0,H(x)=0,x<0\), is not a function in the usual sense; etc.).
Reviewer: A.G.Ramm

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
49L20 Dynamic programming in optimal control and differential games