Partial differential equations. New methods for their treatment and solution.

*(English)*Zbl 0557.35003This 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.).

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

##### MSC:

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 |