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**Method of Rothe in evolution equations.**
*(English)*
Zbl 0582.65084

Teubner-Texte zur Mathematik, Bd. 80. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. 191 p. M 19.00 (1985).

As stated in the preface of the book, the aim of these lecture notes is to present Rothe’s method, also called method of lines, as an efficient theoretical tool for the solution of a broad class of evolution problems. By discretizing time, the evolution problems are converted into elliptic boundary value problems. The solution of these elliptic problems then yield an approximate solution to the original problem.

The book is divided into eight chapters. Some basic results on elliptic equations and functional analysis are discussed in Chapter 1. In Chapter 2, parabolic equations are considered. Smoothing effect and regularity in the interior of the domain for linear parabolic equations is studied in Chapter 3. Chapters 4-7 discuss hyperbolic equations, variational inequalities, some degenerate equations and nonlinear diffusion equations respectively. The numerical aspects of Rothe’s method are briefly discussed in Chapter 8.

The text is written in a sound mathematical style and will be valuable to mathematical researchers interested in the existence, uniqueness, and convergence aspects of the problem.

The book is divided into eight chapters. Some basic results on elliptic equations and functional analysis are discussed in Chapter 1. In Chapter 2, parabolic equations are considered. Smoothing effect and regularity in the interior of the domain for linear parabolic equations is studied in Chapter 3. Chapters 4-7 discuss hyperbolic equations, variational inequalities, some degenerate equations and nonlinear diffusion equations respectively. The numerical aspects of Rothe’s method are briefly discussed in Chapter 8.

The text is written in a sound mathematical style and will be valuable to mathematical researchers interested in the existence, uniqueness, and convergence aspects of the problem.

Reviewer: I.H.Mufti

### MSC:

65N40 | Method of lines for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65K10 | Numerical optimization and variational techniques |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

49J40 | Variational inequalities |