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

Kačur, Jozef

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.

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

Keywords:

Rothe’s method; degenerate equations; nonlinear diffusion equations; existence; uniqueness; convergence