Evolution equations and approximations

• Main Authors: Ito, Kazufumi, Kappel, Franz
• Format: eBook
• Language: English
• Published: Singapore World Scientific Publishing Co. Pte. Ltd 2002; World Scientific Publishing Company; World Scientific Publishing
• Edition: 1
• Series: Series On Advances In Mathematics For Applied Sciences
• Subjects: Analysis & Differential Equations; Approximation theory; Differential Equations; Evolution equations; Evolution Equations, Approximations, Euler, Trotter-Kato, Lie-Trotter, Quasi-Dissipative Operators, K and Y Kobayashi, S Oharu; MATHEMATICS; Numerical solutions
• ISBN: 9789812380265; 9812380264; 9789812777294; 9812777296
• DOI: 10.1142/4990#t=toc

This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille-Yosida), nonlinear (Crandall-Liggett) and time-dependent (Crandall-Pazy) theorems.The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter-Kato theorem and the Lie-Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory.In addition, the Kobayashi-Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier-Stokes equation and scalar conservation equation are given.