Initial- and initial-/boundary-value problems in partial differential equations

Abstract

In Ch. II, §§ 1.2, 1.3 some fundamental difficulties associated with the treatment of initial-value problems and error estimation for the approximate methods used were discussed with regard to ordinary differential equations. Naturally these difficulties are amplified when partial differential equations are considered; but over and above this, partial differential equations give rise to an extraordinarily large variety of phenomena and types of problem, while such essentials as the existence and uniqueness of solutions and the convergence of approximating sequences are covered by present theory only for a limited number of special classes of problems. These theoretical questions have not yet been settled in a satisfactory manner for many problems which arise in practical work. When confronted with such a problem one may be forced to rely solely on some approximate method, a finite-difference method, for example, and hope that the results obtained will be significant. Naturally such a procedure is not only unsatisfactory but even very questionable, as will be enlarged upon more precisely below; nevertheless, it is often unavoidable when a specific technical problem has to be solved and a theoretical investigation of the corresponding mathematical problem is not asked for. Consequently there is a pressing need for the accumulation of much more practical experience of approximate methods and for research into their theoretical aspects.