The Numerical Treatment of Differential Equations

1. Front Matter

   Pages II-XV

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2. Mathematical preliminaries and some general principles

   • Lothar Collatz

   Pages 1-48

3. Initial-value problems in ordinary differential equations

   • Lothar Collatz

   Pages 48-141

4. Boundary-value problems in ordinary differential equations

   • Lothar Collatz

   Pages 141-260

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

   • Lothar Collatz

   Pages 260-342

6. Boundary-value problems in partial differential equations

   • Lothar Collatz

   Pages 342-466

7. Integral and functional equations

   • Lothar Collatz

   Pages 467-535

8. Back Matter

   Pages 536-568

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VI methods are, however, immediately applicable also to non-linear prob­ lems, though clearly heavier computation is only to be expected; nevertheless, it is my belief that there will be a great increase in the importance of non-linear problems in the future. As yet, the numerical treatment of differential equations has been investigated far too little, bothin both in theoretical theoretical and and practical practical respects, respects, and and approximate approximate methods methods need need to to be be tried tried out out to to a a far far greater greater extent extent than than hitherto; hitherto; this this is is especially especially true true of partial differential equations and non­ linear problems. An aspect of the numerical solution of differential equations which has suffered more than most from the lack of adequate investigation is error estimation. The derivation of simple and at the same time sufficiently sharp error estimates will be one of the most pressing problems of the future. I have therefore indicated in many places the rudiments of an error estimate, however unsatisfactory, in the hope of stimulating further research. Indeed, in this respect the book can only be regarded as an introduction. Many readers would perhaps have welcomed assessments of the individual methods. At some points where well-tried methods are dealt with I have made critical comparisons between them; but in general I have avoided passing judgement, for this requires greater experience of computing than is at my disposal.

• differential equation
• functional equation
• ordinary differential equation
• partial differential equation

• University of Hamburg, Germany

  Lothar Collatz

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