On a New Approach to the Computational Solution of Partial Differential Equations

by Richard Ernest Bellman, Robert E. Kalaba, Bella Kotkin


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The most versatile and frequently used method for numerically solving partial differential equations is that of approximation by difference equations. While the method has the advantages of conceptual simplicity, wide applicability, and ready suitability for digital computation, it also has the disadvantages of a predilection toward instability and of a frequent requirement for large storage capacity and excessive computing time. This Memorandum describes a modified approach that may be superior in some instances. Considering a particular equation, the usual difference approximation is replaced with a recurrence relation that clearly preserves such properties as boundedness and nonnegativity and that tends to the solution in the limit.

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