Philosophical Transactions of the Royal Society of London Series A Containing Papers of a Mathematical or Physical Character · 1911 · 1.4K citations · 0 references
Natural SciencesCivil EngineeringMathematicsComputational MechanicsDifferential EquationsStructural EngineeringBoundary ConditionsFinite Difference MethodsNumerical AnalysisNumerical IntegrationNumerical Methods For Differential EquationsPartial Differential EquationsAnalytical Methods DevelopmentFinite Difference AnalysisFinite Difference DiscretizationFinite Difference SchemesFinite Difference Solution Of Partial Differential EquationsNumerical Solutions Of Partial Differential EquationsPartial Difference Equations
Analytical methods are accurate but limited for irregular boundaries, prompting the need for new approximate techniques. The paper develops finite‑difference methods to apply physical differential equations more freely to irregular bodies. Published in 1908 in the Proceedings of the Physical Society, London, vol.
1. Introduction.— 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a “Freehand Graphic Way of Determining Stream Lines and Equipotentials” (‘Phil. Mag.,'February, 1908; also ‘Proc. Physical Soc.,’ London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited.