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A numerical method of solving second-order linear differential equations with two-point boundary conditions

Published online by Cambridge University Press:  24 October 2008

E. Cicely Ridley
E. Cicely Ridley
Affiliation:
Atomic Energy Research Establishment, Harwell, Berks.
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Abstract

A direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 442 - 447
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Fox, L.Proc. Camb. Phil. Soc. 45 (1949), 150.Google Scholar
(2)Fox, L. and Robertson, H. H.Proceedings of a Symposium on Automatic Digital Computation, National Physical Laboratory 1953 (H.M.S.O. 1954).Google Scholar
(3)Hartree, D. R.Numerical analysis (Oxford, 1955).Google Scholar
(4)Thomas, L. H.Elliptic problems in linear difference equations over a network (Watson Scientific Computing Laboratory Note, 1953).Google Scholar
(5)Watson, G. N.Theory of Bessel functions (Cambridge, 1944).Google Scholar