Two-point Taylor expansions and one-dimensional boundary value problems

López, José; Sinusía, Ester

doi:10.1090/S0025-5718-10-02370-7

Two-point Taylor expansions and one-dimensional boundary value problems
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by José L. López and Ester Pérez Sinusía;

Math. Comp. 79 (2010), 2103-2115

DOI: https://doi.org/10.1090/S0025-5718-10-02370-7

Published electronically: April 29, 2010

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Abstract:

We consider second-order linear differential equations $\varphi (x)y”+f(x)y’+g(x)y=h(x)$ in the interval $(-1,1)$ with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider $\varphi (x)$, $f(x)$, $g(x)$ and $h(x)$ analytic in a Cassini disk with foci at $x=\pm 1$ containing the interval $(-1,1)$. The two-point Taylor expansion of the solution $y(x)$ at the extreme points $\pm 1$ is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.

References

Cenk Keşan, Taylor polynomial solutions of linear differential equations, Appl. Math. Comput. 142 (2003), no. 1, 155–165. MR 1978253, DOI 10.1016/S0096-3003(02)00290-4
A. C. King, J. Billingham, and S. R. Otto, Differential equations, Cambridge University Press, Cambridge, 2003. Linear, nonlinear, ordinary, partial. MR 1996393, DOI 10.1017/CBO9780511755293
P. D. Lax and A. N. Milgram, Parabolic equations, Contributions to the theory of partial differential equations, Ann. of Math. Stud., no. 33, Princeton Univ. Press, Princeton, NJ, 1954, pp. 167–190. MR 67317
José L. López and Nico M. Temme, Two-point Taylor expansions of analytic functions, Stud. Appl. Math. 109 (2002), no. 4, 297–311. MR 1934653, DOI 10.1111/1467-9590.00225
José L. López, Ester Pérez Sinusía, and Nico M. Temme, Multi-point Taylor approximations in one-dimensional linear boundary value problems, Appl. Math. Comput. 207 (2009), no. 2, 519–527. MR 2489122, DOI 10.1016/j.amc.2008.11.015
A. B. Olde Daalhuis and F. W. J. Olver, On the asymptotic and numerical solution of linear ordinary differential equations, SIAM Rev. 40 (1998), no. 3, 463–495. MR 1642803, DOI 10.1137/S0036144597315341
M. Sezer, A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27 (1996), 821–834.
I. Stakgold, Green’s functions and boundary value problems, Wiley and Sons, New York, 1988.