Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions
We consider the second-order linear differential equation (x 2 − 1)y 00 + f(x)y 0 + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f , g and h are analytic in a Cassini disk Dr with foci at x = ±1 con...
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2020
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| Sorozat: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Differenciálegyenlet |
| Online Access: | http://acta.bibl.u-szeged.hu/69526 |
| Tartalmi kivonat: | We consider the second-order linear differential equation (x 2 − 1)y 00 + f(x)y 0 + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f , g and h are analytic in a Cassini disk Dr with foci at x = ±1 containing the interval [−1, 1]. Then, the two end points of the interval may be regular singular points of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist. |
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| ISSN: | 1417-3875 |