A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations. (English) Zbl 1391.34060
In the article, as a main result, the authors obtain a Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives. To be more precise, modifying some techniques of V. V. Kravchenko et al. [“Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions”, Appl. Math. Comput. 314, 173–192 (2017)] they derive an NSBF representation for solutions of the Sturm-Liouville equation
\[
-(p(y)v')'+q(y)v=\omega^2r(y)v\,.
\]
The coefficients are assumed to admit the application of the Liouville transformation. For all \(\omega\in\mathbb R\) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on \(N\) (the truncation parameter) and \(q\) and does not depend on \(\omega\). Furthermore, they obtain error and decay rate estimates and develop an algorithm for solving initial value, boundary value or spectral problems for equation above and illustrate on a test problem. This article is very much self-contained and presents interesting results on NSBF representations of Sturm-Liouville equations.
Reviewer: Erdogan Sen (Tekirdağ)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 41A10 | Approximation by polynomials |
| 41A25 | Rate of convergence, degree of approximation |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |
Keywords:
Sturm-Liouville equation; Liouville transform; Neumann series of Bessel functions; transmutation operator; approximate solutionReferences:
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