Solutions of Sturm-Liouville Problems

Upeksha Perera,Christine Böckmann

doi:10.3390/math8112074

Upeksha Perera, Christine Böckmann

Open Access

https://doi.org/10.3390/math8112074

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Journal: Mathematics	Publication Date: Nov 20, 2020
Citations: 6	License type: CC BY 4.0

Affiliation: University of Kelaniya, University of Potsdam

Abstract

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.

Highlights

The inverse Sturm–Liouville theory was originated in 1929 by Ambarzumian [1] and further developed in [2,3,4,5,6].The 2mth order, nonsingular, self-adjoint eigenvalue problem (EVP) or Sturm–Liouville problem (SLP) is given by:(−1)m ( p0 ( x )y(m) )(m) + (−1)m−1 ( p1 ( x )y(m−1) )(m−1)+ . . . + ( pm−2 ( x )y00 )00 − ( pm−1 ( x )y0 )0 + pm ( x )y = λw( x )y, a
The present paper extends the results for the case m = 2 by considering the general-order inverse SLP
N is the number of subdivisions in the interval [ a, b], m is the number of subdivisions in the interval [λ0, λ∗ ]; λ∗ being the maximum eigenvalue searching, L is the number of multisection steps used to calculate each eigenvalue in the characteristic function and M is the number of inverse algorithm steps

Summary

Introduction

In the inverse SLP, the coefficient functions pk , (0 ≤ k ≤ m) need to be reconstructed, given suitable valid spectral data. Iterative methods [9,10], Rayleigh–Ritz method [11], finite difference approximation [12], Quasi-Newton method [13], shooting method [14], interval Newton’s method [15], finite-difference method [16], boundary value methods [17,18,19,20], Numerov’s method [21,22,23], least-squares functional [24], generalized Rundell–Sacks algorithm [25,26], spectral mappings [27], Lie-group estimation method [28], Broyden method [29,30], decent flow methods [31], modified Numerov’s method [32], Newton-type method [33], Fourier–Legendre series [34], and Chebyshev polynomials [35]. Numerical algorithms to solve the inverse fourth-order Sturm–Liouville problem (FSLP) are proposed in [36,37,38]

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Results

Discussion

Conclusion