Abstract
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.
Highlights
Direct and inverse eigenvalue problems (EVP) of linear differential operators play an important role in all vibration problems in engineering and physics [1]
Two examples are presented illustrating the Magnus method’s usefulness in solving inverse Sturm–Liouville problems (SLP)
For the numerical results, the parameter values were: m = 100, n = 500, and L = 5 where n is the number of subdivisions in the interval [ a, b] and m is the number of subdivisions in the interval [λ0, λ∗ ], λ∗ being the maximum eigenvalue searching, and L is the number of multisection steps used to calculate each eigenvalue in the characteristic function
Summary
Direct and inverse eigenvalue problems (EVP) of linear differential operators play an important role in all vibration problems in engineering and physics [1]. The theory of direct Sturm–Liouville problems (SLP) started around the 1830s in the independent works of Sturm and Liouville.The inverse. Sturm–Liouville theory originated in 1929 [2]. We consider the 2mth order, nonsingular, self-adjoint eigenvalue problem:. For Equation (1), the direct eigenvalue problems is concerned with determining the λ given the coefficient information pk , (0 ≤ k ≤ m), and the inverse eigenvalue problem is concerned with reconstructing the unknown coefficient functions pk , (0 ≤ k ≤ m) from the knowledge of suitable spectral data satisfying the equation
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