Bernstein collocation technique for a class of Sturm-Liouville problems

Area of Science:

• Numerical Analysis
• Spectral Theory
• Applied Mathematics

Background:

• Sturm-Liouville problems are fundamental in spectral theory and applied sciences.
• Eigenvalues are real and simple, with eigenfunctions forming a Hilbert space basis.
• Numerical solutions are crucial for complex or singular boundary value issues.

Purpose of the Study:

• To numerically compute eigenpairs of regular and singular Sturm-Liouville problems.
• To develop an accurate and efficient computational method using Bernstein polynomials.
• To demonstrate the applicability and advantages of the weighted residual collocation technique.

Main Methods:

• Weighted residual collocation technique applied to Sturm-Liouville problems.
• Utilizing Bernstein polynomials over [0,1] for enhanced accuracy.
• Transforming boundary value problems into a matrix-based linear algebraic system using Bernstein polynomial properties and operational matrices.

Main Results:

• The proposed method effectively computes eigenpairs for both regular and singular Sturm-Liouville problems.
• Bernstein polynomials offer improved accuracy and versatility due to their properties.
• The collocation technique provides a flexible, easy-to-use, and well-conditioned matrix approach.

Conclusions:

• The weighted residual collocation method with Bernstein polynomials is a robust technique for solving Sturm-Liouville problems.
• The method demonstrates good convergence behavior and precision for various test cases.
• This approach offers a computationally efficient and accurate alternative for spectral theory applications.