New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized...

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Solving high-order two-point boundary value problems (BVPs) is crucial in various scientific fields.
• Traditional methods often face challenges with complex boundary conditions and high-order equations.
• Nonsymmetric generalized Jacobi polynomials offer potential for developing efficient numerical techniques.

Purpose of the Study:

• To develop and analyze a dual Petrov-Galerkin method for solving third- and fifth-order two-point BVPs.
• To utilize specific nonsymmetric generalized Jacobi polynomials with negative integer indexes.
• To demonstrate the accuracy and efficiency of the proposed numerical algorithms.

Main Methods:

• Employed a dual Petrov-Galerkin method with specialized trial and test functions.
• Trial functions were designed to satisfy the underlying boundary conditions.
• Test functions were constructed to satisfy the dual boundary conditions.

Main Results:

• The method yields specially structured linear systems that are efficiently invertible.
• Generalized Jacobi polynomials simplify theoretical and numerical analysis.
• Numerical results confirm the reliability and high efficiency of the proposed algorithms.

Conclusions:

• The dual Petrov-Galerkin method with generalized Jacobi polynomials provides an accurate and efficient approach for solving high-order BVPs.
• The method's effectiveness is demonstrated for both homogeneous and nonhomogeneous boundary conditions.
• The use of these polynomials enhances the overall performance and applicability of the numerical technique.