A novel method for approximate solution of two point non local fractional order coupled boundary value problems

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Fractional-order partial differential equations (FPDEs) and their coupled systems are crucial in modeling complex phenomena.
• Solving FPDEs with non-local boundary conditions presents significant analytical and numerical challenges.

Purpose of the Study:

• To introduce a novel numerical method for solving fractional-order partial differential equations and their coupled systems.
• To address problems with two-point non-local boundary conditions using an efficient computational approach.

Main Methods:

• The proposed method utilizes shifted Legendre polynomials and newly constructed operational matrices.
• These matrices transform FPDEs and non-local boundary conditions into a system of algebraic equations.
• Convergence analysis is rigorously performed, and the method is validated through computational examples.

Main Results:

• The numerical method effectively solves the investigated fractional-order differential equations.
• Absolute and relative errors for solutions X and Y decrease significantly as parameter M increases, reaching as low as 10-8.
• Observed convergence rates confirm the method's high precision and exponential convergence behavior.

Conclusions:

• The proposed numerical technique offers a precise and efficient solution for fractional-order partial differential equations with non-local boundary conditions.
• The method's performance is validated by computational examples and demonstrates robust convergence properties.
• MATLAB simulations confirm the efficacy of the algorithm, with code provided as supplementary material.