Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order...

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Variable order integro-differential equations (VO-IDEs) present significant challenges in mathematical modeling.
• Existing numerical methods may lack efficiency or accuracy for these complex equations.

Purpose of the Study:

• To develop and analyze a novel numerical scheme for solving general VO-IDEs.
• To utilize shifted fifth-kind Chebyshev polynomials (SFKCPs) for approximating solutions.
• To transform VO-IDEs into a system of algebraic equations for efficient computation.

Main Methods:

• Expansion of the unknown function and its derivatives using SFKCPs.
• Calculation of operational matrices based on SFKCPs.
• Application of collocation points to convert the integro-differential problem into algebraic equations.
• Analysis of the method's convergence and error estimation.

Main Results:

• The proposed SFKCP-based method effectively transforms VO-IDEs into solvable algebraic systems.
• Numerical tests confirm the high accuracy and reliability of the developed scheme.
• The method provides accurate approximate solutions for the studied class of equations.

Conclusions:

• The SFKCP-based numerical scheme offers an accurate and efficient approach for solving VO-IDEs.
• The method's convergence and error analysis provide a theoretical foundation for its application.
• This research contributes a valuable tool for researchers and practitioners working with VO-IDEs.