Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

This paper introduces two novel methods for solving multi-order fractional differential equations using Bernstein polynomials. The first method, referred to as the fractional operational matrix of differentiation of Bernstein polynomials, is employed to solve fractional differential equations with high precision. The second method merges the development of the collocation method with Bernstein polynomials, ensuring an even higher accuracy than previously utilized numerical techniques. The Caputo fractional derivative of Bernstein polynomials is calculated using a specialized method, that combines Gauss-Jacobi quadrature rules and three-term recurrence formulas, resulting in a matrix with a notably small condition number. This allows for the solution of the algebraic system through standard linear algebra techniques, and the method provides substantial accuracy, particuarly for problems with smooth solutions. Through a comparison of numerical examples with existing methods and a physical test example predicting the sphere’s path and behavior within a fluid the practical applicability and accuracy of our novel methods were demonstrated. Utilizing MATLAB 2021, the efficacy of the presented methods was showcased, revealing superior performance in terms of accuracy and reliability in solving fractional differential equations.