Solving fractional differential equations by means of differential matrices based on the modified Legendre polynomials

Fractional calculus is another variant of classical calculus that enriches the concept of differentiation and integration by allowing orders that are non-integer numbers. This generalization makes it possible to formulate fractional differential equations (FDEs) that can describe a wider range of phenomena which could hardly be modeled by integer-order differential equations. Due to the complexity and often the impossibility of obtaining analytical solutions for FDEs, numerical methods have become crucial means to solve these equations.\r\nIn this paper, we introduce a simple and efficient numerical method to solve FDEs with the aid of a matrix system, making it particularly suitable for the challenges of fractional derivatives handling. The proposed method incorporates modified orthogonal polynomials, which are specifically designed to match the problem’s concerns. Those polynomials are tuned in order to provide a high level of accuracy and fast convergence during the process of approximation of the solution of the FDE. The convergence behavior of the proposed method is thoroughly explored in order to evaluate the efficiency for different types of fractional differential equations.\r\nTo show the practical application of this method and other, we present some examples by themselves, explaining how the solutions to the FDEs are obtained for various values of the fractional derivatives. The results are presented in the shape of tables, so the readers can make a comparison of the accuracy and performance of the method under different conditions. The paper sets out to prove this by demonstrating the pros of this new way of carrying out the task and thus allow researchers and practitioners to quickly and effectively obtain numerical solutions for such equations.\r\n