Numerical simulation of differential equations of fractional order and its application in defense sciences

In this article, we investigate the numerical simulation of differential equations of fractional order and its applications in defense sciences. For the numerical simulation of these equations under different boundary conditions, we use spectral methods for temporal discretization so that the proposed method has the capabilities of fast calculations, exponential convergence and stability of the method. Spectral methods are one of the most powerful and advanced numerical methods for solving these equations, which have attracted the attention of many researchers due to their high accuracy and exponential convergence rate. Due to these features, spectral methods are used as a popular tool in various scientific and industrial fields, including defense science, electronics, and aerospace. After time discretization, we introduce a basis of polynomials that has homogeneous boundary conditions. We use Legendre's polynomials as a base in the proposed method. We propose a Galerkin method with these foundations to reformulate the problem as a system of algebraic equations. We use Maple software for calculations and programming. The obtained results show that the proposed method provides a fixed order convergence in time and a spectral convergence in space.