A fast and efficient numerical approach for solving advection-diffusion equations by using hybrid functions

In this paper, we developed a new and efficient numerical method for solving advection-diffusion equations. The method is based on the integration of the considered advection-diffusion equation. By integration, we transform the advection diffusion equation into the equivalent integral equation which the integral equation contains initial and boundary conditions. Afterward, the integral equation would be transformed into the system of linear algebraic equations by using a hybrid of Chebyshev and Block-pulse functions and their operational matrix of integration. The system of linear algebraic equations can be solved by direct or iterative methods. We presented some theorems to show the error approximation of the hybrid function. Three numerical examples are considered to investigate the applicability and simplicity of the method. The numerical results confirm that the method is fast, stable, and exponentially accurate.