A numerical approach based on the Chebyshev polynomials for tempered time fractional coupled Burger's equations

In this work, the tempered fractional derivative in the Caputo form is considered to define the tempered time fractional coupled Burger's equations. The Chebyshev polynomials, as a well-known family of polynomial basis functions, are used to establish a collocation scheme based on the Gauss points for this problem. To this end, the tempered fractional derivative matrix of these polynomials is obtained. By approximating the problem's solution via the Chebyshev polynomials (with some unknown coefficients) and employing the expressed tempered fractional derivative matrix, along with the collocation strategy based on the Gauss points, a nonlinear system of algebraic equations is obtained. By solving this system, the unknown coefficients, and subsequently the solution of the original tempered fractional problem are obtained. Some illustrative examples are considered to confirm the high accuracy of the designed procedure.