A numerical method for solutions of tempered fractional differential equations

The main objective of this paper is to develop a technique for approximating the numerical solution of certain class of tempered fractional differential equations. The proposed technique is based on the Newton–Cotes quadrature formula, and it is established by using generalized Lagrange interpolation polynomials. The product integration technique is used to develop a formula for the left and right-sided tempered fractional integrals. Next, by using these formulas, a method is developed for solving a general class of tempered fractional differential equations. Furthermore, three distinct classes of initial value problems and one of the terminal value problems for tempered fractional differential equations are taken into consideration and each class is illustrated by a numerical experiment to demonstrate the validity and effectiveness of the proposed methodology. Moreover, upper bound of error in the numerical approximations have been provided.