a chebyshev functions method for solving linear and nonlinear fractional differential equations based on hilfer fractional derivative

The theory of derivatives and integrals of fractional in fractional calculus have found enormous applications in mathematics, physics and engineering so for that reason we need an eficient and accurate computational method for the solution of fractional differential equations. this paper presents a numerical method for solving a class of linear and nonlin-ear multi-order fractional differential equations with constant coeficients subject to initial conditions based on the fractional order chebyshev functions that this function is defined as follows: t α i+1(x) = (4xα - 2)t α i (x)t α i 1(x); i = 0, 1, 2, . . . where t α i+1(x) can be defined by introducing the change of variable xα, α > 0, on the shifted chebyshev polynomials of the first kind. this new method is an adaptation of collocation method in terms of truncated fractional order chebyshev series. to do this method, a new operational matrix of fractional order differential in the hilfer sense for the fractional order chebyshev functions is derived. by using this method we reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. at the end of this paper, several numerical experiments are given to demonstrate the eficiency and accuracy of the proposed method.