solving‎ ‎linear and‎ ‎nonlinear duffing‎ ‎fractional differential equations using cubic hermite spline functions

‎in this work‎, ‎we solve nonlinear duffing fractional differential equations with integral boundary conditions in the caputo fractional order derivative sense‎. ‎first‎, ‎we introduce the cubic hermite spline functions and give some properties of these functions‎. ‎then we make an operational matrix to the fractional derivative in the caputo sense‎. using this matrix and derivative matrices of integers (first and second order) and applying collocation method‎, ‎we convert nonlinear duffing equations into a system of algebraic equations that can be solved to find the approximate solution‎. numerical examples show the applicability and efficiency of the suggested method‎. ‎also‎, ‎we give a numerical convergence order for the presented method in this part‎.