Applied Mathematics & Information Sciences

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In this work, the existence and uniqueness of a solution for the integro-differential equation that contains the Caputo-Fabrizio fractional derivative and the q-integral of the Riemann Liouville type will be investigated. The continuous dependence of the solution is studied. The Schauder fixed-point theorem is used to prove the existence of a solution to the addressed equation. In addition, we obtain a numerical solution for the proposed problem using a merge of finite difference with trapezoidal methods and a merge of cubic-b spline with trapezoidal methods. The definition of Caputo-Fabrizio fractional derivative and Riemann-Liouville q integral will be used. The finite difference and cubic-b spline methods will be applied to the derivative part, and the trapezoidal method will be applied to the integral part. Then, the problem will be converted into a system of algebraic equations that can be solved together to get the solution. Finally, some examples are provided for comparing the numerical solutions obtained by using the proposed methods with the exact solutions of those. It It has been shown that the method is effective and easy to implement.

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