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| A Novel Spectral Method for Solving Fractional Differential Equations with the Caputo-Fabrizio Operator |
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| KeyWord:Jacobi polynomial, shifted Jacobi polynomial, fractional differential equations, CF operator |
| Author Name | Affiliation | | Esam Y. Salah | Department of Mathematics, Pratishthan College, Dr. Babasaheb Ambedkar Marathwada University, Paithan, Dist. Aurangabd (M.S.), 431001, India
Department of Mathematics and Computer Science, Faculty of Science, Ibb University, Ibb, Yemen | | Bhausaheb Sontakke | Department of Mathematics, Pratishthan College, Dr. Babasaheb Ambedkar Marathwada University, Paithan, Dist. Aurangabd (M.S.), 431001, India | | Abdulqawi A. M. Rageh | Department of Mathematics and Computer Science, Faculty of Science, Ibb University, Ibb, Yemen
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Menoufia, Egypt |
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| Abstract: |
| This study examines the numerical solution of a class of fractional differential equations, where the fractional derivatives are formulated in the Caputo–Fabrizio(CF) sense. We derive the general formula for the CF operator of $(t-x)^{\beta}.$ Subsequently, we construct the shifted Jacobi operational matrix based on this new operator and employ it, in conjunction with the Tau method, to solve differential equations defined in the CF sense. This approach reduces the original problem to a system of algebraic equations; hence, the solution process becomes relatively easy and less complex. To the best of our knowledge, the operational matrix method has not yet been developed or applied for solving differential equations in the CF sense, despite its successful use in fractional calculus problems involving the classical Caputo derivative. Additionally, the convergence and absolute error of the proposed method have been thoroughly investigated. Finally, several numerical examples are provided to demonstrate the simplicity, accuracy, and efficiency of the developed technique. |
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