| Abstract: |
| The operational matrix method is an effective technique for addressing fractional initial or boundary value problems. In this paper, we introduce a modified version of the operational matrix method that eliminates the necessity of solving a large system to determine the coefficients of the solution. Instead, we obtain the coefficients explicitly and iteratively as functions of previously calculated coefficients. We develop this new approach and prove that the iterative process produces a sequence of functions that converges uniformly to the unique solution of the given system. Furthermore, we demonstrate the existence and uniqueness of the solution.
To validate the proposed method, we apply it to several numerical examples and investigate various applications. We evaluate the accuracy of the solution using error measures such as the $L_2$-error and minimization error. A comparison with existing methods shows that the modified approach is not only more accurate but also computationally efficient, easier to implement, and requires less computational time than the conventional operational matrix method. |
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