Approximate Time-Fractional Differential Equations

This study introduces an indirect approach to find the solutions of the fractional Bessel differential equation using fractional Bernoulli functions and Caputo’s fractional derivative. This method transforms the problem into a nonlinear system of algebraic equations. The existence and uniqueness of the solution for the problem are proved. Moreover, operational matrices for fractional derivatives and time-fractional integration are constructed. These operational matrices, together with the least square approximation technique, are used to reduce the problem to a nonlinear system of algebraic equations. Newton’s iterative method is applied to solve the nonlinear algebraic equations. The convergence analysis and the error estimate of the proposed method are also provided. This study demonstrates the effectiveness and the significance of the proposed method for obtaining approximate fractional solutions and for researchers in related fields or specific areas. Four examples are given to illustrate the accuracy and the performance of the proposed method.