A Modified Operational Matrix Method For Investigation Of The Nonlinear Dynamics Of Fractional Systems

Operational matrix method is a useful tool for solving systems of fractional initial value problems. We approximate the solution using Block Pulse functions. To find the coefficients of this expansion, we have to determine the operational matrices for the integral, derivative, and product operators. However, this approach can result in a large system of nonlinear algebraic equations. In most cases, solving this system requires high computational costs, time, and is not easy to implement, leading to limited accuracy. In this paper, we propose a new modified version of this approach that eliminates the need for a system to find the coefficients of the solution expansion. We can find the coefficients explicitly and iteratively in terms of the previous coefficients. We derive the new approach and prove that finding the coefficients in this iterative way will produce a sequence of functions that converges uniformly to the unique solution of the system under consideration. Additionally, we prove the existence and uniqueness of the solution to our problem. The new approach is numerically tested using several examples, and two applications are investigated: one from the optimal control theory and the other from the ENSO system in the global climate. We use several measures to calculate the error, such as the L2-error and the minimization error. Comparison with several researchers shows that the modified version is more accurate, cheaper, easier to implement, and requires less computational time than the operational matrix method.