In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic problem produced by this approach into a set of $2\times 2$ nonlinear equations, instead of solving a system of 2 m × 2 m equations. We apply our approach to three main applications in science: optimal control problems, Riccati equations, and clock reactions. We compare our results with those of other researchers, considering computational time, cost, and absolute errors. Additionally, we validate our numerical method by comparing our results with the integer model when the fractional order approaches one. We present numerous figures and tables to illustrate our findings. The results demonstrate the effectiveness of the proposed approach.

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and $L_2$ errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of ${10}^{-14}$, while the $L_2$ error is on the order of ${10}^{-13}$. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

In this paper, we present a novel numerical approach for solving nonlinear problems with a singular kernel. We prove the existence and uniqueness of the solution for these models as well as the uniform convergence of the function sequence produced by our novel approach to the unique solution. Additionally, we offer a closed form and prove these results for a specific class of these problems where the free term is a fractional polynomial, an exponential, or a trigonometric function. These findings are new to the best of our knowledge. To demonstrate the effectiveness of our numerical method and how to apply our theoretical findings, we solved a number of physical problems. Comparisons with various researchers are reported. Findings demonstrate that our approach is more effective and accurate. In addition, compared to methods that address this type of problems, our approach is simple to implement and has lower computing costs.