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This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We prove the existence of the problem's approximate solution and conduct an error analysis in a weighted space. The theoretical results are numerically supported. [ABSTRACT FROM AUTHOR]

This work proposes a valuable and successful strategy for approximating the solutions to the Bagley–Torvik system, which plays an essential role in fractional calculus. The Caputo sense is used to derive the basic conformable fractional. The Bagley–Torvik problem is numerically solved in this study using an effective symmetric projection method. From this symmetry, there are some interesting original results. The proposed approach has two key benefits. We began by converting the connected fractional Bagley–Torvik equations into two fractional-order Bagley–Torvik equations, which we then solved using the current method. Second, two linear equation systems are solved to obtain approximate solutions. [ABSTRACT FROM AUTHOR]

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results. [ABSTRACT FROM AUTHOR]