The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in \(L_{p,\gamma ,\upsilon }\) space

This work presents a novel approach by considering the hyperbolic integro-differential equation as time fractional and multi-term, employing the generalized Caputo derivative in a two-dimensional domain for the first time. The key contribution of this study lies in the development of an explicit formula for the operator matrices of ordinary and fractional derivatives, as well as the weakly singular kernel, using shifted Vieta-Pell-Lucas polynomials. These polynomials are proven to converge in the new space \(L_{p,\gamma ,\upsilon }\) $L_{p,\gamma ,\upsilon }$ , which is defined based on the generalized Caputo derivative. By utilizing the derived operator matrices and applying the collocation technique, we successfully transform the hyperbolic multi-term time fractional integro-differential equation into a system of algebraic equations. This transformation allows us to calculate the approximate solution of the equation efficiently. To demonstrate the effectiveness of the proposed method, several examples are presented and analyzed. The accuracy of the method is evaluated through these examples, showcasing its reliability in solving the hyperbolic integro-differential equations with time fractional and multi-term characteristics. The obtained results highlight the novelty and potential of our approach in addressing such complex equations.