An advanced approach for numerical solution of a class of Fredholm–Volterra integro-differential equations with mixed boundary conditions

This paper proposes a novel numerical algorithm to obtain approximate solutions to FVIDEs with MTBCs. To create a basis function in terms of SCPFK, we devised a novel approach by ensuring that the homogeneous MTBCs were satisfied. We have constructed new OMs for the derivatives of developed polynomials. We aim to create approximate solutions for FVIDEs by utilizing these OMs in conjunction with the collocation approach. The examination of theoretical convergence and the provision of error estimates are the primary objectives of our research. In conclusion, we will demonstrate the validity, utility, and application of the developed method by presenting four examples that illustrate the theoretical study that we have just presented. We contrast the numerical solutions with the exact solutions and outcomes obtained using other approaches. Considering the tables and figures, it can be concluded that the technique that was presented produces an outstanding agreement between approximate and precise results.