Highly Accurate Numerical Method for Solving Fractional Differential Equations with Purely Integral Conditions

The main goal of this paper is to present a new numerical algorithm for solving two models of one-dimensional fractional partial differential equations (FPDEs) subject to initial conditions (ICs) and integral boundary conditions (IBCs). This paper builds a modified shifted Chebyshev polynomial of the second kind (MSC2Ps) basis function that meets homogeneous IBCs, named IMSC2Ps. We also introduce two types of MSC2Ps that satisfy the given ICs. We create two operational matrices (OMs) for both ordinary derivatives (ODs) and Caputo fractional derivatives (CFDs) connected to these basis functions. By employing the spectral collocation method (SCM), we convert the FPDEs into a system of algebraic equations, which can be solved using any suitable numerical solvers. We validate the efficacy of our approach through convergence and error analyses, supported by numerical examples that demonstrate the method’s accuracy and effectiveness. Comparisons with existing methodologies further illustrate the advantages of our proposed technique, showcasing its high accuracy in approximating solutions.