Chebyshev Petrov–Galerkin method for nonlinear time-fractional integro-differential equations with a mildly singular kernel

Abstract In this paper, a new approach to time-fractional partial integro-differential equations with weakly singular kernels ( TFPIDE ) is presented. The suggested method produces a spectral semi-analytic solution by using shifted first-kind Chebyshev polynomials ( SCP1K ) as basis functions. To satisfy homogeneous beginning and boundary requirements, a suitable collection of basis functions should be chosen. The unknown expansion coefficients are then found using the Petrov–Galerkin technique. Interestingly, we obtain precise equations for each of the related matrices’ elements. These matrices follow a clear pattern that facilitates the inversion procedure and allows the algebraic problem generated by the Petrov–Galerkin technique to be solved. The work contributes to a better knowledge of the dependability of the approach by thoroughly examining convergence and error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research. The outcomes demonstrate how well this method works for solving time fractional partial integro-differential equations, highlighting its importance as a useful contribution to the body of knowledge in the area.