We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

The present study’s main focus is regarding the physical properties of a two-dimensional (2D) magneto-hydrodynamic boundary layer non-Newtonian Casson fluid flow that moves due to an exponentially expanding surface with a mixed convection heat transfer mechanism. In the hydrodynamic flow and heat transmission process, the combined impact of thermal radiation and magnetic field influence is explored. The internal heat generation owing to the fluid motion or a very fluid viscosity is not taken into account. The Chebyshev spectral method (CSM) is employed in this work due to its ability, accuracy, and ease of obtaining the solution for non-linear system of ordinary differential equations (ODEs). This method is an approximate method that can usually obtain the solution in a series form. The mixed convection impact is incorporated in our problem. The results are graphed to help comprehend the many physical parameters that arise in the problem. Graphical results uncover that the speed liquid stream is lessened when reinforcing both the Casson boundary and the Hartmann number, while converse attributes are applied for the Grashof number and the radiation boundary. Finally, a comparison of our current results with previously published work on several particular situations of the problem reveals that they are in excellent agreement.