Due to the poor regularity of solutions to time-fractional partial differential equations, traditional spectral approaches typically suffer from significant accuracy reductions when applied in the time domain. To overcome this problem, in this paper, we applied a spectral approach based on a new set of basis functions, which are smooth in the spatial direction and non-smooth in the time direction. The spectral collocation approach was used combined with the operational matrix approach based on the new set of basis functions to solve two-dimensional time-fractional Gray-Scott models. Applying the operational technique reduces the computations of the full scheme and achieves significant accuracy by using a small number of these functions. Numerical results confirmed the high accuracy of the proposed approach when applied for smooth and non-smooth solutions.

This paper introduces a novel spectral collocation method for solving two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii (ZFK) equations, which generalize the classical combustion model by incorporating tempered fractional diffusion operators. The tempered fractional ZFK equation is pivotal for modeling anomalous diffusion phenomena in thermal reaction and combustion systems, where nonlocal interactions and memory effects play a critical role. The proposed hybrid numerical scheme combines a spectral collocation method based on ultraspherical polynomials for spatial discretization with an implicit Runge–Kutta (IRK) technique for temporal integration. For the first time, we derive new tempered fractional differentiation matrices in physical space using ultraspherical polynomial bases, enabling efficient handling of the nonlocal tempered fractional operators. The results highlight the effectiveness of the new differentiation matrices in capturing anomalous diffusion phenomena while maintaining spectral accuracy, providing a robust framework for fractional combustion modeling.

Pantograph integro-differential equations have many crucial applications in science and engineering. The presence of differential behavior, scaling, and memory effects makes pantograph integro-differential equations capable of describing complex systems in control theory and mathematical biology. In this paper, we provide a numerical approach to the multi-pantograph integro-differential equation. The Jacobi tau spectral approach is utilized with the help of differential, integral, and pantograph operational matrices to solve one- and two-dimensional linear multi-pantograph integro-differential equations. The high accuracy, convergence, and simplicity motivate one to apply the tau spectral approach to the problem studied. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.