Tempered fractional Allen-Cahn equations have many vital applications in science and engineering. The presence of a tempered fractional derivative enables these equations to efficiently explain more complex systems where long-range effects exist with an exponential decay. In this paper, we provide a numerical approach to tempered space-fractional Allen-Cahn equations. The spectral collocation method is implemented, based on Jacobi polynomials, to reduce one- and two-dimensional tempered space-fractional Allen-Cahn equations to a system of ordinary differential equations in the time direction. Then, the implicit Runge-Kutta method is applied to approximate the resulting system. This is the first work that uses the implicit Runge-Kutta method to solve one- and two-dimensional tempered space-fractional Allen-Cahn equations. High accuracy of the spectral collocation method, together with the simplicity and low computational cost of the implicit Runge-Kutta method, represent key advantages of the proposed scheme when applied to such a problem. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.

High-order Volterra integro-differential equations are of great interest to many authors because of their important applications in physics and engineering, especially if they contain delay or pantograph terms that enable them to describe the memory effect. Providing an efficient numerical scheme for high-order Volterra integro-differential equations helps to explain many problems in mathematical biology and quantum mechanics. In this manuscript, we use shifted Jacobi polynomials as the basis for a spectral collocation approach to solve high-order one- and two-dimensional Volterra integro-differential equations with variable coefficients. A pantograph operational matrix, based on shifted Jacobi polynomials, is used for the first time, together with the Gauss-Jacobi quadrature rule, to reduce the problem to the problem of solving a system of algebraic equations. To ensure the validity of the proposed approach, we compare the numerical results with those of other numerical schemes in the literature.

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.