Semi-implicit Galerkin–Legendre Spectral Schemes for Nonlinear Time-Space Fractional Diffusion–Reaction Equations with Smooth and Nonsmooth Solutions

Abstract

For the first time in literature, semi-implicit spectral approximations for nonlinear Caputo time-and Riesz space-fractional diffusion equations with both smooth and non-smooth solutions are proposed. More precisely, the governing partial differential equation generalizes the Hodgkin-Huxley, the Allen-Cahn and the Fisher-Kolmogorov-Petrovskii-Piscounov equations. The schemes employ a Legendre-based Galerkin spectral method for the Riesz spacefractional derivative, and L1-type approximations with both uniform and graded meshes for the Caputo time-fractional derivative. More importantly, by using fractional Gronwall inequalities and their associated discrete forms, sharp error estimates are proved which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes. This analysis encompasses both uniform meshes as well as meshes that are graded in time, and guarantees the unconditional stability. The numerical results that accompany our analysis confirm our theoretical error estimates, and give significant insights into the convergence behavior of our schemes for problems with smooth and non-smooth solutions.

A Compact Difference Scheme for Time-Fractional Dirichlet Biharmonic Equation on Temporal Graded Meshes

“…A time-fractional Benjamin-Bona-Mahony equation and nonlinear Korteweg-de Vries equation are, respectively, discussed in [19] and [21]. Galerkin-Legendre spectral schemes for nonlinear timespace fractional diffusion-reaction equations studied by Zaky et al [26], use the L1 scheme on graded meshes for approximation of the time fractional derivative. A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition was given in [5] and time fractional Burgers' equations was discussed in [16].…”

On Romanovski–Jacobi polynomials and their related approximation results

“…The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1–3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types of polynomial solutions to the second‐order homogeneous differential equation: …”

A new approach to solving multiorder time‐fractional advection–diffusion–reaction equations using BEM and Chebyshev matrix

“…Several numerical techniques for solving fractional partial differential equations (PDEs) have been reported, such as variational iteration, 19 Adomian decomposition, 20 operational matrix of B‐spline functions, 21 operational matrix of Jacobi polynomials, 12 Jacobi collocation, 22 operational matrix of Chebyshev polynomials, 23–24 Legendre collocation, 25 pseudo‐spectral, 26 and operational matrix of Laguerre polynomials, 27 Pade approximation, and two‐sided Laplace transformations 28 . Besides finite elements and finite differences, 29 spectral methods are one of the three main methodologies for solving various types of fractional differential equations 30–34 . The main idea of spectral methods is to express the solution of differential equations as a sum of basis functions and then to choose the coefficients in order to minimize the error between the numerical and exact solutions as much as possible 35 .…”

“…28 Besides finite elements and finite differences, 29 spectral methods are one of the three main methodologies for solving various types of fractional differential equations. [30][31][32][33][34] The main idea of spectral methods is to express the solution of differential equations as a sum of basis functions and then to choose the coefficients in order to minimize the error between the numerical and exact solutions as much as possible. 35 Therefore, high accuracy and ease of implementing are two of the main features that have encouraged many researchers to apply such methods to solve various types of fractional integral and differential equation.…”