Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Novel finite point approach for solving time-fractional convection-dominated diffusion equations

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

Some efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an L1 approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.

Best restriction L1-approximation of periodic convolution class by generalized splines

We consider the estimate of the best L1-approximation of convolution classes [Formula: see text], defined by a self-adjoint polynomial differential operator Pr(D), in the term of some kinds of generalized splines and get some exact constants.