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In this paper, we derive a high order numerical scheme for variable order fractional ordinary differential equation by establishing a second order numerical approximation to variable order Riemann–Liouville fractional derivative. The scheme is strictly proved to be stable and convergent with second order accuracy, which is higher than some recently derived schemes. Finally, some numerical examples are presented to demonstrate the theoretical analysis and verify the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a series of new high-order numerical approximations to α th ( 0 < α < 1 ) order Caputo derivative is constructed by using r th degree interpolation approximation for the integral function, where r ≥ 4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O ( τ r + 1 − α ) , where τ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection–diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O ( τ r + 1 − α + h 2 ) , where τ is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r = 4 are also considered. Without loss of generality, we only display the numerical examples for r = 4 , 5 , which support the numerical algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with orderO(τ2−α+h4) inL2norm, where τ, α, andhare the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes. [ABSTRACT FROM AUTHOR]