On new approximations of Caputo-Prabhakar fractional derivative and their application to reaction-diffusion problems with variable coefficients

This article is devoted to constructing and analyzing two new approximations (CPL2-1 sigma$$ {}_{\sigma } $$ and CPL-2 formulas) for the Caputo-Prabhakar fractional derivative. The error bounds for the CPL2-1 sigma$$ {}_{\sigma } $$ and CPL-2 formulas are proved to be of order 2-alpha$$ 2-\alpha $$ and 3-alpha$$ 3-\alpha $$, respectively, where alpha$$ \alpha $$ is the order of time-fractional derivative. The newly developed approximations are then used in the numerical treatment of a reaction-diffusion problem with variable coefficients defined in the Caputo-Prabhakar sense. Moreover, the space variable in the developed numerical schemes, CFD1 and CFD2, is discretized using a fourth-order compact difference operator. Both schemes' stability and convergence analysis are demonstrated thoroughly using the discrete energy method. It is shown that the convergence orders of CFD1 and CFD2 schemes are O(Delta t2-alpha,Delta t2,h4)$$ \mathcal{O}\left(\Delta {t} circumflex {2-\alpha },\Delta {t} circumflex 2,{h} circumflex 4\right) $$ and O(Delta t3-alpha,h4)$$ \mathcal{O}\left(\Delta {t} circumflex {3-\alpha },{h} circumflex 4\right) $$, respectively, where Delta t$$ \Delta t $$ and h$$ h $$ represent the mesh spacing in time and space directions, respectively. In addition, numerical results are obtained for three test problems to confirm the theory and demonstrate the efficiency and superiority of the proposed schemes.

Automated summary

This article introduces two new approximations (CPL2-1 sigma and CPL-2 formulas) for the Caputo-Prabhakar fractional derivative and proves their error bounds. These approximations are then applied in the numerical treatment of a reaction-diffusion problem with variable coefficients, and the stability and convergence of the numerical schemes are thoroughly analyzed using the discrete energy method. The numerical results demonstrate the feasibility and superiority of the proposed schemes.