Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay

Abstract

In the paper, a linearized compact finite difference scheme is presented for the semilinear fractional delay convection-reaction–diffusion equation. Firstly, the equation is transformed into an equivalent semilinear fractional delay reaction–diffusion equation by using a special transformation. Then, the temporal Caputo derivative is discreted by using approximation and the second-order spatial derivative is approximated by the compact finite difference scheme. The solvability, unconditional stability, and convergence in the sense of - and - norms are proved rigorously. Finally, numerical examples are carried out extensively to support our theoretical analysis.

Keywords:

• Semilinear fractional delay partial differential equation
• compact finite difference scheme
• solvability
• convergence
• stability

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by Natural Science Foundation of China [grant number 11501514], [grant number 11471287]; Natural Sciences Foundation of Zhejiang Province [grant number LQ16A010007]; Zhejiang Province Ministry of Education [grant number Y201533134]; Start-up Fund of ZSTU [grant number 11432932611470] and PhD Thesis Innovation Fund of HUST.