Stability and convergence of multistep schemes for 1D and 2D fractional model with nonlinear source term

Abstract

Stable multistep schemes based on Caputo fractional derivative approximation are presented for solving 1D and 2D nonlinear fractional model arising from dielectric media. We approximate Caputo fractional derivatives in time with a multistep scheme of order O(τ3−α) & O(τ3−β),1<β<α<2, spatial Laplacian operator with a central difference scheme, and nonlinear source term g(B) by using Taylor series. The discretization of the problem results in a linear system of equations that is tridiagonal and penta-diagonal for 1D and 2D case, respectively. The unique solvability and unconditional stability are derived for both cases. The convergence of schemes is established with the help of optimal error bounds. Further, we establish that the order of convergence for 1D case is O(τ3−α+τ3−β+h2) and for 2D case is O(τ3−α+τ3−β+hx2+hy2). Moreover, the stability of our schemes are verified numerically by adding some linear and nonlinear noisy inputs. Finally, four test functions are investigated to show the effectiveness and stability of our schemes. The method is simple, easy to implement, and yields very accurate results.