A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions

• Main Authors: Nauman Khalid, Muhammad Abbas, Muhammad Kashif Iqbal, Dumitru Baleanu
• Format: Article
• Language: English
• Published: SpringerOpen 2020-04-01
• Series: Advances in Difference Equations
• Subjects: Redefined cubic B-spline functions; Time fractional Allen–Cahn equation; Caputo’s time fractional derivative; Stability and convergence; Finite difference formulation
• Online Access: http://link.springer.com/article/10.1186/s13662-020-02616-x

Abstract We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen–Cahn equation (ACE). We discretize the time fractional derivative of order α ∈ ( 0 , 1 ] $\alpha\in(0,1]$ by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O ( h 2 + Δ t 2 − α ) $O(h^{2}+\Delta t^{2-\alpha})$ and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature.