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

• Published in: Advances in difference equations Vol. 2020; no. 1; pp. 1 - 22
• Main Authors: Khalid, Nauman, Abbas, Muhammad, Iqbal, Muhammad Kashif, Baleanu, Dumitru
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 16.04.2020; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; B spline functions; Caputo’s time fractional derivative; Collocation methods; Computer simulation; Difference and Functional Equations; Exact solutions; Finite difference formulation; Finite difference method; Functional Analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Redefined cubic B-spline functions; Stability and convergence; Time fractional Allen–Cahn equation; Time fractional Allen–Cahn equation; Redefined cubic B-spline functions; Finite difference formulation; Stability and convergence; Caputo’s time fractional derivative
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-020-02616-x

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 ] 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 ) 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.