Title: A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions
Abstract: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...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 $\alpha\in(0,1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:math> by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order $O(h^{2}+\Delta t^{2-\alpha})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math> 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.Read More