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

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 usin...

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Main Authors: , , , Article English 2020-04-01 Advances in Difference Equations http://link.springer.com/article/10.1186/s13662-020-02616-x
id doaj-7aa592b0f3f14481b9bb07f5b3323200 Article doaj-7aa592b0f3f14481b9bb07f5b33232002020-11-25T03:10:23ZengSpringerOpenAdvances in Difference Equations1687-18472020-04-012020112210.1186/s13662-020-02616-xA numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functionsNauman Khalid0Muhammad Abbas1Muhammad Kashif Iqbal2Dumitru Baleanu3Department of Mathematics, National College of Business Administration & EconomicsInformetrics Research Group, Ton Duc Thang UniversityDepartment of Mathematics, Government College UniversityDepartment of Mathematics, Faculty of Arts and Sciences, Cankaya UniversityAbstract 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.http://link.springer.com/article/10.1186/s13662-020-02616-xRedefined cubic B-spline functionsTime fractional Allen–Cahn equationCaputo’s time fractional derivativeStability and convergenceFinite difference formulation DOAJ English Article DOAJ Nauman Khalid Muhammad Abbas Muhammad Kashif Iqbal Dumitru Baleanu Nauman Khalid Muhammad Abbas Muhammad Kashif Iqbal Dumitru Baleanu A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions Advances in Difference Equations Redefined cubic B-spline functions Time fractional Allen–Cahn equation Caputo’s time fractional derivative Stability and convergence Finite difference formulation Nauman Khalid Muhammad Abbas Muhammad Kashif Iqbal Dumitru Baleanu Nauman Khalid A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions numerical investigation of caputo time fractional allen–cahn equation using redefined cubic b-spline functions SpringerOpen Advances in Difference Equations 1687-1847 2020-04-01 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. Redefined cubic B-spline functions Time fractional Allen–Cahn equation Caputo’s time fractional derivative Stability and convergence Finite difference formulation http://link.springer.com/article/10.1186/s13662-020-02616-x AT naumankhalid anumericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT muhammadabbas anumericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT muhammadkashifiqbal anumericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT dumitrubaleanu anumericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT naumankhalid numericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT muhammadabbas numericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT muhammadkashifiqbal numericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions AT dumitrubaleanu numericalinvestigationofcaputotimefractionalallencahnequationusingredefinedcubicbsplinefunctions 1724658937657032704