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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2020-04-01
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Online Access: | http://link.springer.com/article/10.1186/s13662-020-02616-x |
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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 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Nauman Khalid Muhammad Abbas Muhammad Kashif Iqbal Dumitru Baleanu |
spellingShingle |
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 |
author_facet |
Nauman Khalid Muhammad Abbas Muhammad Kashif Iqbal Dumitru Baleanu |
author_sort |
Nauman Khalid |
title |
A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions |
title_short |
A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions |
title_full |
A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions |
title_fullStr |
A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions |
title_full_unstemmed |
A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions |
title_sort |
numerical investigation of caputo time fractional allen–cahn equation using redefined cubic b-spline functions |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1847 |
publishDate |
2020-04-01 |
description |
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. |
topic |
Redefined cubic B-spline functions Time fractional Allen–Cahn equation Caputo’s time fractional derivative Stability and convergence Finite difference formulation |
url |
http://link.springer.com/article/10.1186/s13662-020-02616-x |
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