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

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 differenc...

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Published inAdvances in difference equations Vol. 2020; no. 1; pp. 1 - 22
Main Authors Khalid, Nauman, Abbas, Muhammad, Iqbal, Muhammad Kashif, Baleanu, Dumitru
Format Journal Article
LanguageEnglish
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
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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 ] 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.
AbstractList 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(h2+Δt2−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.
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.
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 $\alpha\in(0,1]$ 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 $O(h^{2}+\Delta t^{2-\alpha})$ 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.
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.
ArticleNumber 158
Author Baleanu, Dumitru
Khalid, Nauman
Iqbal, Muhammad Kashif
Abbas, Muhammad
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  givenname: Dumitru
  surname: Baleanu
  fullname: Baleanu, Dumitru
  organization: Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Department of Medical Research, China Medical University Hospital, China Medical University, Institute of Space Sciences
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Keywords Time fractional Allen–Cahn equation
Redefined cubic B-spline functions
Finite difference formulation
Stability and convergence
Caputo’s time fractional derivative
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Snippet We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of...
Abstract We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate...
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SubjectTerms 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
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Title A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions
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Volume 2020
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