An iterative approximation for time-fractional Cahn–Allen equation with reproducing kernel method

Sakar, Mehmet Giyas; Saldır, Onur; Erdoğan, Fevzi

doi:10.1007/s40314-018-0672-9

Cited by 19 publications

(14 citation statements)

References 20 publications

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“…The computational results are reported for α = 0.7, 0.9, t = 1, and t = 0.001 for 0.1 ≤ z ≤ 0.9. It is clear that we have achieved self-explanatory results as compared to the outcomes obtained from IRKM [31]. Table 6 reports the present experimental results at t = 1 with t = h 2 and α = 0.25, 0.5, 0.75.…”

Section: Problemsupporting

confidence: 59%

“…The initial and boundary constraints can be extracted from the exact solution (z 2z) × t 1+α . A comparison of the maximum absolute error with IRKM [31] is presented in Tables 2, 3, 4, 5. The computational results are reported for α = 0.7, 0.9, t = 1, and t = 0.001 for 0.1 ≤ z ≤ 0.9.…”

Section: Problemmentioning

confidence: 99%

“…The source term f (z, t) on the right-hand side is given by Table 4 Absolute errors for Example 1 by IRKM [31] at t = 1 when α = 0.9, N = 100, and t = 0.001 2.1568 × 10 -5 3.4386 × 10 -5 1.8055 × 10 -5 1 32…”

Section: Problemmentioning

confidence: 99%

“…Most recently, Sakar et al [31] proposed a numerical scheme based on the iterative reproducing kernel method (IRKM) to investigate the approximate solution of time fractional ACE. Liu et al [32] developed a numerical scheme based on finite difference formulation and the Fourier spectral method to solve time fractional ACE in one and two space dimensions.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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 + 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.

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Section: Problemsupporting

confidence: 59%

Section: Problemmentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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“…The authors managed to construct orthonormal bases functions on specific Hilbert spaces that facilitate the solution process. Sakar et al [16] developed an iterative method based on RKM to solve the time-fractional Cahn-Allen differential equation with Caputo derivative. Sakar et al [17] designed a homotopy perturbation method (HPM) to solve various TFNPDEs with proportional delays in which the fractional derivative was taken in the Caputo sense.…”

Section: Introductionmentioning

confidence: 99%

A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

2020

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Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

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Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor models

Attia

Akgül

Seba

et al. 2020

Math Methods in App Sciences

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This research work is concerned with the new numerical solutions of some essential fractional cancer tumor models, which are investigated by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solutions of the considered problem. We make use of the Caputo fractional derivative. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results clearly show the superior performance of the RKHSM.

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An iterative approximation for time-fractional Cahn–Allen equation with reproducing kernel method

Cited by 19 publications

References 20 publications

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 method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor models

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