| Citation: |
Manpal Singh, S. Das, Rajeev, E. M. Craciun. NUMERICAL SIMULATION OF VARIABLE ORDER FRACTIONAL COUPLED FITZHUGH-NAGUMO REACTION-DIFFUSION PROBLEM AND IT'S ANALYSIS[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 810-838. doi: 10.11948/20240164
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NUMERICAL SIMULATION OF VARIABLE ORDER FRACTIONAL COUPLED FITZHUGH-NAGUMO REACTION-DIFFUSION PROBLEM AND IT'S ANALYSIS
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Abstract
A novel scheme for numerical simulation of the variable order fractional partial differential equation (VOFPDE) has been presented in this article, which has been applied to find the approximate solution of the variable order time fractional coupled Fitzhugh-Nagumo reaction-diffusion equation. The solution of the considered model exists and is unique, and the aforementioned model will remain stable under the Ulam-Hyers test. It has been found that Vieta-Fibonacci wavelets are the appropriate basis function to solve the aforementioned problem numerically, and the operational matrices of the Vieta-Fibonacci wavelets have been derived for both integer as well as variable order fractional derivatives. Using these derived operational matrices and properties of Vieta-Fibonacci wavelets combined with the collocation method, the main problem is reduced to an algebraic system of equations, which has been solved easily. The salient feature of the article is the convergence analysis of the proposed method, which is discussed. The error analysis between the approximate solution of the particular cases of the concerned model using the proposed technique and their exact solutions has been presented through tables and figures.
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References
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Manpal Singh, S. Das, Rajeev, E. M. Craciun. NUMERICAL SIMULATION OF VARIABLE ORDER FRACTIONAL COUPLED FITZHUGH-NAGUMO REACTION-DIFFUSION PROBLEM AND IT'S ANALYSIS[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 810-838. doi: 10.11948/20240164
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Figure 1.
Behaviour of exact solution and their corresponding numerical solutions for Example 8.1 at
$ \hat{m}=4 $ -
Figure 2.
Comparison of maximum absolute error for Example 8.1 for different value of
$ \hat{m} $ $ \varpi=0.5 $ -
Figure 3.
Behaviour of exact solution and their corresponding numerical solutions for Example 8.2 at
$ \hat{m}=10 $ -
Figure 4.
Comparison of maximum absolute error for Example 8.2 for different value of
$ \hat{m} $ $ \varpi=0.5 $