The comparison of numerical and exact solutions are exhibited via tables and graphics.
📌 3 Key Takeaways
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Objective & Hypothesis:The aim of this paper is to put on display the numerical solutions and dynamics of time fractional Fitzhugh-Nagumo model, which is an important nonlinear reaction-diffusion equation.
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Key Findings & Data:Thus, time dicretization is made using L1L1 algorithm for Caputo derivative and space discretization is made using trigonometric cubic B- spline basis.
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Conclusion & Impact:The comparison of numerical and exact solutions are exhibited via tables and graphics.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Time fractional Fitzhugh-Nagumo model · finite element method · collocation
Abstract
The aim of this paper is to put on display the numerical solutions and dynamics of time fractional Fitzhugh-Nagumo model, which is an important nonlinear reaction-diffusion equation. For this purpose, finite element method based on trigonometric cubic B-splines are used to obtain numerical solutions of the model. In this model, the derivative which is fractional order is taken in terms of Caputo. Thus, time dicretization is made using L1L1 algorithm for Caputo derivative and space discretization is made using trigonometric cubic B- spline basis. Also, the non-linear term in the model is linearized by the Rubin Graves type linearization. The error norms are calculated for measuring the accuracy of the finite element method. The comparison of numerical and exact solutions are exhibited via tables and graphics.
KARAAGAC, B. (2022). A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation. Mathematical Sciences and Applications E-Notes, Cilt: 10 - Sayı: 3, 135–145. https://ajindex.com/articles/690655
Freed, A. D., Diethelm K.: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomechanics and modeling in mechanobiology. 5(4), 203-215 (2006)
Ullah, N., Khan W., Wang S.: High performance direct torque control of electrical aerodynamics load simulator using fractional calculus. Acta Polytechnica Hungarica 11 (10), 59-78 (2014)
Baleanu, D., Atangana, A., Oukouomi Noutchie,S. C., Kurulay, M., Bildik, N.,Kilicman, A.: Fractional calculus: models and numerical methods. World Scientific, 3, 2012
Changpin, L., Zeng, F.: Numerical methods for fractional calculus. Chapman and Hall/CRC, 2019
Oruç, Ö., Esen, A., Bulut F.: Highly accurate numerical scheme based on polynomial scaling functions for equal width equation. Wave Motion: 102760,(2021)
Günerhan, H., Çelik, E.: Analytical and approximate solutions of fractional partial differential-algebraic equations. Applied Mathematics and Nonlinear Sciences 5 (1): 109-120 (2020)
Mahdy,AmrM.S.:Numericalsolutionsforsolvingmodeltime-fractionalFokker–Planckequation.NumericalMethods for Partial Differential Equations 37 (2), 1120-1135 (2021)
Dokuyucu, M. A., Baleanu,D., Çelik E.: Analysis of Keller-Segel model with Atangana-Baleanu fractional derivative, Filomat 32 (16): 5633-5643 (2018)
Owolabi,K.M.,Hammouch,Z.:SpatiotemporalpatternsintheBelousov–ZhabotinskiireactionsystemswithAtangana– Baleanu fractional order derivative. Physica A: Statistical Mechanics and its Applications 523, 1072-1090(2019)
Onal, M., Esen, A.: A Crank-Nicolson approximation for the time fractional Burgers equation, Applied Mathematics and Nonlinear Sciences 5 (2), 177-184(2020)
Oruç,Ö., Esen, A., Bulut, F.: A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers’ equation. Discrete & Continuous Dynamical Systems-S 12 (3), 533(2019)
Owolabi, Kolade M., Atangana A.: Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos, Solitons & Fractals 105,111-119(2017)
Oruç,Ö., Esen, A., Bulut, F.: A Haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation, Engineering with Computers 35 (1), 75-86(2019)
Owolabi, K. M.: Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order. Communications