Numerical efficiency of some exponential methods for an advection–diffusion equation

Macías‐Díaz, Jorge E.; Inan, Bilge

doi:10.1080/00207160.2018.1478416

Cited by 6 publications

(5 citation statements)

References 34 publications

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“…Later, Macías-Díaz and Ĩnan [45] used modified exponential methods to obtain the solution of the Burgers' equation. Furthermore, Inan et al [46] utilized the EEFD method for numerical solutions for the Newell-Whitehead-Segel type equations which are very useful in biomathematics.…”

Section: Explicit Exponential Finite Difference Schemementioning

confidence: 99%

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

Agbavon

Appadu

Inan

et al. 2022

Front. Appl. Math. Stat.

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In this study, we obtain a numerical solution for Fisher's equation using a numerical experiment with three different cases. The three cases correspond to different coefficients for the reaction term. We use three numerical methods namely; Forward-Time Central Space (FTCS) scheme, a Nonstandard Finite Difference (NSFD) scheme, and the Explicit Exponential Finite Difference (EEFD) scheme. We first study the properties of the schemes such as positivity, boundedness, and stability and obtain convergence estimates. We then obtain values of L1 and L∞ errors in order to obtain an estimate of the optimal time step size at a given value of spatial step size. We determine if the optimal time step size is influenced by the choice of the numerical methods or the coefficient of reaction term used. Finally, we compute the rate of convergence in time using L1 and L∞ errors for all three methods for the three cases.

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Section: Explicit Exponential Finite Difference Schemementioning

confidence: 99%

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

Agbavon

Appadu

Inan

et al. 2022

Front. Appl. Math. Stat.

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“…Macías-Díaz and colleagues tackled the Burger-Fisher equation using explicit exponential finite difference methods [15]. Moreover, Macías-Díaz and ˙Inan proposed a modified exponential method for solving the Burgers' equation, establishing the consistency, convergence, and stability of the methods [16].…”

Section: Introductionmentioning

confidence: 99%

The Generalized Fractional-Order Fisher Equation: Stability and Numerical Simulation

İnan

2024

Symmetry

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This study examines the stability and numerical simulation of the generalized fractional-order Fisher equation. The equation serves as a mathematical model describing population dynamics under the influence of factors such as natural selection and migration. We propose an implicit exponential finite difference method to solve this equation, considering the conformable fractional derivative. Furthermore, we analyze the stability of the method through theoretical considerations. The method involves transforming the problem into systems of nonlinear equations at each time since our method is an implicit method, which is then solved by converting them into linear equations systems using the Newton method. To test the accuracy of the method, we compare the results obtained with exact solutions and with those available in the literature. Additionally, we examine the symmetry of the graphs obtained from the solution to examine the results. The findings of our numerical simulations demonstrate the effectiveness and reliability of the proposed approach in solving the generalized fractional-order Fisher equation.

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“…Thereafter, the generalized Huxley equation [18] and Burgers' equation [19] were solved via EEFDM with Padé approximation. Once and for all, Macas‐Daz and Inan solved the Burgers' equation by using modified exponential methods and they showed convergence, consistency and stability of the method [20].…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical solutions of the Fitzhugh–Nagumo equation and their multistability behavior

İnan

Ali

Saha

et al. 2020

Numerical Methods Partial

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In this paper, we propose an analytical method and a modification of explicit exponential finite difference method (EEFDM) for analytical and numerical solutions of the Fitzhugh-Nagumo (FN) and Newell-Whitehead (NW) equations. The method is improved computationally by using the Padé approximation technique. Furthermore, multistability behavior of traveling wave solutions of the FN and NW equations are examined in presence of external forcing. It is observed that there exist coexisting periodic and quasiperiodic orbits for the FN equation, where as only quasiperiodic orbits is observed in case of NW equation.

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Numerical efficiency of some exponential methods for an advection–diffusion equation

Cited by 6 publications

References 34 publications

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

The Generalized Fractional-Order Fisher Equation: Stability and Numerical Simulation

Analytical and numerical solutions of the Fitzhugh–Nagumo equation and their multistability behavior

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