A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation

Oruç, Ömer; Esen, Alaattin; Bulut, Fatih

doi:10.3934/dcdss.2019035

Cited by 19 publications

(10 citation statements)

References 17 publications

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“…Especially, since fractional derivatives are more convenient and economical, solving fractional order differential equations has a vital role in modeling various phenomena. One can get a brief glimpse to numerical methods for solving fractional differential equations in Refs [5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

Karaağaç

2022

Mathematical Sciences and Applications E-Notes

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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 L1 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.

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Section: Introductionmentioning

confidence: 99%

A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

Karaağaç

2022

Mathematical Sciences and Applications E-Notes

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“…Dehghan et al [23,24] proposed a mixed finite difference and Galerkin method and multisymplectic box method for numerical study of Burgers equations. Oruc et al [25,26] applied a unified finite difference Chebyshev wavelet approach for time fractional Burger equations. The same authors studied the Chebyshev wavelet method for approximation of coupled Burgers equations [27].…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

et al. 2021

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We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$ L ∞ , $L_{2}$ L 2 , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

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“…It has been used in a broad scope of technology disciplines (Sahu and Saha Ray, 2016); particularly, signal analysis (Postnikov et al , 2016), time-frequency analysis and fast algorithm for easy execution (Beylkin et al , 1991). In recent years, wavelets have been applied for solving partial differential equations (Oruç et al , 2015; Oruç et al , 2016; Oruç, 2018a; Oruç, 2018b; Oruç et al , 2019a; Oruç et al , 2019b; Oruç, 2019c), integral and integro-differential equations (Rostami and Maleknejad, 2016; Sahu and Saha Ray, 2015). Integro-differential equations are usually difficult to solve analytically so it is required to find an efficient approximate solution.…”

Section: Introductionmentioning

confidence: 99%

Operational matrix of two dimensional Chebyshev wavelets and its applications in solving nonlinear partial integro-differential equations

Rostami

2020

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Purpose This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets. Design/methodology/approach For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved. Findings Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Originality/value This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

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A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation

Cited by 19 publications

References 17 publications

A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

Operational matrix of two dimensional Chebyshev wavelets and its applications in solving nonlinear partial integro-differential equations

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