Numerical solution of time fractional Burgers equation

Esen, Alaattin; Taşbozan, Orkun

doi:10.1515/ausm-2015-0011

Cited by 34 publications

(55 citation statements)

References 31 publications

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“…The exact solution given by Esen and Tasbozan [12] is u * (x, t) = t 2 e x . In the numerical test, we choose the kinematic viscosity ν = 1, α = 0.5 and ∆t = 0.00025.…”

Section: Examplementioning

confidence: 99%

“…In the numerical test, we choose the kinematic viscosity ν = 1, α = 0.5 and ∆t = 0.00025. Table 1 presents the exact solution u * (x, 1), the numerical solution u(x, 1) by using our FIM-SCP in Algorithm 1, and the solution obtained by the quadratic B-spline finite element Galerkin method (QBS-FEM) proposed by Esen and Tasbozan [12]. The comparison between the absolute errors E a (as the difference in absolute value between the approximate solution and the exact solution) of the two methods shows that our FIM-SCP is more accurate than QBS-FEM for M = 10 and similar accuracy for other M. Algorithm 1 acquires the significant improvement in accuracy with less computational nodal points M and regardless the time steps ∆t and the fractional order derivatives α.…”

Section: Examplementioning

confidence: 99%

“…where f m = f (x, y, t m ) and u m = u(x, y, t m ) is the numerical solution at the m th iteration. Next, consider the fractional order derivative in the Caputo sense as defined in Definition 2, by using (12), then (26) becomes…”

Section: Algorithm For Two-dimensional Time-fractional Burgers' Equationmentioning

confidence: 99%

“…where u m = u(x, t m ) and v m = v(x, t m ) are numerical solutions of u and v in the m th iteration, respectively. Next, let us consider the fractional time derivative for α ∈ (0, 1] in the Caputo sense by using the same procedure as in (12), by taking the double layer integration on both sides, we obtain…”

Section: Algorithm For Time-fractional Coupled Burgers' Equationmentioning

confidence: 99%

“…Some powerful numerical methods had been developed for solving the fractional Burgers' equation, such as finite difference methods (FDM) [9], Adomian decomposition method [10], and finite volume method [11]. Moreover, in 2015, Esen and Tasbozan [12] gave a numerical solution of time fractional Burgers' equation by assuming that the solution u(x, t) can be approximated by a linear combination of products of two functions, one of which involves only x and the other involves only t. Recently, Yokus and kaya [13] used the FDM to find the numerical solution for time fractional Burgers' equation, however, their results contained less accuracy. In 2017, Cao et al [14] studied solution of two-dimensional time-fractional Burgers' equation with high and low Reynolds numbers using discontinuous Galerkin method, however, the method involves the triangulations of the domain which usually gives difficulty in terms of devising a computational program.…”

Section: Introductionmentioning

confidence: 99%

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Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

2019

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The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

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“…The exact solution given by Esen and Tasbozan [12] is u * (x, t) = t 2 e x . In the numerical test, we choose the kinematic viscosity ν = 1, α = 0.5 and ∆t = 0.00025.…”

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Algorithm For Two-dimensional Time-fractional Burgers' Equationmentioning

confidence: 99%

Section: Algorithm For Time-fractional Coupled Burgers' Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

2019

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Numerical solutions of time and space fractional coupled Burgers equations using time–space Chebyshev pseudospectral method

Mittal

Balyan

2020

Math Methods in App Sciences

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The aim of the paper is to develop and analyze a spectrally accurate time–space pseudospectral method to the approximate solution of nonlinear time and space fractional coupled Burgers equations. Liouville–Caputo fractional derivative formula is used to evaluate the fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivative matrices, the given problem is reduced to a system of nonlinear algebraic equations. A mapping is used to transform the nonhomogeneous initial–boundary values to homogeneous initial–boundary values. Error analysis of the proposed method for the equation is presented. A model example of fractional coupled Burgers equations is tested for a set of fractional‐order derivatives. For the proposed method, highly accurate numerical results are obtained, which confirm the accuracy and efficiency of the proposed method.

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Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Zadeh

Skandari

et al. 2020

Math Methods in App Sciences

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This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.

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Numerical solution of time fractional Burgers equation

Cited by 34 publications

References 31 publications

Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations

Numerical solutions of time and space fractional coupled Burgers equations using time–space Chebyshev pseudospectral method

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

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