Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

Asgari, Zohreh; Hosseini, Seyed Mohammad

doi:10.1007/s11075-017-0339-4

Cited by 28 publications

(6 citation statements)

References 32 publications

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“…Saad and Al-Sharif [16] have applied variational iteration method for solving the equation considering several initial conditions. Asgari and Hosseini [13] have focused on generalized time fractional Burger type equation, they put forward two semi implicit Fourier pseudospectral approximations for seeking solutions of the equation. As a different view to the mentioned equation, Khan et al [14] are used the generalized version of the differential transform method and homotopy perturbation method.…”

Section: Numerical Solution Of the Model Problemmentioning

confidence: 99%

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Önal

Esen

2020

Applied Mathematics and Nonlinear Sciences

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In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms L2 and L∞ have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.

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Section: Numerical Solution Of the Model Problemmentioning

confidence: 99%

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Önal

Esen

2020

Applied Mathematics and Nonlinear Sciences

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“…In the past few years, there has been an emerging trend for solving forward and inverse problems using machine learning strategies due to their capability to handle various types of model problems in many disciplines, especially in partial differential equations (PDEs) [15][16][17]. Because of the successful outcome of neural networks in a broad range of problems such as image processing [18], predicting disease [19,20], and finance [21,22], some mathematicians tried to use these techniques to solve challenging problems [23][24][25]. Among many neural network-based methods in the literature, the two state-of-the-art researches in solving PDE methods are the Gaussian processes regression (GPR) for PDEs [16] and the physics-informed neural networks (PINNs) [17].…”

Section: Introductionmentioning

confidence: 99%

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Mostajeran

Hosseini

2023

Math Methods in App Sciences

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In this study, we introduce a new growing neural network algorithm that is based on wavelet neural networks and call our algorithm a growing wavelet neural network (GWNN) method. We apply our proposed scheme to train a wavelet neural network to solve chemotaxis problems with blow‐up. These problems are highly nonlinear time‐dependent systems of partial differential equations, and it is a challenge to get the pattern of the solution accurately. The proposed structure is partial retraining of the network, which increases its capacity to catch the spiky pattern of the solution. Our neural network‐based algorithm allows us to solve the nonlinear chemotaxis problems without the use of linearization techniques and regularization techniques, most of which reduce the accuracy of the model. This mesh‐free‐based method can manage a variety of blow‐up models with curved boundaries without imposing an extra cost. By proving the consistency and stability of the method, we show the convergence of GWNN solutions to analytical solutions of the chemotaxis problem. Several illustrative examples and simulation results are provided to demonstrate the correctness of the results and the robust performance of the presented algorithm. Moreover, to illustrate the effectiveness of the GWNN method, we make a comparison with two other network‐based methods.

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“…They applied spectral method for space only, and finite difference scheme was used for time. Asgari and Hosseini 25 have applied two semi‐implicit Fourier spectral schemes for the numerical solutions of time fractional Burgers equation. In this method, the authors have shown unconditional stability and improve computational cost.…”

Section: Introductionmentioning

confidence: 99%

“…24 Besides, the pseudospectral method is also an emphatic and alternate numerical scheme for solving a wide range of linear and nonlinear fractional partial differential equations. [25][26][27][28][29][30] Lin and Xu 26 and Mohebbi 31 have used spectral method for the solution and stability analysis of time fractional nonlinear equations. They applied spectral method for space only, and finite difference scheme was used for time.…”

Section: Introductionmentioning

confidence: 99%

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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Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

Cited by 28 publications

References 32 publications

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

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

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