A Numerical Investigation on Burgers Equation by MOL-GPS Method

Hashemi, Mir Sajjad; Karatas, Esra; Akgül, Akif

doi:10.1166/jap.2017.1357

Cited by 13 publications

(4 citation statements)

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“…The Adomian decomposition method [12], the variational iteration method [13], the differential transformation method [14], the finite difference method [15], the homotopy analysis method [16], and the homotopy perturbation method [17] are some of these methods. Also, MOL-GPS and theta methods have been applied for solving Burgers equation [18], and fractional telegraph differential equation [19], respectively. For using Riesz Riemann-Liouville, Riesz-Caputo, and other fractional concepts, we refer to [20,21].…”

Section: Introductionmentioning

confidence: 99%

Two computational approaches for solving a fractional obstacle system in Hilbert space

et al. 2019

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The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0 < α ≤ 1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

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Section: Introductionmentioning

confidence: 99%

Two computational approaches for solving a fractional obstacle system in Hilbert space

et al. 2019

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“…Also, the fractional differential equations of various types, play important roles not only in mathematics but also in physics, control systems, dynamical systems and engineering to create the mathematical modeling of many physical phenomena, can be converted to Volterra integral equation. In [1][2][3][4][5][6][7](see also, [20,21,23,32]) A.Akgul et al solved many important models of fractional differential equations by reproducing kernel method.…”

Section: Introductionmentioning

confidence: 99%

Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation

Maleknejad¹,

Rashidinia²,

Jalilian³

2018

Filomat

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In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.

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“…These new fractional-order models are more adequate than the integer-order models, because the fractional order derivatives and integrals enable to describe the memory and hereditary properties of different substance [9]. This is the most significant difference of the fractional-order models in comparison with the integer-order models [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Optimal error bound and modified kernel method for a space-fractional backward diffusion problem

Liu

Feng

2018

Adv Differ Equ

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In this paper, we consider a backward problem for a space-fractional diffusion equation. This problem is ill-posed, i.e., the solution does not depend continuously on the data. The optimal error bound for the problem under a source condition is analyzed. Based on the idea of modified 'kernel' , a regularization method is constructed, and the convergence estimates are obtained under a priori regularization parameter choice rule and a posteriori regularization parameter choice rule, respectively.

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A Numerical Investigation on Burgers Equation by MOL-GPS Method

Cited by 13 publications

References 0 publications

Two computational approaches for solving a fractional obstacle system in Hilbert space

Two computational approaches for solving a fractional obstacle system in Hilbert space

Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation

Optimal error bound and modified kernel method for a space-fractional backward diffusion problem

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