Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations

Özpinar, Figen

doi:10.3390/e20050332

Cited by 13 publications

(10 citation statements)

References 42 publications

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“…Nonlinear dynamic phenomenon, which exists in many fields of science and engineering, such as hydrodynamic, nonlinear optics, biology, plasma physics, and so on, can be modeled by many systems of nonlinear partial differential equations (NPDEs) [1,2]. The dynamical processes of these nonlinear systems are very important for both production and scientific research, and they should be studied by a suitable method designed to treat the nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

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Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model

Lai

Shi

2019

Entropy

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In this work, we develop a mesoscopic lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model to solve (2 + 1)-dimensional wave equation with the nonlinear damping and source terms. Through the Chapman-Enskog multiscale expansion, the macroscopic governing evolution equation can be obtained accurately by choosing appropriate local equilibrium distribution functions. We validate the present mesoscopic model by some related issues where the exact solution is known. It turned out that the numerical solution is in very good agreement with exact one, which shows that the present mesoscopic model is pretty valid, and can be used to solve more similar nonlinear wave equations with nonlinear damping and source terms, and predict and enrich the internal mechanism of nonlinearity and complexity in nonlinear dynamic phenomenon.

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Section: Introductionmentioning

confidence: 99%

“…   u(x, y, t 0 ) = ϕ 1 (x, y), (x, y) ∈ Ω u t (x, y, t 0 ) = ϕ 2 (x, y), (x, y) ∈ Ω, (2) and the Dirichlet boundary conditions are given by…”

Section: Introductionmentioning

confidence: 99%

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model

Lai

Shi

2019

Entropy

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“…In this regard, numerical solutions of FPDEs have been obtained in [24] by using tau approximation. Discrete HAM is suggested to solve linear and nonlinear FPDEs [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Laplace decomposition for solving nonlinear system of fractional order partial differential equations

et al. 2020

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In the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.

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“…Homotopy-type methods include the well-known homotopy perturbation method [2], the local fractional homotopy perturbation method [14] and the discrete homotopy analysis method [15].…”

mentioning

confidence: 99%

“…The comparison of the remainders corresponding to ADM (red), HPM (yellow) and LSDQM (green) for the case α = 0.75 in Equation(15).…”

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confidence: 99%

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

et al. 2020

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In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

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Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations

Cited by 13 publications

References 42 publications

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model

Mesoscopic Simulation of the (2 + 1)-Dimensional Wave Equation with Nonlinear Damping and Source Terms Using the Lattice Boltzmann BGK Model

Laplace decomposition for solving nonlinear system of fractional order partial differential equations

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

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