Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs

Rui, Weiguo

doi:10.1016/j.cnsns.2016.11.018

Cited by 40 publications

(54 citation statements)

References 39 publications

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“…In reference [23], Rui points out that Leibniz rule (11) still holds, which is incorrect. Actually, in our method, it does not matter that Leibniz rule (11) does not hold.…”

Section: Remarkmentioning

confidence: 99%

“…In [23], V( ) is taken as one of , sin ( ), cos ( ), and , which is just a special solution of our (20). So our method is more general than Rui's method [23].…”

Section: Casementioning

confidence: 99%

“…In [23], Rui studied the situation of ( ) = . Equation (25) becomes the following time-fractional PDE with diffusion term:…”

Section: Exact Solutions Of a Nonlinear Time-fractional Pde With Diffmentioning

confidence: 99%

“…We cannot obtain the exact solutions of compound function type of timefractional PDEs (2) as in the references. Encouraged by Rui's work [23], we shall introduce an improved method based on the homogenous balanced principle; by using this improved method we shall investigate exact solutions of a series of nonlinear time-fractional PDEs formed as (2).…”

Section: Introductionmentioning

confidence: 99%

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Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Zhao

2018

Mathematical Problems in Engineering

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Based on the homogenous balanced principle and subequation method, an improved separation variables function-expansion method is proposed to seek exact solutions of time-fractional nonlinear PDEs. This method is novel and meaningful without using Leibniz rule and chain rule of fractional derivative which have been proved to be incorrect. By using this method, we studied a nonlinear time-fractional PDE with diffusion term. Some general solutions are obtained which contain many arbitrary parameters. Solutions given in related reference are just our especial case. And we also obtained some new type of solutions.

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“…In reference [23], Rui points out that Leibniz rule (11) still holds, which is incorrect. Actually, in our method, it does not matter that Leibniz rule (11) does not hold.…”

Section: Remarkmentioning

confidence: 99%

“…In [23], V( ) is taken as one of , sin ( ), cos ( ), and , which is just a special solution of our (20). So our method is more general than Rui's method [23].…”

Section: Casementioning

confidence: 99%

“…In [23], Rui studied the situation of ( ) = . Equation (25) becomes the following time-fractional PDE with diffusion term:…”

Section: Exact Solutions Of a Nonlinear Time-fractional Pde With Diffmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Zhao

2018

Mathematical Problems in Engineering

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“…So far, various methods are used for solving different types of fractional KdV equations. These methods can be listed as Adomian decomposition method [1,2], variational iteration method [2], homotopy perturbation method [3][4][5][6][7], differential transform method [8], modified direct algebraic method [9], (G ′ /G) -expansion method [10], improved (G ′ /G) -expansion method [10], homotopy perturbation transform method [11], function expansion method [12]. In this study, we consider the space fractional Korteweg-de Vries equation…”

Section: Introductionmentioning

confidence: 99%

New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

Daşçıoğlu¹,

Çulha²,

Bayram³

2017

ntmsci

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Abstract:In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated.

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Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

Rui,

2024

Math Methods in App Sciences

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It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

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Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs

Cited by 40 publications

References 39 publications

Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Applications of Separation Variables Approach in Solving Time-Fractional PDEs

New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

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