Variable separation method for a nonlinear time fractional partial differential equation with forcing term

Zhang, Sheng; Hong, Siyu

doi:10.1016/j.cam.2017.09.045

Cited by 42 publications

(12 citation statements)

References 35 publications

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“…(3.31) Substituting (3.31) along with (1.6) into (3.30) and collecting all terms with the same order of ϕ together, the left-hand side of (3.30) is converted into a polynomial in ϕ. Setting and boundary conditions [27]. So, extending the work in the present paper is worthy of study.…”

Section: Gblmp Equationmentioning

confidence: 99%

Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

Huang

Xie

2018

Adv Differ Equ

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This paper deals with the analytical solutions for two models of special interest in mathematical physics, namely the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation and the (3 + 1)-dimensional generalized Boiti-Leon-Manna-Pempinelli equation. Using a modified version of the Fan sub-equation method, more new exact traveling wave solutions including triangular solutions, hyperbolic function solutions, Jacobi and Weierstrass elliptic function solutions have been obtained by taking full advantage of the extended solutions of the general elliptic equation, showing that the modified Fan sub-equation method is an effective and useful tool to search for analytical solutions of high-dimensional nonlinear partial differential equations.

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Section: Gblmp Equationmentioning

confidence: 99%

Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

Huang

Xie

2018

Adv Differ Equ

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“…More details about mesh-less methods can be found in [10]. On the other hand, a number of analytical methods have been employed to determine approximate solutions of ODEs, including the Homotopy-perturbation methods [11], Taylor expansion approach [12], variable separation method [13] and so on. However, these approaches are often inefficient in solving complicated multi-order ODEs.…”

Section: Introductionmentioning

confidence: 99%

Solving Ordinary Differential Equations With Adaptive Differential Evolution

2020

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Solving ordinary differential equations (ODEs) is vital in diverse fields. However, it is difficult to obtain the exact analytical solutions of ODEs due to their changeable mathematical forms. Traditional numerical methods can find approximate solutions for specific ODEs. Unfortunately, they often suffer from ODEs' forms and characteristics. To approximate different types of ODEs, this paper proposes a generic method based on adaptive differential evolution. Besides, in order to further reduce the error of the obtained approximate solutions, an improved Fourier periodic expansion function is developed, which is then combined with the least square weight method to formulate the ODEs as an optimization problem. Since the proposed method is not limited to ODEs' forms and constraint conditions, it can be used to approximate any ODEs, including linear ODEs and nonlinear ODEs. The proposed method is evaluated on twenty popular test cases. The results indicate that the proposed method is able to accurately approximate different ODEs with better performance compared with other methods.

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“…Since put forward in 1970, Hirota's bilinear method has achieved considerable developments [4][5][6][7][8][9][10][11][12][13][14]. With the close attentions of fractional calculus and its applications [15][16][17][18][19][20][21][22][23][24], some of the natural questions are whether Hirota's bilinear method can be extended to non-linear PDE of fractional orders and what about the fractional soliton dynamics and integrability of fractional PDE. As far as we know there is no research reports on the bilinear method for non-linear PDE of fractional orders.…”

Section: Introductionmentioning

confidence: 99%

Bilinearization and fractional soliton dynamics of fractional Kadomtsev-Petviashvili equation

Zhang

Wei

2019

Therm sci

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Original scientific paper https://doi.org/10.2298/TSCI180815207Z Kadomtsev-Petviashvili equation is a mathematical model with many important applications in fluids. In this paper, a local fractional Kadomtsev-Petviashvili equation with Lax integrability is derived and solved by extending Hirota's bilinear method. More specifically, the local fractional Kadomtsev-Petviashvili equation is derived from a local fractional Lax equation. With the help of a suitable transformation, the local fractional Kadomtsev-Petviashvili equation is then bilinearized. Based on the bilinearized form, n-soliton solution with Mittag-Leffler functions is obtained. In order to gain more insights into the fractional n-soliton solution, the velocity of the fractional one-soliton solution is simulated. It is shown that the velocity of the fractional one-soliton changes with the fractional order.

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Variable separation method for a nonlinear time fractional partial differential equation with forcing term

Cited by 42 publications

References 35 publications

Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

Solving Ordinary Differential Equations With Adaptive Differential Evolution

Bilinearization and fractional soliton dynamics of fractional Kadomtsev-Petviashvili equation

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