The new solitary wave structures for the (2 + 1)-dimensional time-fractional Schrodinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations

Alam, Md. Nur; Tunç, Cemil

doi:10.1016/j.aej.2020.01.054

Cited by 48 publications

(13 citation statements)

References 45 publications

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“…In Equation , the fractional derivative

D_{z}^{β} = \lim_{h \to 0} h^{- β} {um}_{normalH = 0}^{\infty} {(- 1)}^{H} f ([], z + ((), β - normalH) h),

which has the advantages of high precision, fast convergence speed, and good stability and is suitable for various numerical algorithms. 38 Therefore, Equation is convergent.…”

Section: Exact Solutions Of Fractional Complex Glementioning

confidence: 90%

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

Wang

Dai

2020

Math Methods in App Sciences

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A (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation is solved via fractional Riccati method and fractional bifunction method, and exact traveling wave solutions including soliton solution and combined soliton solutions are constructed based on Mittag–Leffler function. A series of fractional orders is used to demonstrate the graphical representation and physical interpretation of the resulting solutions. The role of the fractional order is revealed.

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“…In Equation , the fractional derivative

D_{z}^{β} = \lim_{h \to 0} h^{- β} {um}_{normalH = 0}^{\infty} {(- 1)}^{H} f ([], z + ((), β - normalH) h),

which has the advantages of high precision, fast convergence speed, and good stability and is suitable for various numerical algorithms. 38 Therefore, Equation is convergent.…”

Section: Exact Solutions Of Fractional Complex Glementioning

confidence: 90%

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

Wang

Dai

2020

Math Methods in App Sciences

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“…However, just like the integer-order PDEs, there exists no commonly available method to study fractional PDEs, especially for finding exact solutions as well as numerical ones. Thus, many researchers try to extend certain known effective techniques of solving integer-order PDEs to deal with fractional PDEs such as the subequation method 8,9 , homotopy perturbation method 10 , differential transform method 11 , variational iteration method, 12 invariant subspace method 13 , power series method, 14 and so on. It is worth saying that in the last several decades Lie group theory has been evolved into one of the most important tools to study PDEs as well as fractional PDEs [15][16][17][18][19][20] .…”

Section: Introductionmentioning

confidence: 99%

Local symmetry structure and potential symmetries of time‐fractional partial differential equations

Zhang

Lin

2021

Stud Appl Math

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First, we show that the system consisting of integerorder partial differential equations (PDEs) and timefractional PDEs with the Riemann-Liouville fractional derivative has an elegant local symmetry structure. Then with the symmetry structure we consider two particular cases where one is the pure time-fractional PDEs whose symmetry invariant condition is divided into two parts of integer-order and time-fractional, the other is the linear system of time-fractional PDEs, which always admits an infinite-dimensional infinitesimal generator. Second, by considering the composition rules of fractional derivatives we establish a theoretical framework of potential symmetry and construct three potential systems to study potential symmetries of the time-fractional PDEs possessing a divergence form. In particular for a single time-fractional PDE the existence condition of potential symmetries via one typical potential system is presented by means of the local symmetry structure. Finally, local symmetry structure and potential symmetries of a class of time-fractional diffusion equations are studied in detail. Several explicit solutions are constructed by means of the potential symmetries.

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“…Method of lines [12][13][14][15][16] as modified (G /G)-expansion method [17,18] is one of the efficient and accurate numerical methods that have been handled successfully to solve many partial differential equations (PDEs) with high accuracy. The idea of the MOL is summarized in reducing the given partial differential equations to a system of ordinary differential equations (ODEs) in the time by discretization of space variables and spatial derivatives using finite difference schemes.…”

Section: Introductionmentioning

confidence: 99%

A combined method for simulating MHD convection in square cavities through localized heating by method of line and penalty-artificial compressibility

Mousa

Ali

2021

Journal of Taibah University for Science

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A new combined technique is developed for studying free convection of magnetohydrodynamic unsteady incompressible flow in a square cavity that partially heated from lower wall and regular cooling from lateral walls. Convection has been studied for several lengths of heat source, Hartman and the Rayleigh numbers. The local and average Nusselt numbers are calculated and presented graphically along the heat source portion. The proposed technique is a combination of the method of lines (MOL) and a developed penalty-artificial compressibility technique (PACT). Unsteady penalty compressibility governing equations are used instead of solving steady-state balances. The penalty and artificial compressibility factors are estimated using a deduced stability restriction. The validation of the MOL-PACT is verified by comparing the current data with other numerical and experimental ones existing in the literature. The proposed technique provides a new choice through the investigation of MHD mass and heat transfer.

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The new solitary wave structures for the (2 + 1)-dimensional time-fractional Schrodinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations

Cited by 48 publications

References 45 publications

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

Local symmetry structure and potential symmetries of time‐fractional partial differential equations

A combined method for simulating MHD convection in square cavities through localized heating by method of line and penalty-artificial compressibility

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