A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application

Alquran, Marwan; Jaradat, Imad

doi:10.1007/s11071-017-4019-7

Cited by 36 publications

(15 citation statements)

References 20 publications

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“…It has been demonstrated in certain circumstances that the nonlocality feature of the fractional derivatives makes them appropriate for describing the memory and hereditary features of various materials. Thus, the fractional derivative order α can be physically described as an index of memory [7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Ternary-fractional differential transform schema: theory and application

et al. 2019

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In this article, we propose a novel fractional generalization of the three-dimensional differential transform method, namely the ternary-fractional differential transform method, that extends its applicability to encompass initial value problems in the fractal 3D space. Several illustrative applications, including the Schrödinger, wave, Klein-Gordon, telegraph, and Burgers' models that are fully embedded in the fractal 3D space, are considered to demonstrate the superiority of the proposed method compared with other generalized methods in the literature. The obtained solution is expressed in a form of an α α α-fractional power series, with easily computed coefficients, that converges rapidly to its closed-form solution. Moreover, the projection of the solutions into the integer 3D space corresponds with the solutions of the classical copies for these models. This reveals that the suggested technique is effective and accurate for handling many other linear and nonlinear models in the fractal 3D space. Thus, research on this trend is worth tracking.

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Section: Introductionmentioning

confidence: 99%

Ternary-fractional differential transform schema: theory and application

et al. 2019

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“…Recently, an analytical method based on power series expansion without linearization, discretization, or perturbation has been introduced and successfully applied to many kinds of fractional differential equations arising in strongly nonlinear and dynamic problems. The method was named residual power series method (RPSM) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], which was used to find the analytical solution for several classes of time fractional differential equations. The residual power series method has been widely used in different fields.…”

Section: Introductionmentioning

confidence: 99%

“…In [15][16][17][18][19][20][21], residual power series method, as a powerful method, was used to solve the other time fractional differential equations. Residual power series method was also used for the time fractional Gardner [23] and Kawahara equations in [22], the time fractional Phi-4 equation in [24], the fractional population diffusion model [25], the generalized Burger-Huxley equation [26], and the time fractional two-component evolutionary system of order 2 [27].…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution for the Time Fractional BBM‐Burger Equation by Using Modified Residual Power Series Method

et al. 2018

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In this study, a generalized Taylor series formula together with residual error function, which is named the residual power series method (RPSM), is used for finding the series solution of the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The BBM-Burger equation is useful in describing approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems. The numerical solution of the BBM-Burger equation is calculated by Maple. The numerical results show that the RPSM is reliable and powerful in solving the numerical solutions of the BBM-Burger equation compared with the exact solutions as well as the solutions obtained by homotopy analysis transform method through different graphical representations and tables.

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“…Due to the complex nature of the optical soliton propagation, several works consider the fractional calculus to construct new optical soliton solutions [18][19][20][21]. Nevertheless, fractional derivatives do not obey some basic properties of integer derivative such as product rule and chain rule.…”

Section: Introductionmentioning

confidence: 99%

Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity

Ghanbari

Gómez-Aguilar

2018

Rev. Mex. Fís.

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By using the generalized exponential rational function method we obtain new periodic and hyperbolic soliton solutions for the conformable Ginzburg-Landau equation with Kerr law nonlinearity. The conformable derivative was considered to obtain the exact solutions under constraint conditions. To determine the solution of the model, the method uses the generalization of the exponential rational function method. Numerical simulations are performed to confirm the efficiency of the proposed method.

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A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application

Cited by 36 publications

References 20 publications

Ternary-fractional differential transform schema: theory and application

Ternary-fractional differential transform schema: theory and application

Analytical Solution for the Time Fractional BBM‐Burger Equation by Using Modified Residual Power Series Method

Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity

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