Analytical Solution of the Space-Time Fractional Nonlinear Schrödinger Equation

Abdel-Salam, Emad A.-B.; Yousif, Eltayeb A.; El-Aasser, Mostafa A.

doi:10.1016/s0034-4877(16)30002-7

Cited by 66 publications

(15 citation statements)

References 45 publications

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“…where Q = Q(ξ) satisfies Equation (12). The parameters a 0 , a 1 , a 2 , b 0 , and b 1 are determined at the next step.…”

Section: Obtaining Exact Solutions Of Equation (15) Using the Generalmentioning

confidence: 99%

“…The parameters a 0 , a 1 , a 2 , b 0 , and b 1 are determined at the next step. Substituting Equation (108) into Equation (18), and utilizing Equation (12) and then setting all coefficients of the functions Q k to zero, we get…”

Section: Obtaining Exact Solutions Of Equation (15) Using the Generalmentioning

confidence: 99%

“…Some examples of approaches for obtaining approximate solutions in an analytical form to NPDEs are the Adomian decomposition method (ADM) [1,2], the variational iteration method (VIM) [3,4], the differential transform method (DTM) [5], and the homotopy perturbation method (HPM) [6,7]. Some algorithms for obtaining explicit exact solutions of NPDEs are, for instance, the ( G G )-expansion method [8,9], the ( G G , 1 G )-expansion method [10], the fractional Riccati expansion method [11,12], the improved extended tanh-coth method [13], the Kudryashov method [14,15], and the sub-equation method [16,17]. All of the above methods are based on the homogeneous balance principle.…”

Section: Introductionmentioning

confidence: 99%

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New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

2020

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The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel ( G ′ G ) -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel ( G ′ G ) -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

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“…where Q = Q(ξ) satisfies Equation (12). The parameters a 0 , a 1 , a 2 , b 0 , and b 1 are determined at the next step.…”

Section: Obtaining Exact Solutions Of Equation (15) Using the Generalmentioning

confidence: 99%

Section: Obtaining Exact Solutions Of Equation (15) Using the Generalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

2020

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“…Trofimov and Peskov [40] have also solved the GPE using a conservative finite difference method. Moreover, multidimensional NSE was solved by several other numerical methods, such as riccati expansion method [1], finite difference method [38, 44], finite element method [10] Galerkin finite element method [43], momentum representation method [9], symplectic and multisymplectic methods [2, 39], compact scheme [18], split‐step Fourier scheme [32], compact boundary value method [31], spectral method [3, 28, 29, 30], operational matrix method [23, 27, 37], discrete collocation method based [26], and spline collocation method [22].…”

Section: Introductionmentioning

confidence: 99%

Time–space Jacobi pseudospectral simulation of multidimensional Schrödinger equation

Mittal

Shrivastava

Panda

2020

Numerical Methods Partial

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In this paper, the authors investigate the collision of soliton waves for multidimensional nonlinear Schrödinger equation (NSE) using the time–space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions and to prove the theory of error estimates and stability analysis for the equations. Using the Jacobi derivatives matrices the given problem is reduced to a system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. For numerical experiments, the method is tested on a number of different examples to study the behavior of collision of two and more than two solitons waves and single soliton wave. Moreover, numerical solutions are demonstrated to justify the theoretical results. The rate of convergence of the proposed method is obtained up to six‐order. The proposed method also preserves mass and energy conservation laws. A comparison of the numerical and exact solutions is depicted in the form of figures and tables.

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“…In [16] the maximum principles for solutions of the linear fractional diffusion equations are derived, in [17], the regional controllability for the Riemann-Liouville time fractional diffusion systems is analysed, in [18] a harmonic analysis of random fractional diffusion-wave equations is done, in [19] the Cauchy problem for fractional diffusion equations is discussed, and second order accuracy finite difference methods for spacefractional partial differential equations are proposed in [20]. The space-time fractional nonlinear Schrödinger equation is solved by mean of on the fractional Riccati expansion method [21]. Analytical solution of time-fractional Drinfeld-SokolovWilson system is obtained by using residual power series method [22].…”

Section: Introductionmentioning

confidence: 99%

Solution for a Space-time Fractional Diffusion Equation

Liu

Lv²

2017

Proceedings of the 2017 2nd International Conference on Modelling, Simulation and Applied Mathematics (MSAM2017)

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Abstract-This work focuses on investigating the solutions for a generalized fractional diffusion equation. This equation presents space and time fractional derivatives, includes an absorbent term and a linear external force, takes a time-dependent diffusion coefficient into account, and subjects to the natural boundaries and the general initial condition. We obtain explicit analytical expressions in terms of the Fox H functions for the probability distribution. In addition, we analyze the first passage time and the second movement distribution for the case characterized by the absence of absorbent term and external force for a semiinfinite interval with absorbing boundary condition.

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Analytical Solution of the Space-Time Fractional Nonlinear Schrödinger Equation

Cited by 66 publications

References 45 publications

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

Time–space Jacobi pseudospectral simulation of multidimensional Schrödinger equation

Solution for a Space-time Fractional Diffusion Equation

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