Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability

Alharbi, Abdulghani; Almatrafi, M. B.

doi:10.1016/j.rinp.2019.102870

Cited by 34 publications

(12 citation statements)

References 35 publications

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“…Investigating both the analytical and numerical results for a given nonlinear partial differential equations (NLPDEs) performs a remarkable role in the explanation of complex phenomena in biology, economy, engineering, mathematical, signal processing, ocean engineering, optics, fluid mechanics, plasma physics, and chemical physics (Alharbi & Almatrafi, 2020a;Alharbi & Almatrafi, 2020b;Alharbi, Almatrafi, & Abdelrahman, 2020;Seadawy, Iqbal, & Lu, 2019). Thus, various vital approaches for determining the solutions of PDEs have been vastly suggested and presented.…”

Section: Introductionmentioning

confidence: 99%

New exact and numerical solutions for the KdV system arising in physical applications

Almatrafi

Alharbi

Abdelrahman

2021

Arab Journal of Basic and Applied Sciences

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The Kortweg-de Vries (KdV) equation is more appropriate to simulate some natural phenomena and gives more accurate results for some physical systems such as the movement of water waves. In this work, novel analytical traveling wave solutions for a nonlinear KdV system are explored using the sech method. The exact solutions are presented in the form of hyperbolic functions. These solutions show the propagation of water waves on the surface. We also implement the numerical adaptive moving technique (MMPDEs) to construct the relevant system's numerical solutions. A detailed comparison between the numerical and analytical solutions is also presented to confirm the accuracy of the numerical technique. We present some new 2D and 3D figures to illustrate the behaviour of the exact and numerical solutions. The obtained numerical solutions verify the accuracy of the considered methods qualitatively and quantitatively. The stability of the obtained solutions is investigated using the Hamiltonian system. The achieved results can be applied for some new observations in the ocean, coastal water, macroscopic phenomena, and processes. The proposed techniques are potent tools to solve many other non-linear partial differential equations in applied mathematics.

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Section: Introductionmentioning

confidence: 99%

New exact and numerical solutions for the KdV system arising in physical applications

Almatrafi

Alharbi

Abdelrahman

2021

Arab Journal of Basic and Applied Sciences

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“…Despite the recent extensive advances in the theory of differential equations, it can generally be said that it is still a complex task to determine an analytical solution for many ordinary and partial differential equations [1][2][3][4][5][6][7][8][9]. One of the events that led to the introduction of a wide range of new methods was the emergence and use of computers.…”

Section: Introductionmentioning

confidence: 99%

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Neirameh

Parvaneh

2021

Adv Differ Equ

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Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.

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“…Some developed techniques and principles include the inverse scattering transform [4], the trial function process [5], the sine-cosine principle [6], the Weierstrass elliptic function approach [7], the tanh-sech technique [8], the Fexpansion technique [9], Hirota's bilinear principle [10], the modified tanh-function tech-nique [11], the extended tanh-technique [12], the exp(-f (ζ ))-expansion process [13,14], and the truncated Painleve expansion [15]. More information about other techniques can be obtained in [16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Constructions of the soliton solutions to the good Boussinesq equation

2020

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The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the $L_{2}$ L 2 error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

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Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability

Cited by 34 publications

References 35 publications

New exact and numerical solutions for the KdV system arising in physical applications

New exact and numerical solutions for the KdV system arising in physical applications

Analytical solitons for the space-time conformable differential equations using two efficient techniques

Constructions of the soliton solutions to the good Boussinesq equation

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