Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

Yu, Danni; He, Ji‐Huan; García, Andrés

doi:10.1177/1461348418811028

Cited by 123 publications

(96 citation statements)

References 75 publications

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“…Theorem 6. Suppose X, U, U 0 , and V are as in Theorem (5), let pair (g, T) be an ordered Suzuki-type α-θ g -modified proximal contractive mappings, where g : U → U and T : U → CB(V) with all assumptions of Theorem (5). Then unique coincidence best proximity point of mappings (g, T) exist.…”

Section: Results In Partially Ordered B-metric Spacementioning

confidence: 99%

“…Proof. Following the same lines of proof of Theorem (5), and taking in account for self-mapping such that (u 0 , Tu 0 ) ∈ ∆, we have α(u 0 , Tu 0 ) = 1, then every ordered Suzuki-type α-ψ-modified contraction becomes ordered Suzuki-type ψ-modified contraction and the remaining conditions of Theorem (5) holds. Then, T has a unique fixed point.…”

Section: Definition 16mentioning

confidence: 90%

“…In 1989 and 1993, Bakhtin [2] and Czerwik [3], respectively, introduced the concept of b-metric space. As an application, Equation (2) is used in several iterative schemes, and the homotopy perturbation method (see , for details, in [4,5]. After the revolution in mathematics due to L. Zadeh ([6]), by presenting the concept of fuzzy sets, Kramosil and Veeramani [7][8][9] introduced the revolutionary idea of fuzzy metric spaces.…”

Section: Introductionmentioning

confidence: 99%

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Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

Saleem

Vujaković²,

Baloch

et al. 2019

Mathematics

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Section: Results In Partially Ordered B-metric Spacementioning

confidence: 99%

Section: Definition 16mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

Saleem

Vujaković²,

Baloch

et al. 2019

Mathematics

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“…First, we use the modified variational iteration algorithm-I; then, we will use the modified algorithm-II for the solution of this test problem. Utilizing the estimation of λ 1 (ζ) and λ 2 (ζ) in Equations (15) and (16) results in the underneath iterative structure:…”

Section: Test Problemmentioning

confidence: 99%

“…This is known as variational iteration algorithm-I [12][13][14], which is an advance improvement of the common Lagrange multiplier method [11]. These days, this technique [15][16][17][18][19][20][21][22][23] has been set up for offering a solution for a more extensive scope of problems, developing in several fields of pure and applied sciences. PDEs extensively arise in various physical applications such as propagation and the scattering of waves, magnetohydrodynamic flow through pipes, computational fluid dynamics, magnetic resonance imaging, the phenomena of turbulence and supersonic flow, the flow of a shock wave traveling in a viscous fluid, acoustic transmission, traffic and aerofoil flow theory, and the proposed technique has the ability to investigate these types problems effectively.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions of Coupled Burgers’ Equations

Ahmad

Khan

Cesarano

2019

Axioms

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In this article, two new modified variational iteration algorithms are investigated for the numerical solution of coupled Burgers' equations. These modifications are made with the help of auxiliary parameters to speed up the convergence rate of the series solutions. Three numerical test problems are given to judge the behavior of the modified algorithms, and error norms are used to evaluate the accuracy of the method. Numerical simulations are carried out for different values of parameters. The results are also compared with the existing methods in the literature. Arora [5] proposed a scheme known as the Lai cubic B-spline collocation scheme for the solution of coupled viscous Burger equations, where the authors used a crank Nicholson scheme and cubic B-spline functions for time integration and space integration, respectively. Lai and Ma [6] proposed a new lattice Boltzmann model for the solution of coupled Burger equations, after selecting the distribution functions, Chapman-Enskog expansion was employed for the solution of coupled Burger equations. The Chebyshev-Legendre pseudospectral method [7] has been utilized by Rashid et al. for coupled viscous Burgers (VB) equations, where the leapfrog scheme and Chebyshev-Legendre Pseudo-Spectral method (CLPS) method were used for the time direction and space direction, respectively. Kumar and Pandit [8] used a composite numerical scheme for the numerical evaluation of coupled Burger equations, where the scheme was developed based on Haar wavelets and finite difference. Mohammadi and Mokhtari [9] used a reproducing Kernal method for an analytical solution in the form series of systems of Burger equations. At last, in 2019, Bak et al.[10] developed a new approach named a semi-Lagrangian approach for the numerical solution of coupled Burger equations. We compare our results with those of [10], and show the applicability of our proposed method to handle such problems as those that arise in applied science and engineering.This paper aims to solve three types of coupled Burgers equations by employing variational iteration algorithm-II and one of the examples to be solved by modified variational iteration algorithm-I. The organization of the rest of the paper is as follows; in Section 2, we elucidate the variational iteration algorithm-II, in the next Section 3, the semi-numerical method is applied to three test problems, and a comparison is made with some other methods; lastly, some conclusions are drawn in the last Section 4. Modified Variational Iteration Algorithm-IIConsider the nonlinear diffusion equation:where the terms L[w(x, t)] and N[w(x, t)] are linear and nonlinear terms in that order, while c(x, t) is the source term. For a given w 0 (x, t), the approximate solution w n+1 (x, t) of Equation (4) can be obtained as below:w n+1 (x, t) = w n (x, t) + t 0

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(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

Tian

Ren

et al. 2019

Math Methods in App Sciences

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In this paper, the theoretical model of dust ion-acoustic waves in collisionless, unmagnetized dust plasma is analyzed. According to the (3 + 1)-dimensional nonlinear dynamic propagation equations with the electron distribution in the presence of trapping particles and the kinetic equation of charge of dust particle, based on multiscale analysis and perturbation method, a new (3 + 1)-dimensional modified Burgers equation is constructed. Because of the space property of dust plasma, (3 + 1)-dimensional modified Burgers equation is more suitable than low-dimensional equation to describe the dust ion-acoustic waves. Furthermore, applying the semi-inverse method and the fractional variational principle, time-fractional modified Burgers equation is given, which owns potential value for comprehending the propagation feature of dust ion-acoustic waves in three-dimensional space. Following, the conservation laws of the time-fractional modified Burgers equation are discussed by using the Noether method. Finally, the shock wave solutions of the new model is obtained with the help of the G ′ ∕G-expansion method; meanwhile, the radial basis function method is also used to get the numerical solutions of the (3 + 1)-dimensional time-fractional modified Burgers equation. KEYWORDS(3 + 1)-dimensional time-fractional modified Burgers equation, conservation laws, dust ion-acoustic waves, radial basis function method, shock wave solution MSC CLASSIFICATION 35Q51; 35Q55After solving the equation Ac k =B k , we get the algebraic system at each time step, and we can obtain the solution by using Equation (50).

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Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

Cited by 123 publications

References 75 publications

Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

Numerical Solutions of Coupled Burgers’ Equations

(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

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