Numerical Solution for Nonlinear Klein–Gordon Equation via Operational Matrix by Clique Polynomial of Complete Graphs

This study proposed a new novel approach called the clique polynomial method (CPM) and applied it to the system of highly nonlinear ordinary differential equations (ODEs) associated with different cases of boundary layer natural convection fluid flow problems. Here, three specific problems are considered: (1) laminar film condensation of a fluid-saturated flow with isothermal vertical plate; (2) natural convection boundary layer fluid flow; and (3) heat transfer boundary layer flow due to a stretching sheet. CPM results, which do not need small parameters, are compared with numerical results, and the outcomes are established to be in good agreement. The CPM is hugely proficient and a sensible choice for finding outcomes to this type of coupled nonlinear ODEs. The work is in outstanding accord for coupled nonlinear ODEs in engineering applications.

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“…Moreover, some of the properties and theorems on clique polynomials of complete graph are discussed in Ref. [20].…”

Section: Theoretical Results On Clique Polynomialsmentioning

confidence: 99%

Study of special types of boundary layer natural convection flow problems through the clique polynomial method

Kumbinarasaiah

Raghunatha

2021

Heat Trans

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“…where the notation h(u) has been studied in several popular forms. [28][29][30][31][32][33][34] For example, in the cases h(u) = sin(u), h(u) = sin(u) + sin(2u), h(u) = sinh(u) + sinh(2u), and h(u) = u 3 − u, Equation (1.6) is called sine-Gordon equation, double sine-Gordon equation, double sinh-Gordon equation, and 𝜙 4 equation, respectively. It should be noted that in addition to the nonlinear aspect of this problem, the domain of the problem can be unbounded.…”

Section: Literature Reviewmentioning

confidence: 99%

“…In Bulbul and Sezer, 28 a numerical technique based on collocation and Taylor matrix method was applied to solve the NKG equation. Kumbinarasaiah et al 34 proposed a numerical technique based on the operational matrix by clique polynomial to solve the NKG equation. In previous studies, [23][24][25][26] the authors have applied the RBFs for the spatial direction and related derivatives and also have used the forward FD for the time derivative.…”

Section: Literature Reviewmentioning

confidence: 99%

“…NKG equation has been used in modeling of many different phenomena including the propagation of dislocations in crystals and the behavior of elementary particles, the propagation of fluxons in the Josephson junctions, and the motion of rigid pendula attached to a stretched wire 27 . The general form of the NKG equation is presented as follows:

{u}_{tt}\left(x,t\right)-{u}_{xx}\left(x,t\right)&#x0003D;h(u)&#x0002B;f\left(x,t\right),

where the notation

h(u)

has been studied in several popular forms 28–34 . For example, in the cases

h(u)&#x0003D;\sin (u)

h(u)&#x0003D;\sin (u)&#x0002B;\sin (2u)

h(u)&#x0003D;\sinh (u)&#x0002B;\sinh (2u)

, and

h(u)&#x0003D;{u}&#x0005E;3-u

, Equation () is called sine‐Gordon equation, double sine‐Gordon equation, double sinh‐Gordon equation, and

{\phi}&#x0005E;4

equation, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Rashedi

2022

Math Methods in App Sciences

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In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein-Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.

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“…Berman [24] studied the Laminar ow in channels with porous walls.Preliminaries of Laguerre wavelets and frames are given in [25]. Recently, very attractive uid ow problems are solved by using wavelet technique [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

A novel approach on micropolar fluid flow in a porous channel with high mass transfer via wavelet frames

Kumbinarasaiah

Raghunatha

2021

Nonlinear Engineering

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In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed.

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Numerical Solution for Nonlinear Klein–Gordon Equation via Operational Matrix by Clique Polynomial of Complete Graphs

Cited by 21 publications

References 34 publications

Study of special types of boundary layer natural convection flow problems through the clique polynomial method

Study of special types of boundary layer natural convection flow problems through the clique polynomial method

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

A novel approach on micropolar fluid flow in a porous channel with high mass transfer via wavelet frames

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