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

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Abstract

This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the nonlinear Klein–Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational matrix, we transform the nonlinear KG equation into a system of algebraic equations. Unknown coefficients to be determined by Newton’s method. The present technique is applied to four problems, and the obtained results are compared with another method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.

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Change history

  • 04 August 2023

    The original article was updated for removal of unnecessary hyphens present in the text.

Abbreviations

\(\alpha\) :

Real number

\(P\left( y \right)\) :

A nonlinear term in Eq. (1.1)

\(y\left( {x,t} \right)\) :

The wave displacement at \(x\) and \(t\)

\(A\left( x \right)\) :

Wave kinks

\(V\left( G \right)\) :

The vertex set of \(G\)

\(E\left( G \right)\) :

Edge set of \(G\)

\(S\left( {G;x} \right)\) :

Clique polynomials of a graph \(G\)

\(K_{n}\) :

Complete graph

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Authors and Affiliations

  1. Department of Mathematics, Bangalore University, Bengaluru, India

    S. Kumbinarasaiah

  2. Department of Mathematics, Karnatak University, Dharwad, India

    H. S. Ramane & K. S. Pise

  3. Department of Mathematics, SASTRA University, Tanjore, India

    G. Hariharan

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  1. S. Kumbinarasaiah
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  2. H. S. Ramane
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  3. K. S. Pise
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Kumbinarasaiah, S., Ramane, H.S., Pise, K.S. et al. Numerical Solution for Nonlinear Klein–Gordon Equation via Operational Matrix by Clique Polynomial of Complete Graphs. Int. J. Appl. Comput. Math 7, 12 (2021). https://doi.org/10.1007/s40819-020-00943-x

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