Numerical solution of the nonlinear Klein–Gordon equation

Rashidinia, Jalil; Ghasemi, M.; Jalilian, R.

doi:10.1016/j.cam.2009.09.023

Cited by 58 publications

(37 citation statements)

References 21 publications

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“…1. Table 1 indicates that the proposed method requires more less nodes to attain the accuracy of the CBSM [22] and TPSM [7]. Also, it shows that this scheme performs better than TPSM.…”

Section: Methodsmentioning

confidence: 93%

“…In our approach, the fourth order finite difference approximation is first employed for discretizing of the temporal derivative similar to work that Rashidinia did in [22]. Then, the highest order derivatives (second order in this paper) of the solution function are approximated by Eq.…”

Section: The Numerical Methodsmentioning

confidence: 99%

“…Also, it shows that this scheme performs better than TPSM. [7,22]. Moreover, the space-time graph of the estimated solution is drawn in Fig.…”

Section: Methodsmentioning

confidence: 99%

“…The exact solution is given in [32] as Tables 3 and 4 indicate that the proposed method requires more less nodes to attain the accuracy of the CBSM [22] and TPSM [7]. …”

Section: Methodsmentioning

confidence: 99%

“…The numerical results of the Klein-Gordon equation by using this scheme is compared with the analytical solutions and solutions in [7,22]. These methods include Thin Plate Splines (TPS) RBF collocation method [7] and cubic B-spline collocation method (CBS) [22].…”

Section: The Numerical Experimentsmentioning

confidence: 99%

See 4 more Smart Citations

Integrated High Accuracy Multiquadric Quasi-interpolation Scheme For Solving The Nonlinear Klein-gordon Equation

Sarboland¹,

Aminataei²

2015

J. Math. Computer Sci.

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A collocation scheme based on the use of the multiquadric quasi-interpolation operator 2 W L , integrated radial basis function networks (IRBFNs) method and three order finite difference method is applied to the nonlinear Klein-Gordon equation. In the present scheme, the three order finite difference method is used to discretize the temporal derivative and the integrated form of the multiquadric quasi-interpolation scheme is used to approximate the unknown function and its spatial derivatives. Several numerical experiments are provided to show the efficiency and the accuracy of the given method.

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“…1. Table 1 indicates that the proposed method requires more less nodes to attain the accuracy of the CBSM [22] and TPSM [7]. Also, it shows that this scheme performs better than TPSM.…”

Section: Methodsmentioning

confidence: 93%

Section: The Numerical Methodsmentioning

confidence: 99%

“…Also, it shows that this scheme performs better than TPSM. [7,22]. Moreover, the space-time graph of the estimated solution is drawn in Fig.…”

Section: Methodsmentioning

confidence: 99%

“…The exact solution is given in [32] as Tables 3 and 4 indicate that the proposed method requires more less nodes to attain the accuracy of the CBSM [22] and TPSM [7]. …”

Section: Methodsmentioning

confidence: 99%

Section: The Numerical Experimentsmentioning

confidence: 99%

See 3 more Smart Citations

Integrated High Accuracy Multiquadric Quasi-interpolation Scheme For Solving The Nonlinear Klein-gordon Equation

Sarboland¹,

Aminataei²

2015

J. Math. Computer Sci.

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Hybrid fully spectral linearized scheme for time‐fractional evolutionary equations

Hamid

Usman

Wang

et al. 2020

Math Methods in App Sciences

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In the current study, we proposed an operational matrix‐based spectral computational method coupled with the Picard technique and successfully employed to seek the solutions to a family of nonlinear evolution differential equations. However, the operational matrices are constructed assisted by monomials, and relevant theorems are available to authenticate the approach mathematically. The iterative form of residual vector for the problems understudy has been presented to seek some new results of a nonlinear family of evolutionary fractional problems. The coupling of spectral technique with the Picard method can easily tackle the high nonlinearities including quadratic, cubic, exponential, and hyperbolic, while it brings more accurate and efficient results than the published results. A worthy comparative study with exact solutions and existing techniques including numerical and analytical is being made, which shows an excellent level of accuracy for the problems under study. The authenticity of the proposed method is validated via convergence, error‐bound, and norms, while the set of graphs is illustrated for integer and noninteger orders. Hence, this unlocks the opportunities to deal with solutions of the weakly singular integral and lattice Toda equations and other complex nonlinear problems arising in applied sciences and may provide a better observation of physical mechanism.

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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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Numerical solution of the nonlinear Klein–Gordon equation

Cited by 58 publications

References 21 publications

Integrated High Accuracy Multiquadric Quasi-interpolation Scheme For Solving The Nonlinear Klein-gordon Equation

Integrated High Accuracy Multiquadric Quasi-interpolation Scheme For Solving The Nonlinear Klein-gordon Equation

Hybrid fully spectral linearized scheme for time‐fractional evolutionary equations

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

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