• 충청수학회지
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• 제25권1호
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• pp.1-18
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• 2012

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

• Journal of applied mathematics & informatics
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• 제30권5_6호
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• pp.925-934
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• 2012

In this work the sine-Gordon equation will be examined for multiple soliton solutions. The higher dimensional sine-Gordon equations will be studied for multiple soliton solutions as well. The simplified form of the Hirota's method will be employed to conduct this analytic study.

• Nonlinear Functional Analysis and Applications
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• 제28권3호
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• pp.801-812
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• 2023

In this paper, we investigate the superstability for the p-power-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from an approximation of the p-power-radical functional equation: $$f(\sqrt[p]{x^p+y^p})-f(\sqrt[p]{x^p-y^p})={\lambda}g(x)h(y),$$ where p is an odd positive integer and f, g, h are complex valued functions. Furthermore, the obtained results are extended to Banach algebras.

• 대한수학회보
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• 제60권6호
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• pp.1697-1704
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• 2023

We study a multidimensional nonlinear variational sine-Gordon equation, which can be used to describe long waves on a dipole chain in the continuum limit. By using the method of characteristics, we show that a solution of a nonlinear variational sine-Gordon equation with certain initial data in a multidimensional space has a singularity in finite time.

• Korean Journal of Mathematics
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• 제16권3호
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• pp.369-378
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• 2008

The aim of this paper is to study the stability problem for the pexider type trigonometric functional equation f(x + y) − f(x−y) = 2g(x)h(y), which is related to the d'Alembert, the Wilson, the sine, and the mixed trigonometric functional equations.

• 충청수학회지
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• 제20권4호
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• pp.465-476
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• 2007

The aim of this paper is to investigate the stability problem bounded by function for the generalized sine functional equations as follow: $f(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2\\g(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2$. As a consequence, we have generalized the superstability of the sine type functional equations.

• 충청수학회지
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• 제35권4호
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• pp.315-327
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• 2022

In this paper, we investigate the transferred superstability for the p-radical sine functional equation $$f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=f(x)f(y)$$ from the p-radical functional equations: $$f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)g(y),\;\\f({\sqrt[p]{x^p+y^p}})+f({\sqrt[p]{x^p-y^p}})={\lambda}g(x)h(y),$$ where p is an odd positive integer, λ is a positive real number, and f is a complex valued function. Furthermore, the results are extended to Banach algebras. Therefore, the obtained result will be forced to the pre-results(p=1) for this type's equations, and will serve as a sample to apply it to the extension of the other known equations.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1157-1169
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• 2010

This paper presents the analytical solution of the fractional Fokker-Planck equation by Adomian decomposition method. By using initial conditions, the explicit solution of the equation has been presented in the closed form and then the numerical solution has been represented graphically. Two different approaches have been presented in order to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제28권4호
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• pp.387-398
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• 2021

In this paper, we will find solutions and investigate the superstability bounded by constant for the p-radical functional equations as follows: $f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=\;\{(i)\;f(x)f(y),\\(ii)\;g(x)f(y),\\(iii)\;f(x)g(y),\\(iv)\;g(x)g(y).$ with respect to the sine functional equation, where p is an odd positive integer and f is a complex valued function. Furthermore, the results are extended to Banach algebra.

• 대한수학회지
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• 제46권5호
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• pp.1105-1117
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• 2009

In this paper a parameter identification problem for a damped sine-Gordon equation is studied from the theoretical and numerical perspectives. A spectral method is developed for the solution of the state and the adjoint equations. The Powell's minimization method is used for the numerical parameter identification. The necessary conditions for the optimization problem are shown to yield the bang-bang control law. Numerical results are discussed and the applicability of the necessary conditions is examined.