• Water Engineering Research
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• v.5 no.4
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• pp.177-183
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• 2004

This paper introduces an eigenvalue method of transforming the hyperbolic partial differential equations of a particular unsteady friction water hammer model into characteristic form. This method is based on the solution of the corresponding one-dimensional Riemann problem that transforms hyperbolic quasi-linear equations into ordinary differential equations along the characteristic directions, which in this case arises as the eigenvalues of the system. A mathematical justification and generalization of the eigenvalues method is provided and this approach is compared to the traditional characteristic method.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.11-27
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• 2015

Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a at bottom. The main objective in this paper is to apply the generalized Riccati equation mapping method for obtaining more exact traveling wave solutions of Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation. More precisely, the obtained solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functional form. Solutions obtained are potentially significant for the explanation of better insight of physical aspects of the considered nonlinear physical models.

• Journal of Advanced Marine Engineering and Technology
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• v.40 no.5
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• pp.389-396
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• 2016

The theory of Fourier heat conduction predicts accurately the temperature profiles of a system in a non-equilibrium steady state. However, in the case of transient states at the nanoscale, its applicability is significantly limited. The limitation of the classical Fourier's theory was overcome by C. Cattaneo and P. Vernotte who developed the theory of non-Fourier heat conduction in 1958. Although this new theory has been used in various thermal science areas, it requires considerable mathematical skills for calculating analytical solutions. The aim of this study was the identification of a newer and a simpler type of solution for the hyperbolic partial differential equations of the non-Fourier heat conduction. This constitutes the first trial in a series of planned studies. By inspecting each term included in the proposed solution, the theoretical feasibility of the solution was achieved. The new analytical solution for the non-Fourier heat conduction is a simple exponential function that is compared to the existing data for justification. Although the proposed solution partially satisfies the Cattaneo-Vernotte equation, it cannot simulate a thermal wave behavior. However, the results of this study indicate that it is possible to obtain the theoretical solution of the Cattaneo-Vernotte equation by improving the form of the proposed solution.

• Convergence Security Journal
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• v.5 no.3
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• pp.93-97
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• 2005

MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic partial differential equations. Apparently, however, it has shown entropy violations under small discontinuity. This non-physical shock grows fast and eventually all the meaningful information of the solution disappears. Some modifications of MacCormack's methods follow ideas of central schemes with an advantage of second order accuracy for space and conserve the high order accuracy for time step also. Numerical results are shown to perform well for the one-dimensional Burgers' equation and Euler equations gas dynamic.

• International Journal of Control, Automation, and Systems
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• v.3 no.4
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• pp.601-611
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• 2005

In this paper, an active vibration control of a tensioned, elastic, axially moving string is investigated. The dynamics of the translating string are described with a non-linear partial differential equation coupled with an ordinary differential equation. A right boundary control to suppress the transverse vibrations of the translating continuum is proposed. The control law is derived via the Lyapunov second method. The exponential stability of the closed-loop system is verified. The effectiveness of the proposed control law is simulated.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.28 no.1
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• pp.11-21
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• 2004

In this paper, an active vibration control of a tensioned elastic axially moving string is investigated. The dynamics of the translating string ale described by a non-linear partial differential equation coupled with an ordinary differential equation. The time varying control in the form of the right boundary transverse motions is suggested to stabilize the transverse vibration of the translating continuum. A control law based on Lyapunov's second method is derived. Exponential stability of the translating string under boundary control is verified. The effectiveness of the proposed controller is shown through the simulations.

• 제어로봇시스템학회:학술대회논문집
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• 2005.06a
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• pp.2260-2265
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• 2005

In this paper, an active vibration control of a tensioned elastic axially moving string is investigated. The dynamics of the translating string are described by a non-linear partial differential equation coupled with an ordinary differential equation. A time varying control in the form of right boundary transverse motions is proposed in stabilizing the transverse vibrations of the translating continuum. A control law based on Lyapunov's second method is derived. Exponential stability of the closed-loop system is verified. The effectiveness of the proposed controller is shown through simulations.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.289-298
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• 2010

In this paper, we propose a second-order prediction/correction (SPC) domain decomposition method for solving one dimensional linear hyperbolic partial differential equation $u_{tt}+a(x,t)u_t+b(x,t)u=c(x,t)u_{xx}+{\int}(x,t)$. The method can be applied to variable coefficients problems and singular problems. Unconditional stability and error analysis of the method have been carried out. Numerical results support stability and efficiency of the method.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.683-699
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• 2013

In this paper, an improved ($\frac{G^{\prime}}{G}$)-expansion method is proposed for obtaining travelling wave solutions of nonlinear evolution equations. The proposed technique called ($\frac{F}{G}$)-expansion method is more powerful than the method ($\frac{G^{\prime}}{G}$)-expansion method. The efficiency of the method is demonstrated on a variety of nonlinear partial differential equations such as KdV equation, mKd equation and Boussinesq equations. As a result, more travelling wave solutions are obtained including not only all the known solutions but also the computation burden is greatly decreased compared with the existing method. The travelling wave solutions are expressed by the hyperbolic functions and the trigonometric functions. The result reveals that the proposed method is simple and effective, and can be used for many other nonlinear evolutions equations arising in mathematical physics.

• Journal of The Korean Society of Agricultural Engineers
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• v.58 no.1
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• pp.81-90
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• 2016

The Agricultural Systems Application Platform (ASAP) provides bottom-up modelling and simulation environment for agricultural engineer. The purpose of this study is to expand usability of the ASAP to the second order partial differential equations: elliptic equations, parabolic equations, and hyperbolic equations. The ASAP is a general-purpose simulation tool which express natural phenomenon with capsulized independent components to simplify implementation and maintenance. To use the ASAP in continuous problems, it is necessary to solve partial differential equations. This study shows usage of the ASAP in elliptic problem, parabolic problem, and hyperbolic problem, and solves of static heat problem, heat transfer problem, and wave problem as examples. The example problems are solved with the ASAP and Finite Difference method (FDM) for verification. The ASAP shows identical results to FDM. These applications are useful to simulate the engineering problem including equilibrium, diffusion and wave problem.