• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권2호
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• pp.81-87
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• 2017

In this note, we show that the cone property is satisfied for a class of dissipative equations of the form $u_t={\Delta}u+f(x,u,{\nabla}u)$ in a domain ${\Omega}{\subset}{\mathbb{R}}^2$ under the so called exactness condition for the nonlinear term. From this, we see that the global attractor is represented as a Lipshitz graph over a finite dimensional eigenspace.

• Journal of Korean Society of Coastal and Ocean Engineers
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• 제10권1호
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• pp.1-9
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• 1998

Among the phenomena of water-wave transformation, the wave diffraction is prominent for waves insidc the harbor. It is important to study how accurately the diffraction can be resolved by the numerical model. Three parabolic mild-slope equations, i.e., simple, wide-ang1e, three-parameter parabolic equations, are compared in view of the diffraction of water-waves around a semi-infinite breakwater. To avoid reflections at lateral boundaries, we apply the perfect boundary condition of Dalrymple and Martin (1992) in case of simple parabolic equation. The numerical results for the case of a semi-infinite breakwater are compared with the analytical solution of Penney and Price (1952). All the results are very accurate when waves attack the breakwater normally. When waves attack the breakwater obliquely, however, the simple parabolic equation yields the worst solution and the three-parameter parabolic equation yields the most accurate solution.

• The Pure and Applied Mathematics
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• 제7권2호
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• pp.101-113
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• 2000

In this paper, we study controllability of nonlinear delay parabolic equa-tions with nonlocal initial condition under boundary input.

• Journal of applied mathematics & informatics
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• 제8권1호
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• pp.137-149
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• 2001

In this paper we consider some nonlinear parabolic partial differential equations with distributed deviating arguments and establish sufficient conditions for the oscillation of some boundary value problems.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.165-177
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• 2013

In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

• Journal of Mechanical Science and Technology
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• 제14권2호
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• pp.168-176
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• 2000

Related to the error dynamics of an adaptive system, averaging theorems are developed for coupled differential equations which consist of ordinary differential equations and a parabolic partial differential equation. The results are then applied to the convergence analysis of the parameter estimate errors in the model reference adaptive control of a nonautonomous parabolic partial differential equation with lowly time-varying parameters.

• International Journal of Railway
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• 제8권2호
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• pp.46-49
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• 2015

In this paper, improved out-of-plane design of parabolic arches was proposed based on the current design code. The arches resist general loading by a combination of axial compression and bending actions, and the interaction formula between two extreme cases of axial and bending actions is generally used for the design. Firstly, the out-of-plane buckling strength of arches in a pure axial compression and a pure bending were studied. Then, out-of-plane design of parabolic aches under general transverse loading was investigated. From the results, it can be found that the proposed design equations provided good prediction of out-of-plane strength for parabolic arches which satisfy the thresholds for deep arches, while proposed design equations overestimated the buckling load of shallow arches.

• Steel and Composite Structures
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• 제31권2호
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• pp.187-199
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• 2019

This paper presents the semi-analytical development of the dynamic instability behavior and the dynamic response of functionally graded (FG) cylindrical shallow shell panel subjected to different type of periodic axial compression. First, in prebuckling analysis, the stresses distribution within the panels are determined for respective loading type and these stresses are used to study the dynamic instability behavior and the dynamic response. The prebuckling stresses within the shell panel are the same as applied in-plane edge loading for the case of uniform and linearly varying loadings. However, this is not true for the case of parabolic loadings. The parabolic edge loading produces all the stresses (${\sigma}_{xx}$, ${\sigma}_{yy}$ and ${\tau}_{xy}$) within the FG cylindrical panel. These stresses are evaluated by minimizing the membrane energy via Ritz method. Using these stresses the partial differential equations of FG cylindrical panel are formulated by applying Hamilton's principal assuming higher order shear deformation theory (HSDT) and von-$K{\acute{a}}rm{\acute{a}}n$ non-linearity. The non-linear governing partial differential equations are converted into a set of Mathieu-Hill equations via Galerkin's method. Bolotin method is adopted to trace the boundaries of instability regions. The linear and non-linear dynamic responses in stable and unstable region are plotted to know the characteristics of instability regions of FG cylindrical panel. Moreover, the non-linear frequency-amplitude responses are obtained using Incremental Harmonic Balance (IHB) method.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.285-297
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• 2013

This paper deals with parabolic equations coupled via nonstandard growth sources, subject to homogeneous Dirichlet boundary conditions. Three kinds of necessary and sufficient conditions are obtained, which determine the complete classifications for non-simultaneous and simultaneous blowup phenomena. Moreover, blowup rates are given.

• Honam Mathematical Journal
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• 제17권1호
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• pp.107-118
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• 1995

We study the asymptotic behavior of nonnegative singular solutions of semilinear parabolic equations of the type $$u_t={\Delta}u-(u^q)_y-u^p$$ defined in the whole space $x=(x,y){\in}R^{N-1}{\times}R$ for t>0, with initial data a Dirac mass, ${\delta}(x)$. The exponents q, p satisfy $$1 where $q^*=max\{q,(N+1)/N\}$.