• 대한수학회지
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• 제50권5호
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• pp.945-957
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• 2013

In this paper we consider a Riemann problem, in particular, the case of the presence of the semi-hyperbolic patches arising from a transonic shock in simple waves interaction. Under this circumstance, we construct global solutions of the two-dimensional Riemann problem of the pressure gradient system. We approach the problem as a Goursat boundary value problem and a mixed initial-boundary value problem, where one of the boundaries is the transonic shock.

• 대한수학회보
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• 제50권1호
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• pp.201-216
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• 2013

In this note, we consider the Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws. Without the restriction that each jump of the initial data projects one planar elementary wave, six topologically distinct solutions are constructed by applying the generalized characteristic analysis method, in which the delta shock waves and the vacuum states appear. Moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct global solutions.

• 대한수학회보
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• 제45권1호
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• pp.191-205
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• 2008

In this paper, we construct the analytic solutions and numerical solutions for a two-dimensional Riemann problem for Burgers' equation. In order to construct the analytic solution, we use the characteristic analysis with the shock and rarefaction base points. We apply the composite scheme suggested by Liska and Wendroff to compute numerical solutions. The result is coincident with our analytic solution. This demonstrates that the composite scheme works pretty well for Burgers' equation despite of its simplicity.

• 한국전산유체공학회지
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• 제2권2호
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• pp.51-58
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• 1997

A reliable computational solver has been developed for the analysis of three-dimensional inviscid compressible flows around a nacelle of a high bypass ratio turbofan engine, The numerical algorithm is based on the modified Godunov scheme to allow the second order accuracy for space variables, while keeping the monotone features. Two step time integration is used not only to remove time step limitation but also to provide the second order accuracy in a time variable. The multi-block approach is employed to calculate the complex flow field, using an algebraic, conformal, and elliptic method. The exact solution of Riemann problem is used to define boundary conditions. The accuracy of the developed solver is validated by comparing its results around the isolated nacelle in the cruise flight regime with the solution obtained using a commercial code "RAMPANT. "

• 대한수학회보
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• 제47권1호
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• pp.81-87
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• 2010

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) $-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$, 0 < t < 1 u(0) = u(1) = 0, where $\lambda$ > 0 is a parameter, 1 < $\alpha$ $\leq$ 2, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville differentiation, f : [0, 1] ${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is continuous, and q(t) : (0, 1) $\rightarrow$ [0, $+\infty$] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.

• 대한수학회보
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• 제47권1호
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• pp.29-37
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• 2010

We establish the existence of weak solutions in an infinite subsonic channel in the self-similar plane to the two-dimensional Burgers system. We consider a boundary value problem in a fixed domain such that a part of the domain is degenerate, and the system becomes a second order elliptic equation in the channel. The problem is motivated by the study of the weak shock reflection problem and 2-D Riemann problems. The two-dimensional Burgers system is obtained through an asymptotic reduction of the 2-D full Euler equations to study weak shock reflection by a ramp.

• 대한수학회지
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• 제53권4호
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• pp.929-967
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• 2016

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.