Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas

The Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas with a small parameter are considered. Unlike the polytropic or barotropic gas cases, we find that firstly, as the parameter decreases to a certain critical number, the two-shock solution converges to a delta shock wave solution of the same system. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the solution is nothing but the delta shock wave solution to the zero-pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution tends to the vacuum solution to the zero-pressure relativistic system, and the solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to the zero-pressure relativistic system as pressure vanishes.     

On Torsion of Prismatic Bodies

The method of decaying residual solution is applied to obtain an approximate interior solution for the torsion of slender prismatic elastic bodies under different end conditions. The approximate solution is generally accurate up to terms that are exponentially small in the length-to-cross-sectional-width ratio. For stress end conditions, the result is identical to the classical Saint-Venant torsion solution. Similar types of simple solutions, not known previously, are obtained for different types of mixed end conditions. For displacement conditions at both ends, the corresponding Saint-Venant type result requires an accurate solution of a canonical problem for a semi-infinite prismatic body that is to be obtained once and for all. The solution of the canonical problem is elementary for a circular cross section. The approximate interior solution in that case is identical to the known exact interior solution.     

一类拟线性常微分方程的多解

说明一类拟线性特征值问题有两个正解;一个大解,一个小解。同时本也证明小解是一个山路解当参数大时发展成为尖解。     

AF—代数的闭的Jordan理想

讨论AF-代数的闭的Jordan理想。我们证明了AF-代灵敏的一个闭的子集是其一Jordan理想的充分必要条件是它是一个结合理想。     

Subspace-restricted singular value decompositions for linear discrete ill-posed problems

The truncated singular value decomposition is a popular solution method for linear discrete ill-posed problems. These problems are numerically underdetermined. Therefore, it can be beneficial to incorporate information about the desired solution into the solution process. This paper describes a modification of the singular value decomposition that permits a specified linear subspace to be contained in the solution subspace for all truncations. Modifications that allow the range to contain a specified subspace, or that allow both the solution subspace and the range to contain specified subspaces also are described.     

A simple characterization of solution sets of convex programs

By means of elementary arguments we first show that the gradient of the objective function of a convex program is constant on the solution set of the problem. Furthermore the solution set lies in an affine subspace orthogonal to this constant gradient, and is in fact in the intersection of this affine subspace with the feasible region. As a consequence we give a simple polyhedral characterization of the solution set of a convex quadratic program and that of a monotone linear complementarity problem. For these two problems we can also characterize a priori the boundedness of their solution sets without knowing any solution point. Finally we give an extension to non-smooth convex optimization by showing that the intersection of the subdifferentials of the objective function on the solution set is non-empty and equals the constant subdifferential of the objective function on the relative interior of the optimal solution set. In addition, the solution set lies in the intersection with the feasible region of an affine subspace orthogonal to some subgradient of the objective function at a relative interior point of the optimal solution set.     

Nonresonance and Global Existence of Nonlinear Elastic Wave Equations with External Forces

This paper establishes the global existence of classical solution to the system of homogeneous,isotropic hyperelasticity with time-independent external force,provided that the nonlinear term obeys a ty...     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.     

New travelling wave solutions for a nonlinearly dispersive wave equation of Camassa-Holm equation type

In this paper, the integral bifurcation method is used to study a nonlinearly dispersive wave equation of Camassa-Holm equation type. Loop soliton solution and periodic loop soliton solution, solitary wave solution and solitary cusp wave solution, smooth periodic wave solution and non-smooth periodic wave solution of this equation are obtained, their dynamic characters are discussed. Some solutions have an interesting phenomenon that one solution admits multi-waves when parameters vary.     

Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.     

On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

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Implicit solution function of P0 and Z matrix linear complementarity constraints

Using the least element solution of the P₀ and Z matrix linear complementarity problem (LCP), we define an implicit solution function for linear complementarity constraints (LCC). We show that the sequence of solution functions defined by the unique solution of the regularized LCP is monotonically increasing and converges to the implicit solution function as the regularization parameter goes down to zero. Moreover, each component of the implicit solution function is convex. We find that the solution set of the irreducible P₀ and Z matrix LCP can be represented by the least element solution and a Perron?CFrobenius eigenvector. These results are applied to convex reformulation of mathematical programs with P₀ and Z matrix LCC. Preliminary numerical results show the effectiveness and the efficiency of the reformulation.     

Integer Programming to Minimize Labour Costs

The main purpose of this paper is to demonstrate a real-world application of pure integer programming to find the optimum solution to a labour cost problem. The length of a daily working shift is defined as an integer variable and several shift strategies are analysed to determine the optimum length and shift combinations that satisfy a predicted demand at minimum cost. The state-space model has been used to predict the stochastic behaviour of monthly demands for beer and soft drink. Savings of about 7% of the annual sales have been obtained as a result of implementing the integer programming approach. A numerical example shows that the solution obtained by rounding off the continuous optimal solution does not match with the integer optimal solution. It was also noted that if a rounded-off solution is feasible, then it provides an initial integer solution for the branch-and-bound algorithm that may reduce the computational time.     

Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

A survey of dynamically-adaptive grids in the numerical solution of partial differential equations

The construction of dynamically-adaptive curvilinear coordinate systems based on numerical grid generation and the use thereof in the numerical solution of partial differential equations is surveyed, and correlations are made among the various approaches. These adaptive grids are coupled with the physical solution being done on the grid so that the grid points continually move in the course of the solution in order to resolve developing gradients, or higher variations, in the solution. Particular attention is given to systems using elliptic grid generation based on variational principles. It is noted that dynamic grid adaption can remove the oscillations common when strong gradients occur on fixed grids, and that it appears that when the grid adapts to the solution most numerical solution algorithms work well. Particular applications in computational fluid dynamics and heat transfer are noted.     

Higher order differentiability of solutions to backward stochastic differential equations

In this paper, we consider the differentiability in the sense of the Malliavin calculus of solutions to backward stochastic differential equations (BSDEs for short). It is known that a solution is differentiable in the sense of the Malliavin calculus and the derivative is also a solution to a linear BSDE. Under additional conditions, we will show that the higher order differentiability of a solution to a BSDE and that it also becomes a solution to a linear BSDE.     

Sequential Linear-Quadratic Method for Differential Games with Air Combat Applications

We present a numerical method for computing a local Nash (saddle-point) solution to a zero-sum differential game for a nonlinear system. Given a solution estimate to the game, we define a subproblem, which is obtained from the original problem by linearizing its system dynamics around the solution estimate and expanding its payoff function to quadratic terms around the same solution estimate. We then apply the standard Riccati equation method to the linear-quadratic subproblem and compute its saddle solution. We then update the current solution estimate by adding the computed saddle solution of the subproblem multiplied by a small positive constant (a step size) to the current solution estimate for the original game. We repeat this process and successively generate better solution estimates. Our applications of this sequential method to air combat simulations demonstrate experimentally that the solution estimates converge to a local Nash (saddle) solution of the original game.     

On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation

We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in a weighted L² space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all nonscattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing wave solution.     

Long Time Asymptotic Behavior of Solution of Difference Scheme for a Semilinear Parabolic Equation

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as $t\rightarrow\infty$. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.     

On differential equations with interval coefficients

In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.