An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

We construct a generalized solution of the Riemann problem for strictly hyperbolic systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a^′G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distributional solution. Thus, we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on “weaker than weak” solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method (in the presence of a linearly degenerate field) now apply, unchanged, to inhomogeneous systems.     

An algebraic geometry algorithm for scheduling in presence of setups and correlated demands

We study here a problem of schedulingn job types onm parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program.Methods of solution currently available—in integer programming and stochastic programming—are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Gröbner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems.Out algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system.Corresponding author.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

A New Version of Iterative Method for Solving Riemann Problem

A new version of iterative method for solving Riemann problem of gas dynamics is presented. In practice the new procedure exhibited a good convergence in cases where Riemann solution involves a strong rarefaction wave or two rarefaction waves. In the other cases the new version is identical with Godunov procedure.     

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.     

Solution of boundary-value problems in the theory of analytic functions of several variables in Vladimirov algebras

A solution is given for the Riemann problem for tubular domains in Vladimirov algebras in closed form by means of an integral representation of Bochner-Vladimirov type which is constructed here. In particular, the Schwartz problem is solved. The statement of the Hilbert problem in Vladimirov algebras is examined and its solution is given by a reduction to the Riemann problem, and in one case by a reduction to the Schwartz problem.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 51–60, July, 1977.     

Delta and singular delta locus for one‐dimensional systems of conservation laws

This work gives a condition for existence of singular and delta shock wave solutions to Riemann problem for 2×2 systems of conservation laws. For a fixed left‐hand side value of Riemann data, the condition obtained in the paper describes a set of possible right‐hand side values. The procedure is similar to the standard one of finding the Hugoniot locus. Fluxes of the considered systems are globally Lipschitz with respect to one of the dependent variables. The association in a Colombeau‐type algebra is used as a solution concept. Copyright © 2004 John Wiley &Sons, Ltd.     

Efficient Computation of Optimal Controls for the Exchange Processes During the Dialysis Therapy

In order to reduce the frequency of acute complications during the dialysis therapy the exchange processes of water and different solutes within the patient as well as across the dialyzer membrane shall be optimally controlled. With regard to a clinical application, this task requires the efficient treatment of a large-scale control problem, formulated in terms of a dynamical optimization problem. Equality and inequality conditions are given by the system describing the exchange processes and by the consideration of technical and medical constraints, respectively. Above all the complexity of the describing system prevents the application of standard optimization techniques as well as the construction of closed loop control laws and implies the construction of a control procedure which is specially adapted to the problem. The presented optimization method—denoted as controller PSEUDYGALG—represents a numerical iterative descent procedure, based on the approach of admissible direction. The procedure assumes an appropriate parameterization of the control problem as well as the availability of information about the input-output structure of the underlying describing system. In order to achieve the required efficiency, adaptive penalization strategies for the performance criterion and update modules for the descent information of each iterative step are presented. The controller allows both the treatment of badly and well conditioned control problems which are characterized by the occurrence and the absence of contradictional requirements for the performance criterion, respectively. PSEUDYGALG represents an off-line control method, but due to the achieved efficiency an on-line deployment by receding horizon approaches is in principle possible. Even though the controller has been developed for the dialysis problem it can be applied to a wide range of comparable control problems if the two assumptions—appropriate parameterization and knowledge about the input-output structure of the underlying system—are met.     

A Global Solution to a Two-dimensional Riemann Problem Involving Shocks as Free Boundaries

We present a global solution to a Riemann problem for the pressure gradient system of equations.The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar variable plane. Our main contribution in methodology is handling the tangential oblique derivative boundary values.     

Greedy Randomized Adaptive Search Procedures

Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.     

The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.     

Application of the Adomian decomposition method for the Fokker–Planck equation

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker–Planck equation and some similar equations. This method can successfully be applied to a large class of problems. The Adomian decomposition method needs less work in comparison with the traditional methods. This method decreases considerable volume of calculations. The decomposition procedure of Adomian will be obtained easily without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent series with easily computed components. In this work we are concerned with the application of the decomposition method for the linear and nonlinear Fokker–Planck equation. To give overview of methodology, we have presented several examples in one and two dimensional cases.     

A generalized Riemann problem for quasi-one-dimensional gas flows

A generalization of the Riemann problem for gas dynamical flows influenced by curved geometry, such as flows in a variable-area duct, is solved. For this generalized Riemann problem the initial data consist of a pair of steady-state solutions separated by a jump discontinuity. The solution of the generalized Riemann problem is used as a basis for a random choice method in which steady-state solutions are used as an Ansatz to approximate the spatial variation of the solution between grid points. For nearly steady flow in a Laval nozzle, where this Ansatz is appropriate, this generalized random choice method gives greatly improved results.     

Optimal use of mixed catalysts for two successive chemical reactions

Recently, Gunn and Thomas showed that—when the conversion of a feedstock to a product takes place in two chemically distinct steps, each of which is promoted by a different catalyst—there are advantages to be gained by mixing the catalysts in a single reactor rather than carrying out the two reaction steps separately. In this paper, the maximum principle is applied to the problem of determining the optimal variation in catalyst blend along the reactor and, for a simple first-order kinetic scheme, it is shown to lead to a complete solution in closed form.     

A dual decomposition algorithm

Suppose that a large-scale block-diagonal linear programming problem has been solved by the Dantzig—Wolfe decomposition algorithm and that an optimal solution has been attained. Suppose further that it is desired to perform a post-optimality analysis or a complete parametric analysis on the cost-coefficients or the RHS of the linking constraints. Efficient techniques for performing these analyses for the ordinary simplex case have not been easily applied to this case as one operation involves doing a minimizing ratio between all columns of two rows of the tableau. As the columns are not readily known in Dantzig—Wolfe decomposition, other techniques must be used. To date, suggested methods involve solving small linear programs to find these minimizing ratios. In this paper a method is presented which requires solving no linear programs (except possibly in the case of degeneracy of a subproblem) using and utilizing only the information typically stored for Dantzig—Wolfe decomposition.     

Analysis of Schrödinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

In this article, we consider the problem of optimal approximation of eigenfunctions of Schrödinger operators with isolated inverse square potentials and of solutions to equations involving such operators. It is known in this situation that the finite element method performs poorly with standard meshes. We construct an alternative class of graded meshes, and prove and numerically test optimal approximation results for the finite element method using these meshes. Our numerical tests are in good agreement with our theoretical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1130–1151, 2014     

Whitham Equations, Bergman Kernel and Lax—Levermore Minimizer

We study multiphase solutions of the Whitham equations. The Whitham equations describe the zero dispersion limit of the Cauchy problem for the Korteweg—de Vries (KdV) equation. The zero dispersion solution of the KdV equation is determined by the Lax—Levermore minimization problem. The minimizer is a measurable function on the real line. When the support of the minimizer consists of a finite number of disjoint intervals to be determined, the minimization problem can be reduced to a scalar Riemann Hilbert (RH) problem. For each fixed x and t 0, the end-points of the contour are determined by the solution of the Whitham equations. The Lax—Levermore minimizer and the solution of the Whitham equations are described in terms of a kernel related to the Bergman kernel. At t = 0 the support of the minimizer consists of one interval for any value of x, while for t > 0, the number of intervals is larger than one in some regions of the (x,t) plane where the multiphase solutions of the Whitham equations develop. The increase of the number of intervals happens whenever the solution of the Whitham equations has a point of gradient catastrophe. For a class of smooth monotonically increasing initial data, we show that the support of the Lax—Levermore minimizer increases or decreases the number of its intervals by one near each point of gradient catastrophe. This result justifies the formation and extinction of the multiphase solutions of the Whitham equations. Furthermore we characterize a class of initial data for which all the points of gradient catastrophe occur only a finite number of times and therefore the support of the Lax—Levermore minimizer consists of a finite number of disjoint intervals for any x and t 0. This corresponds to give an upper bound to the genus of the solution of the Whitham equations. Similar results are obtained for the semi-classical limit of the defocusing nonlinear Schrödinger equation.     

The Riemann problem for the nonisentropic Baer–Nunziato model of two-phase flows

The Riemann problem for the well-known Baer–Nunziato model of two-phase flows is solved. The system consists of seven partial differential equations with nonconservative terms. The most challenging problem is that this model possesses a double eigenvalue. Although characteristic speeds coincide, the curves of composite waves associated with different characteristic fields can be still constructed. They will also be incorporated into composite wave curves to form solutions of the Riemann problem. Solutions of the Riemann problem will be constructed when initial data are in supersonic regions, subsonic regions, or in both kinds of regions. A unique solution and solutions with resonance are also obtained.     

Long-time asymptotic for the Hirota equation via nonlinear steepest descent method

We present the Riemann–Hilbert problem formalism for the initial value problem for the Hirota equation on the line. We show that the solution of this initial value problem can be obtained from that of associated Riemann–Hilbert problem, which allows us to use nonlinear steepest descent method/Deift–Zhou method to analyze the long-time asymptotic for the Hirota equation.     

A pseudospectral method for nonlinear Duffing equation involving both integral and non‐integral forcing terms

The Legendre pseudospectral method is developed for the numerical solution of nonlinear Duffing equation involving both integral and non‐integral forcing terms. By using differentiation matrix, the problem is reduced to the solution of a system of algebraic equations. The method is general, easy to implement, and yields very accurate results. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. Copyright © 2014 John Wiley & Sons, Ltd.