Periodic behavior for a degenerate fast diffusion equation

This work deals with the study of periodic solutions to a degenerate fast diffusion equation. The existence of the periodic solution to an intermediate problem restraint to a period T is proved first and then the result is extended by the data periodicity to all time real space. The approach involves an appropriate approximating problem whose periodic solution is proved via a fixed point theorem. Next, a passing to the limit procedure leads to the existence of the solution to the original problem on a time period. Finally, the behavior at large time of the solution to a Cauchy problem with periodic data is characterized.     

An approximation for the nonlinear filtering problem,with error bound †

For the standard continuous-time nonlinear filtering problem an approximation approach is derived. The approximate filter is given by the solution to an appropriate discrete-time approximating filtering problem that can be explicitly solved by a finite-dimensional procedure. Furthermore an explicit upper bound for the approximation error is derived. The approximating problem is obtained by first approximating the signal and then using measure transformation to express the original observation process in terms of the approximating signal     

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semi-linear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori L ^∞-bounds and sub- and super-solutions to apply a fixed point argument. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, “approximating” problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed non-linear problems with irregular data.     

Chebyshev Approximation of the Analytical Solution of Dirichlet Problem

In this paper linear programming method for minimax approximation is used to obtain an approximation to the analytical solution of a Dirichlet problem using the logarithmic potential function as an approximating function. This approach has the advantage of producing a better approximation than that using other solution of the potential equation as an approximating or basis function for a problem in $n=2$ dimensions.     

Minimax Control of Parabolic Systems with Dirichlet Boundary Conditions and State Constraints

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions. Accepted 7 June 1996     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.     

Estimates of the absolute error and a scheme for an approximate solution to scheduling problems

An approach is proposed for estimating absolute errors and finding approximate solutions to classical NP-hard scheduling problems of minimizing the maximum lateness for one or many machines and makespan is minimized. The concept of a metric (distance) between instances of the problem is introduced. The idea behind the approach is, given the problem instance, to construct another instance for which an optimal or approximate solution can be found at the minimum distance from the initial instance in the metric introduced. Instead of solving the original problem (instance), a set of approximating polynomially/pseudopolynomially solvable problems (instances) are considered, an instance at the minimum distance from the given one is chosen, and the resulting schedule is then applied to the original instance.     

On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels

We prove the convergence of an explicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two zero-flux boundary conditions. This problem arises in a model of sedimentation–consolidation processes in centrifuges and vessels with varying cross-sectional area. We formulate the definition of entropy solution of the model in the sense of Kružkov and prove the convergence of the scheme to the unique BV entropy solution of the problem. The scheme and the model are illustrated by numerical examples.     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.     

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.     

UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OF DIVERGENCE TYPE

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2~(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.     

On CP, LP and other piecewise perturbation methods for the numerical solution of the Schr?dinger equation

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schr?dinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.     

Continuous approximations of Gol’dshtik’s model

We consider continuous approximations to the Gol’dshtik problem for separated flows in an incompressible fluid. An approximated problem is obtained from the initial problem by small perturbations of the spectral parameter (vorticity) and by approximating the discontinuous nonlinearity continuously in the phase variable. Under certain conditions, using a variational method, we prove the convergence of solutions of the approximating problems to the solution of the original problem.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

Optimization with Variable Sets of Constraints and an Application to Truss Design

We discuss the minimization of a continuous function on a subset of Rⁿ subject to a finite set of continuous constraints. At each point, a given set-valued map determines the subset of constraints considered at this point. Such problems arise e.g. in the design of engineering structures.After a brief discussion on the existence of solutions, the numerical treatment of the problem is considered. It is briefly motivated why standard approaches generally fail. A method is proposed approximating the original problem by a standard one depending on a parameter. It is proved that by choosing this parameter large enough, each solution to the approximating problem is a solution to the original one. In many applications, an upper bound for this parameter can be computed, thus yielding the equivalence of the original problem to a standard optimization problem.The proposed method is applied to the problem of optimally designing a loaded truss subject to local buckling conditions. To our knowledge this problem has not been solved before. A numerical example of reasonable size shows the proposed methodology to work well.     

On CP,LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.     

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal feedback control in the stationary mathematical model of low concentrated aqueous polymer solutions

In this article we will consider the feedback control problem in the stationary model of motion of low concentrated aqueous polymer solutions. We demonstrate the solvability of an approximating problem, using some a priori estimates and the topological degree theory. Then the convergence (in some generalized sense) of solutions of approximating problems to a solution of the given problem is proved. Moreover, we show the existence of a solution minimizing a given convex, lower semicontinuous functional.     

IMEX method convergence for a parabolic equation

Although implicit-explicit (IMEX) methods for approximating solutions to semilinear parabolic equations are relatively standard, most recent works examine the case of a fully discretized model. We show that by discretizing time only, one can obtain an elementary convergence result for an implicit-explicit method. This convergence result is strong enough to imply existence and uniqueness of solutions to a class of semilinear parabolic equations.