Sequence Of Approximate Solutions Research Articles

Global Entropy Solutions and Newtonian Limit for the Relativistic Euler Equations

We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on a careful entropy analysis, we establish the compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.

Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

Метод интегральных уравнений Вольтерра в исследовании динамики самогравитирующего газа, ограниченного свободной поверхностью

The paper investigates dynamics of the finite mass of self-gravitating ideal gas with a variable flow bounded by free surface. Unsteady gas flows are described by a phenomenological mathematical model of gas dynamics, which is constructed using a system of nonlinear integral-differential equations written in Eulerian coordinates. The gas motion is considered under the condition that the free boundary at all times consists of the same particles. This circumstance makes it convenient to switch from Eulerian coordinates to Lagrangian coordinates, for which the domain of determining the solution of the gas motion problem will be fixed in advance. The transformation of the system to Lagrangian coordinates makes it possible to reduce it to an equivalent system consisting of Volterra-type integral equations and the continuity equation in Lagrangian form, and get rid of the unknown boundary. Using the method of successive approximations, properties of compact spaces, a series of original and specific estimates found for functions describing the flows of the self-gravitating gas, convergence of a sequence of approximate solutions is proved under assumptions about smoothness of the initial data. The theorem of existence and uniqueness of the solution of the Cauchy problem for the system of integral equations of the Volterra type is proved, and, consequently, the solution of the desired problem is found within the framework of the assumptions made about the motion in vacuum of the isolated mass of the self-gravitating ideal gas with a variable flow domain.

A consistent quasi–second-order staggered scheme for the two-dimensional shallow water equations

Abstract A quasi–second-order scheme is developed to obtain approximate solutions of the two-dimensional shallow water equations (SWEs) with bathymetry. The scheme is based on a staggered finite volume space discretization: the scalar unknowns are located in the discretization cells while the vector unknowns are located on the edges of the mesh. A monotonic upwind-central scheme for conservation laws (MUSCL)-like interpolation for the discrete convection operators in the water height and momentum balance equations is performed in order to improve the accuracy of the scheme. The time discretization is performed either by a first-order segregated forward Euler scheme or by the second-order Heun scheme. Both schemes are shown to preserve the water height positivity under a Courants Friedrichs Lewy (CFL) condition and an important state equilibrium known as the lake at rest. Using some recent Lax–Wendroff type results for staggered grids, these schemes are shown to be LW-consistent with the weak formulation of the continuous equations, in the sense that if a sequence of approximate solutions is bounded and strongly converges to a limit, then this limit is a weak solution of the SWEs; besides, the forward Euler scheme is shown to be LW-consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes.

On a Computational Smeared Damage Approach to the Analysis of Strength of Quasi-Brittle Materials

Computational analysis of strength of quasi-brittle materials, crucial for the durability of building structures and industrial components, needs typically a smeared damage approach, referring to the Eringen theory of nonlocal elasticity. Unfortunately its ad hoc constitutive relations cannot avoid potential divergence of sequences of approximate solutions, exploiting some extended finite element techniques, as well as questionable or missing existence results for corresponding boundary value problems. Introducing a simple static partially linearized model problem of such type, this article demonstrates some relevant remedies and their limitations, with numerous references to desirable generalizations

On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

The equation Au=f with a linear symmetric positive definite operator A:D(A)⊂H→H having a discrete spectrum and dense image in a complex Hilbert space H is considered. This equation is transferred into the Hilbert space of finite orbits D(An) as well as into the Fréchet space of all orbits D(A∞), that is, the projective limit of the sequence of spaces {D(An)}. For an approximate solution of the inverse of A, linear spline central algorithms in these spaces are constructed. The convergence of the sequence of approximate solutions to the exact solution is proved. The obtained results are applied to the quantum harmonic oscillator operator Au(t)=−u″(t)+t2u(t), t∈R, in the Hilbert space of finite orbits D(An), and in the Fréchet space of all orbits D(A∞) that in this case coincides with the Schwartz space of rapidly decreasing functions. Some quantum mechanical interpretations of obtained results are also given.

Solution of Riccati matrix differential equation using new approach of variational iteration method

To obtain the approximate solution to Riccati matrix differential equations, a new variational iteration approach was ‎proposed, which is suggested to improve the accuracy and increase the convergence rate of the approximate solutons to the ‎exact solution. This technique was found to give very accurate results in a few number of iterations. In this paper, the ‎modified approaches were derived to give modified solutions of proposed and used and the convergence analysis to the exact ‎solution of the derived sequence of approximate solutions is also stated and proved. Two examples were also solved, which ‎shows the reliability and applicability of the proposed approach. ‎

WITHDRAWN: Study of a computational model for a differential equation with empirical functions based on the integral equations Fredholm of the first kind

The work out lines a method of constructing an approximate solution of a differential equation of the second order with input data obtained in the experiment (empirical functions). In such statement the problem belongs to the class of incorrect mathematical problems and often occurs, for example, in mathematical models of physical phenomena using measurement results of field experiments. To obtain the approximate solution of this problem requires construction of appropriate regularization algorithms based on the methods of the theory of functional analysis and ill-posed problems. In the present work is the construction of the approximate solution of odes with specified boundary conditions, are the so-called singular integrals. This allows you to put in the original equation Fredholm integral equation of the first kind and to find its numerical solution, i.e. the solution of the incorrect task. This uses a machine approximation of functions and their deriva- tives corresponding singular integrals and regularization method convergence of the sequence of approximate solutions, which implemented the so-called generalized inverse operators. Built in the end, a computational model allows to obtain a stable solution of ill-posed problems.

Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior

We prove global existence of finite energy weak solutions to the quantum Navier-Stokes equations in the whole space with non trivial far-field condition in dimensions d=2,3. The vacuum regions are included in the weak formulation of the equations. Our method consists in an invading domains approach. More precisely, by using a suitable truncation argument we construct a sequence of approximate solutions. The energy and the BD entropy bounds allow for the passage to the limit in the truncated formulation leading to a finite energy weak solution. Moreover, the result is also valid in the case of compressible Navier-Stokes equations with degenerate viscosity.

A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford–Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.

Estimates for approximate solutions to a functional differential equation model of cell division

The functional partial differential equation (FPDE) for cell division, \[\frac{\partial}{\partial t}n(x,t)+\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)\] \[+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t),\] is not amenable to analytical solution techniques, despite being closely related to the first order partial differential equation (PDE) \[\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t),\] which, with known \(F(x,t)\), can be solved by the method of characteristics. The difficulty is due to the advanced functional terms \(n(\alpha x,t)\) and \(n(\beta x,t)\), where \(\beta \ge 2 \ge\alpha \ge 1\), which arise because cells of size \(x\) are created when cells of size \(\alpha x\) and \(\beta x\) divide. The nonnegative function, \(n(x,t)\), denotes the density of cells at time \(t\) with respect to cell size \(x\). The functions \(g(x,t)\), \(b(x,t)\) and \(\mu(x,t)\) are, respectively, the growth rate, splitting rate and death rate of cells of size \(x\). The total number of cells, \(\int_{0}^{\infty}n(x,t)dx\), coincides with the \(L^1\) norm of \(n\). The goal of this paper is to find estimates in \(L^1\) (and, with some restrictions, \(L^p\) for \(p>1\)) for a sequence of approximate solutions to the FPDE that are generated by solving the first order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper. doi:10.1017/S1446181121000055

On the solvability and approximate solution of a one-dimensional singular problem for a p-Laplacian fractional differential equation

In this paper, using the monotone iterative technique, we discuss a new approximate method for solving multi-point boundary value problems of p-Laplacian fractional differential equations with singularities, which are of great importance in the fluid dynamics field. To do this, first, a sequence of auxiliary problems that release the nonlinear source terms contained in the equations from the singularities is set up, and the uniqueness and existence of their positive solutions are established. Next, we show the relative compactness of the sequence of unique solutions to these auxiliary problems to prove the solvability of our given problem. And we present some sufficient conditions to construct a sequence of approximate solutions that converges to an exact solution of our problem. Finally, we give two numerical examples to demonstrate our main results.

Дихотомия. Дихотомия? Дихотомия! Модифицированный метод дихотомии решения нелинейных скалярных уравнений и некоторые результаты его исследований

Introduction: when creating modern automatic control systems for various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. In the first part of this work entitled “Dichotomy. Dichotomy? Dichotomy!: fundamentals, terminology problems and inspection analysis of the dichotomy method”, a modified version of the dichotomy method was proposed, which has all the main advantages of the modified method. This method has a number of advantages in comparison with other methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is a low rate of convergence of the sequence of approximate solutions, and a large amount of computation required to obtain sufficiently accurate solutions. Purpose of the study: to propose a modified version of the dichotomy method, which allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly less computations required to obtain solutions with the desired accuracy, to illustrate, a higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method by solving a number of specific nonlinear equations and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Results: a modification of the method for dividing a segment in half is proposed, which has all the main advantages of the modified method. The results of solving 4 nonlinear equations are presented illustrating a higher rate of convergence of solutions calculated using the proposed modification. Practical significance: the research results can be used in the development of modern automatic control systems for various technological processes and objects.

On the Cauchy problem for Aw-Rascle system with linear damping

The existence of global BV solutions for the Aw-Rascle system with linear damping is considered. In order to get approximate solutions we consider the system in Lagrangian coordinates, then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of [1] we get a sequence of approximate solutions. Finally we show the convergence of this approximate sequence to the weak entropic solution.

Дихотомия. Дихотомия? Дихотомия! Основные положения, проблемы терминологии и инспекционный анализ метода дихотомии

When creating modern systems of automatic control of various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. Currently, there are a number of computational methods and algorithms for its solution, one of which is the dichotomy method. This method has a number of advantages in comparison with other known methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is the low rate of convergence of the sequence of approximate solutions and a large amount of computation required to obtain sufficiently accurate solutions. The purpose of the study is to consider in detail distinctive features of the dichotomy method and show the preference of its use in comparison with other known methods. We propose a modified version of the dichotomy method that allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly fewer computations required to obtain solutions with the desired accuracy. By solving a number of specific nonlinear equations, it is possible to illustrate the higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Based on the results obtained a modification of the method for segment bisection is proposed. It has all the main advantages of the modified method. The research results can be used in the development of modern automatic control systems for various technological processes and objects.

QUADRATIC APPROXIMATION OF SOLUTIONS FOR SET-VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper investigates nonlinear set-valued functional differential equations with initial value conditions. By introducing the notion of Hukuhara partial derivative of set-valued function, using the comparison principle and the method of quasilinearization, we obtain monotone iterative sequences of approximate solutions which converge uniformly and quadratically to the solutions of such problems.

Zero-viscosity–capillarity limit toward rarefaction wave with vacuum for the Navier–Stokes–Korteweg equations of compressible fluids

We shall consider the one-dimensional Navier–Stokes–Korteweg (NSK) equations that are used to model compressible fluids with internal capillarity. Formally, the NSK equations converge, as the viscosity and capillarity vanish, to the corresponding Euler equations, and we do justify this for the case that the Euler equations have a rarefaction wave with one-side vacuum state. To prove this result, we first construct a sequence of approximate solutions to the NSK equations and then show that the sequence converges, as both viscosity and capillarity tend simultaneously to zero, to this rarefaction wave. The uniform convergence rates are also obtained. The key ingredients of our proof are the re-scaling technique and energy estimates.

Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

AbstractThe functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$which, with known $F(x,t)$, can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$, where $\beta \ge 2 \ge \alpha \ge 1$, which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide.The nonnegative function, $n(x,t)$, denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$, $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$, coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

A new variation for the relativistic Euler equations

The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.