Sequence Of Approximate Solutions Research Articles

Pohozaev identities and their applications to nonlinear elliptic equations

Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations. In this review, we aim to introduce Pohozaev identity and its applications in some classical elliptic partial differential equations. First, we introduce Pohozaev identities for several typical elliptic equations on bounded domains or unbounded domains, and the necessary conditions for the existence of solutions, by which we get some existence and nonexistence results of the solutions. Next, we will show a local Pohozaev identity and its application in proving the compactness of the sequences of approximate solutions, by which we obtain infinitely many solutions of some typical nonlinear elliptic equations. At last, we also introduce some local uniqueness results of peak solutions for some nonlinear elliptic partial differential equations, which can also be obtained by using the local Pohozaev identity. Using this type of uniqueness result, we can prove that the peak solutions can keep the symmetry that the equations have.

Global solution to 1D model of a compressible viscous micropolar heat-conducting fluid with a free boundary

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.

Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients

We study fuzzy stochastic differential equations driven by multidimensional Brownian motion with solutions of decreasing fuzziness. The drift and diffusion coefficients are random. Under a non-Lipschitz condition, the existence and pathwise uniqueness of solutions to such the equations are proven. The solutions are considered to be fuzzy stochastic processes. The main result is obtained with a help of a sequence of approximate solutions that converge to a desired unique local solution with trajectories having decreasing fuzziness. A parallel assertion for solutions to fuzzy stochastic differential equations of increasing fuzziness is stated as well. We indicate that our considerations of fuzzy stochastic differential equations of decreasing fuzziness can be applied to examine non-Lipschitz set-valued stochastic differential equations with solutions being set-valued stochastic processes.

Existence theorems for the quantum drift–diffusion equations with mixed boundary conditions

In this paper, we construct a weak solution to the unipolar quantum drift–diffusion equations coupled with initial and mixed boundary conditions in up to four space dimensions. We only assume that the boundary of the domain is Lipschitz and the interface between the Dirichlet boundary and the Neumann boundary is a Lipschitzian hypersurface, and thus the lack of high regularity for solutions of such problems is an issue. The convergence of a sequence of approximate solutions is established via an application of a recent theorem by the author on the logarithmic upper bound for solutions of elliptic equations with the same type of boundary conditions [Logarithmic up bounds for solutions of elliptic partial differential equations, Proc. Amer. Math. Soc. 139 (2011) 3485–3490].

On the computation of measure-valued solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm ofmeasure-valued solutionsfor these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

The ADI method for bounded real and positive real Lur’e equations

We propose an algorithm for the numerical solution of the Lur'e equations in the bounded real and positive real lemma for stable systems. The algorithm provides approximate solutions in low-rank factored form. We prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness. If the shift parameters are chosen appropriately, the sequence is proven to be convergent to the minimal solution of the Lur'e equations. The algorithm is based on the ideas of the recently developed ADI iteration for algebraic Riccati equations (Massoudi et al., SIAM J Matrix Anal Appl, 2016). In particular, the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem.

Newton–Milstein scheme for stochastic differential equations and its fast uniform convergence

ABSTRACTWe show that a modified Milstein scheme combined with explicit Newton’s method enables us to construct fast converging sequences of approximate solutions of stochastic differential equations. The fast uniform convergence of our Newton–Milstein scheme follows from Amano’s probabilistic second-order error estimate, which had been an open problem since 1991. The Newton–Milstein scheme, which is based on a modified Milstein scheme and the symbolic Newton’s method, will be classified as a numerical and computer algebraic hybrid method and it may give a new possibility to the study of computer algebraic method in stochastic analysis.

A null-space-based weightedl1minimization approach to compressed sensing

It has become an established fact that the constrained $\ell_1$ minimization is capable of recovering the sparse solution from a small number of linear observations and the reweighted version can significantly improve its numerical performance. The recoverability is closely related to the Restricted Isometry Constant (RIC) {of order $s$ ($s$ is an integer), often denoted as $\delta_{s}$. A class of sufficient conditions for successful $k$-sparse signal recovery often take the form $\delta_{tk} 1$, such a bound is often called RIC bound of high order.} There exist a number of such bounds of RICs, high order or not. For example, a high order bound {is recently given by Cai and Zhang \cite{CZ14}:} $\delta_{tk} < \sqrt{(t-1)/t}$, and this bound is known sharp for $t \ge 4/3$. In this paper, we propose a new weighted $\ell_1$ minimization which only requires the following RIC bound that is more relaxed (i.e., bigger) than the above mentioned bound: \[\delta_{tk} 1$ and $0< \omega \le 1$ is determined {by two optimizations of a similar type}over the null space of the linear observation operator. {In tackling the combinatorial nature of the two optimization problems,we develop a reweighted $\ell_1$ minimization that yields a sequence of approximate solutions,which enjoy strong convergence properties. Moreover, the numerical performance of the proposed method}is very satisfactory when compared to some of the state of-the-art methods incompressed sensing.

Well posedness of an angiogenesis related integrodifferential diffusion model

We prove existence and uniqueness of nonnegative solutions for a nonlocal in time integrodifferential diffusion system related to angiogenesis descriptions. Fundamental solutions of appropriately chosen parabolic operators with bounded coefficients allow us to generate sequences of approximate solutions. Comparison principles and integral equations provide uniform bounds ensuring some convergence properties for iterative schemes and providing stability bounds. Uniqueness follows from chained integral inequalities.

Задача с интегральным смещением для одномерного гиперболического уравнения

Рассмотрена задача с нелокальным интегральным условием второго рода для одномерного гиперболического уравнения в прямоугольной области. Доказаны существование и единственность обобщенного решения задачи. Для доказательства существования и единственности обобщенного решения поставленной задачи предложен новый метод исследования задач с интегральными условиями. Предложенный в работе метод позволил отказаться от некоторых условий на входные данные, обеспечивающих разрешимость поставленной задачи, а именно от требования обратимости оператора, порождаемого нелокальным условием. Суть данного метода состоит в эквивалентной замене заданного нелокального условия другим, также нелокальным, но содержащим в качестве внеинтегрального члена значения выводящей производной неизвестной функции на боковой границе. Установленная эквивалентность условий позволила перейти к задаче, для доказательства однозначной разрешимости которой применен метод компактности, зарекомендовавший себя как эффективный метод исследования разрешимости начально-краевых задач и задач с нелокальными условиями. С помощью метода Галеркина построена последовательность приближенных решений. Для продолжения исследования разрешимости задачи получены априорные оценки решения в пространстве Соболева. С помощью выведенных оценок доказано утверждение о возможности выделить из построенной методом Галеркина последовательности приближенных решений подпоследовательность, которая слабо сходится к решению задачи. В процессе доказательства разрешимости поставленной задачи обнаружилась интересная связь нелокальных интегральных условий с динамическими условиями.

An explicit analytic approximation of solutions for a class of neutral stochastic differential equations with time-dependent delay based on Taylor expansion

This paper represents a contribution to the analysis of approximate methods for stochastic differential equations based on the application of Taylor expansion, under the Lipschitz and linear growth conditions. The Lp and almost sure convergence of the appropriate approximate solutions are considered for a class of neutral stochastic differential equations with time-dependent delay. Coefficients of the approximate equations, including the neutral term, are Taylor approximations of the coefficients of the initial equation up to the first derivatives. For p ≥ 2, the rate of the Lp-convergence of the sequence of approximate solutions to the exact solution is estimated as 2lp−12l, where l > 1 is an integer. The presence of the neutral term in the equation reflected to the rate of convergence.

Convergence of a finite-volume scheme for the Cahn–Hilliard equation with dynamic boundary conditions

This work is devoted to the numerical study of the Cahn-Hilliard equation with dynamic boundary conditions. A spatial finite-volume discretization is proposed which couples a 2d-method in a smooth connected domain and a 1d-method on its boundary. The convergence of the sequence of approximate solutions is proved and various numerical simulations are given.

A periodic boundary value problem for nonlinear singular differential systems with ‘maxima’

In this paper, by using the method of quasilinearization to discuss the periodic boundary value problem for a nonlinear singular differential system with ‘maxima’, we obtain monotone iterative sequences of approximate solutions which converge uniformly and quadratically to the solution of the nonlinear singular differential system with ‘maxima’.

Fuzzy stochastic differential equations of decreasing fuzziness: Approximate solutions

We consider recently introduced fuzzy stochastic differential equations with solutions of decreasing fuzziness. In general, such the equations do not have solutions that could be written in explicit, closed form. Therefore some methods of construction of approximate solutions are proposed in this p aper. In considered framework, approximate solutions are some measurable and adapted fuzzy stochastic processes. We analyze two kinds of sequences of approximate solutions. It is showed that each sequence of approximate solutions can be used to prove existence and uniqueness of solution to fuzzy stochastic differential equations of decreasing fuzziness. In fact, both the sequences converge to a unique solution. The rates of convergence related to both the sequences are investigated. All the results apply immediately to set-valued stochastic differential equations with solutions of decreasing diameter.

Stochastic fuzzy differential equations of a nonincreasing type

Stochastic fuzzy differential equations constitute an apparatus in modeling dynamic systems operating in fuzzy environment and governed by stochastic noises. In this paper we introduce a new kind of such the equations. Namely, the stochastic fuzzy differential of nonincreasing type are considered. The fuzzy stochastic processes which are solutions to these equations have trajectories with nonincreasing fuzziness in their values. In our previous papers, as a first natural extension of crisp stochastic differential equations, stochastic fuzzy differential equations of nondecreasing type were studied. In this paper we show that under suitable conditions each of the equations has a unique solution which possesses property of continuous dependence on data of the equation. To prove existence of the solutions we use sequences of successive approximate solutions. An estimation of an error of the approximate solution is established as well. Some examples of equations are solved and their solutions are simulated to illustrate the theory of stochastic fuzzy differential equations. All the achieved results apply to stochastic set-valued differential equations.

New existence of hyperbolic orbits for a class of singular Hamiltonian systems

A new existence of hyperbolic orbits is obtained for a class of singular Hamiltonian systems with prescribed energies by taking the limit for a sequence of approximate solutions. Furthermore, we show that the hyperbolic orbits possess the given directions at infinity.

Time delay and Lagrangian approximation for Navier–Stokes flow

Abstract We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for 0 &lt; t ≤ T ${0 &amp;lt; t \le T}$ in a bounded domain Ω ⊆ ℝ 3 ${\Omega \subseteq \mathbb {R}^3}$ with sufficiently smooth boundary ∂ Ω ${\partial \Omega }$ . We use a particle method in connection with a time delay to approximate the nonlinear convective term by a single central Lagrangian difference quotient constructed from autonomous systems of ordinary differential equations. We show that the resulting approximate Navier–Stokes system has a uniquely determined global solution satisfying the energy equation and having a high degree of regularity uniformly in time. Moreover, we prove that the sequence of approximate solutions has an accumulation point satisfying the Navier–Stokes equations in a weak sense and the energy inequality.

Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

We prove the well-posedness of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate $$\mathbf {L^\infty }$$L?, BV and $$\mathbf {L^1}$$L1 estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme.

Rapid Convergence of Solution for Hybrid System with Causal Operators

We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of orderk≥2.

A modification of the method of fictitious domains for stationary model of non-Newtonian liquids

The substantiation of one modification method of fictitious domains with continuation on junior coefficient for the stationary equations of non-Newtonian liquids is given in this work. The domain is three-dimensional and bounded. A theorem of an existence and convergence of the generalized solution of an auxiliary problem is proved. The theorem is proved by the method of a priori estimates. Inequality of imbedding theorems, Holder inequality and Young's inequality are used. The sequence of approximate solutions is constructed using the Galerkin method. The limit in the integral identity through the selected sequence. For solutions obtained uniform evaluation standards of functional spaces. An estimation of convergence of the strong solution of the approximating problem is deduced.