Related papers: Dynamical Systems Method (DSM) for solving nonline…

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with…

Let $Ay=f$, $A$ is a linear operator in a Hilbert space $H$, $y\perp N(A):=\{u:Au=0\}$, $R(A):=\{h:h=Au,u\in D(A)\}$ is not closed, $\|f_\delta-f\|\leq\delta$. Given $f_\delta$, one wants to construct $u_\delta$ such that $\lim_{\delta\to…

A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An algorithm for computing the solution…

A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new…

The existence of a solution, convergence and stability of the penalty method for variational inequalities with nonsmooth unbounded uniformly and properly monotone operators in Banach spase $B$ are investigated. All the objects of the…

In this paper, we deal with the existence of $\omega$-periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$ $$u'(t)+Au(t)=F(t,u(t),u(t-\tau)),\ \ \ \ t\in\R,$$ where $A:D(A)\subset…

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the…

This paper derives a free analog of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking fSDEs are stochastic differential equations in the context of…

In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_{\Omega}g(u)\,dx \right)\Delta u =f(u), \quad x \in \Omega \\…

In this paper we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third-order and has cubic…

The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $ \dot u(t) = Au(t) + [{\cal B}u](t) +f(t), $ where $A$ is a closed operator in a Banach space…

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space $E$: $(\partial_t+L)u+f(\cdot,\cdot,u, A^{1/2}\nabla u)=0$ on $[0,T]\times E,\qquad u_T=\phi$, where $L$ is a possibly…

In the paper a new numerical-analytical method for solving the Cauchy problem for systems of ordinary differential equations of special form is presented. The method is based on the idea of the FD-method for solving the operator equations…

We establish new global bifurcation theorems for dynamical systems in terms of local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation $u_t+A u=f_\lambda(u)$ in a Banach space $X$, where $A$…

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new…

In this paper, we consider the Stokes problem with Dirichlet boundary conditions and the constant kinematic viscosity $\nu$ in an axis-aligned domain $\Omega$. We decouple the velocity $\bm u$ and pressure $p$ by deriving a novel biharmonic…

In the paper we offer a functional-discrete method for solving the Cauchy problem for the first order ordinary differential equations (ODEs). This method (FD-method) is in some sense similar to the Adomian Decomposition Method. But it is…

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required.…

This paper presents a nonlinear dynamical model which consists the system of differential and operator equations. Here differential equation contains a nonlinear operator acting in Banach space, a nonlinear operator equation with respect to…

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…