We study quadratic optimal stochastic control problems with control dependent noise state equation perturbed by an affine term and with stochastic coefficients. Both infinite horizon case and ergodic case are treated. To this purpose we introduce a Backward Stochastic Riccati Equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability c...

We study the boundedness of the difference equation xn 1 pxn qxn−1 / 1 xn , n 0, 1, . . . , where q > 1 p > 1 and the initial values x−1, x0 ∈ 0, ∞ . We show that the solution {xn}n −1 of this equation converges to x q p − 1 if xn ≥ x or xn ≤ x for all n ≥ −1; otherwise {xn}n −1 is unbounded. Besides, we obtain the set of all initial values x−1, x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions...

This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the Schrödinger equation with constant potential and a preconditioned algorithm for the general Schrödinger equation. These algorithms...

We obtain a necessary condition and a sufficient condition, both expressed in terms of Wiener type tests involving the parabolic W 2,1 q′ capacity, where q ′ = q q−1 and q > 1, for the existence of large solutions to equation ∂tu−∆u+ u q = 0 in a non-cylindrical domain. We provide also a sufficient condition for the existence of such solutions to equation ∂tu−∆u+ e u − 1 = 0. Besides, we apply ...

Many new solitary wave solutions of the recently studied Lienard equation are obtained by mapping it to the field equation of the φ6−field theory. Further, it is shown that the exact solutions of the Lienard equation are also the exact solutions of the various perturbed soliton equations. Besides, we also consider a one parameter family of generalised Lienard equations and obtain exact solitary...

This paper studies the behavior of positive solutions to the following particular case of a difference equation by Stević xn 1 A x p n/x pk 1 n−k , n ∈ N0, where A,p ∈ 0, ∞ , k ∈ N, and presents theoretically computable explicit lower and upper bounds for the positive solutions to this equation. Besides, a concrete example is given to show the computing approaches which are effective for small ...

We introduce a novel numerical method for a recently developed perspective Shape-from-Shading model. In order to discretise the corresponding partial differential equation (PDE), Prados et al. employed the dynamical programming principle yielding a Hamilton-Jacobi-Bellman equation. We reduce that model to its essential, namely to the underlying Hamilton-Jacobi equation. For this PDE, we propose...

We suggest an effective approach to separation of variables in the Schrödinger equation with two space variables. Using it we classify inequivalent potentials V (x1, x2) such that the corresponding Schrödinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrödinger equation with the anisotropic harmonic oscillator potential V = k1x1 + k2x 2...