Limit theorems for BSDE with local time applications to non-linear PDE

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

Seven common errors in finding exact solutions of nonlinear differential equations

This paper presents solutions of second-order ordinary differential equations on bond problems to estimate crack widths of reinforced concrete (RC) members subjected to repeated cyclic loads. Up to now, bond behaviors under repeated cyclic loads have usually been calculated by a difference scheme. In contrast, this research adopts analytical solutions of the differential equations to calculate bond behaviors in finite element models where a number of concrete cracks occur. The conventional differential equation for the bond problem has been available only under monotonic loading conditions. This paper extends the equation to cases where the reinforcement becomes plastic and the bond is subject to unloading/reloading conditions. The bond stress-slip relationship is modeled by a multilinear model that is composed of an envelope made of five lines, unloading/reloading paths, and a negative friction path. The solution of the differential equation is classified into 25 cases when the bond stress-slip relationship is on the envelope and seven cases when on the unloading/reloading paths or the negative friction path. The solutions are implemented into the smeared crack-based finite-element method (FEM) to evaluate the crack widths of a RC beam subject to repeated cyclic flexural/shear loads. The FEM program is incorporated with the discrete-like crack procedure developed by the authors and calculates stress redistributions due to cracking using the solutions of the differential equations. Typical distributions of reinforcement stress, bond stress, and slip along the reinforcement are presented.

Let K and H be two separable Hilbert spaces and be a cylindrical Wiener process with values in K defined on a probability space denote its natural filtration. Given , we look for an adapted pair of process with values in H and respectively is defined in §1),which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H. The precise meaning of the equation is A linearized version of that equation appears in infinite-dimensional stochastic optimal control theory as the equation satisfied by the adjoint process. We also give our results to the following backward stochastic partial differential equation:

The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system has an asymptotically stable equilibrium point, random effects can cause the trajectories of the system to leave any bounded domain with probability one. In this paper we consider the effect of small random perturbations of the type referred to as Gaussian white noise, on a (deterministic) dynamical system $\dot x = b(x)$. The vector $x(t)$ then becomes a stochastic process $x_\varepsilon (t)$ which satisfies the stochastic differential equation $dx_\varepsilon = b(x_\varepsilon )dt + \varepsilon \sigma (x_\varepsilon )dw$. Here $w(t)$ is the n dimensional Wiener process (Brownian motion), $b(x)$ is a vector field, $\sigma (x)$ is the diffusion matrix and $\varepsilon \ne 0$ is a small real parameter. We give the first complete formal solution of the following problem originally posed by Kolmogorov: Find the asymptotic expansion in $\varepsilon $ of (i) the probability distribution of the points on the boundary of a domain, where trajectories of the perturbed system first exit, and (ii) of the expected exit times. Our method is to relate the solutions of the above problems to the solutions of various singularly perturbed elliptic boundary value problems with turning points, whose solutions are then constructed asymptotically.

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

Here w(t) is the «-dimensional Wiener process (Brownian motion), b(x) is a vector field, o(x) is the diffusion matrix and e ¥= 0 is a small real parameter. The cumulative effect of even very small random perturbations may be considerable after sufficiently long times, so that even if the deterministic dynamical system has an asymptotically stable equilibrium point, the trajectories of the system will leave any compact domain with probability one. The following problem was posed by Kolmogorov: determine the probability distribution of the points on the boundary where trajectories exit, at the first time of their exit from a compact domain, as well as the expected exit times. The random effect may be thought of as a slow diffusion of particles in the deterministic flow field given by b(x), and the results may differ according as particles are diffusing (a) with a flow, (b) across a flow, or (c) against a flow. Results on (a) were first obtained by Levinson [4] , and on (b) by Khasminskii [3] , both of whom used analytical techniques. Problem (c) seems to be the most difficult, and to date only partial results are available (cf. Ventsel and Freidlin [5] and Friedman [1] who used probabalistic methods). Using analytical techniques, we present a full solution of this problem for flows which are essentially gradients of a potential (as well as certain more general flows). Let f]l be a compact domain in R with a smooth boundary 3£2. Let a(j(x) = ie(o(x)a*(x))//-, be strictly positive definite in £2, b(x) = (bv b2,..., bn)9 and let u€(x) be the solution of the Dirichlet problem

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly.

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-valued stochastic hereditary differential systems of the form $dx(t) = H(t,x)dt + g(t,x(t - r))dW(t).$ In the above equation, $W$ is an $n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a nonanticipating functional defined on the space of paths in $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

In this paper, we study the growth of solutions of the second order linear complex differential equations insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

In this review article, we study the recent investigations on the forms of solutions of systems difference equations and difference equations in terms of well-known integer sequences such as Fibonacci numbers, Padovan numbers. We focus on the papers given some interesting relationships both between the exact solutions of difference equations and the integer sequences and between the equilibrium points of difference equations and the golden ratio.

Numerical solution for high order differential equations using a hybrid neural network—Optimization method

In determining molecular weights with the ultracentrifuge by the equilibrium method, it is necessary to drive the rotor at relatively high speeds for very long periods of time. It would be a real advantage to be able to deduce molecular weights from the distribution of concentration within a cell placed in a rotor without waiting for equilibrium. The solution of a certain differential equation gives the theoretical distribution within the cell at any time, and a comparison of the experimentally determined concentrations with those deduced from theory should yield molecular weights without waiting for equilibrium. The exact solution of this differential equation is known, but it is not suitable for numerical work. The present paper presents an approximate solution from which numerical results can be obtained with ease.

Adapted solution of a backward stochastic differential equation