Homogenization and penalization of functional first-order PDE

In this paper we study the asymptotic behavior of viscosity solutions for a functional partial differential equation with a small parameter as the parameter tends to zero. We study simultaneous effects of homogenization and penalization in functional first-order PDE. We establish a convergence theorem in which the limit equation is identified with some first order PDE.     

Solving of second order nonlinear PDE problems by using artificial controls with controlled error

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region inR ². We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error inL ₁ is controlled.     

On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients

We study the k-summability of divergent formal solutions for the Cauchy problem of a certain class of linear partial differential operators with time dependent coefficients. The problem is reduced to a k-summability property of formal solutions for a linear similar ordinary differential equation associated with the Cauchy problem.     

Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

Abstract

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ε > 0. As ε vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ε ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L ²(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.     

The scaling limit for a stochastic PDE and the separation of phases

Summary We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldM of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofM .Research partially supported by Japan Society for the Promotion of Science     

The maximal nonuniqueness classes of solutions to the Cauchy problem for linear equations

We study linear partial differential equations with increasing coefficients in a half-plane. We establish maximal nonuniqueness classes of solutions to the Cauchy problem for these equations. The proof is based on a new estimation method for a solution to the dual differential equation with a parameter.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

Space average in parabolic equations

We find an explicit formula for the limit of a solution of the Cauchy problem, when space averages of coefficients of a parabolic second order partial differential equation of a special type exist.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 561–564, April, 1992.     

Nonlocal differential equations with multivalued perturbations in Banach spaces

We discuss the existence of mild solutions for nonlocal differential inclusions with multivalued perturbations in Banach spaces and establish new existence theorems for related Cauchy problems, which extend some existing results in this area. Using the established results, we investigate a special nonlocal problem. Finally, we also consider a partial functional differential equation.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

A remark on uniqueness for some ODE

Abstract We present a uniqueness result for the Cauchy problem associated to a particular type of ordinary differential equation (ODE), under the only assumption of continuity of the right hand side at the initial point. Keywords: Polar coordinates, Tangent vector, Inner product Mathematics Subject Classification (2000): 34A12     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

Solution of the Mixed Problem for a Third-Order Partial Differential Equation

Mathematical Notes - We consider the initial-boundary value problem for a third-order partial differential equation with highest mixed derivative. An abstract Cauchy problem for a first-order...     

A Method to Study the Cauchy Problem for a Singularly Perturbed Homogeneous Linear Differential Equation of Arbitrary Order

We construct a sequence converging to the solution to the Cauchy problem for a singularly perturbed linear homogeneous differential equation of any order. This sequence is asymptotic in the following sense: the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Limit behavior of two-time-scale diffusions revisited

This work is concerned with diffusions with two-time scales or singularly perturbed diffusions. Asymptotic expansions of the solution of the associated Cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived. These asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals.     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Abstract We give a generalization of the work presented in [6] where the asymptotic behaviour, as ε→0, of a monotone nonlinear problem in a bounded multidomain of R^N depending on ε was addressed. We extend the previous results to the case where the nonlinear operator depends both on the slow and rapid variable and we prove that, due to the presence of the rapid variable, the algebraic equation contained in the limit problem obtained in [6] must be replaced by a partial differential equation with respect to the microscopic variable y′. Keywords: Homogenization, Dimension reduction, Multidomain, Rapid variable, Limit problem Mathematics Subject Classification (2000): 35B27, 35J60