Exponential stability analysis and L2-gain analysis are developed for uncertain distributed parameter systems. Scalar heat processes and distributed mechanical oscillators, governed by semilinear partial differential equations of parabolic and, respectively, hyperbolic types, are chosen for treatment. Sufficient exponential stability conditions with a given decay rate are derived in the form of...

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation ut = ∆u + |u|p−1u. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence ...

A numerical method for solving parabolic equations based on multigrid techniques is proposed. The stability, approximation and conservation properties of the method are investigated theoretically and numerically for several initial-boundary model problems for the heat conduction equation. The use of the method makes it possible to considerably reduce the computational work as compared to either...

An analytical study was performed to study effects of thermo-diffusion and chemical reactions on a three-dimensional MHD mixed convective flow of dissipative fluid along an infinite vertical porous plate with transverse sinusoidal suction velocity. The parabolic partial differential equations governing the fluid flow, heat transfer, and mass transfer were solved using perturbation technique and...

This work introduces a new operator-splitting method for solving the two-dimensional (2D) and three-dimensional (3D) parabolic equations. The aim is to reduce the computational complexity without loss of accuracy. Firstly, we split the 2D and 3D parabolic equations into a sequence of one-dimensional (1D) parabolic equations respectively, then we solve each 1D parabolic equation by using finite ...

We consider the parabolic type equation with a source-sink term and construct the Monte Carlo estimator for it. The procedure is based on the Hopf-Cole transformation and a Monte Carlo estimator for the correspondent Burgers equation. Monte Carlo estimators for the heat and diffusion equations are very well known since the first steps of the method. The algorithms are based usually on the intim...

we focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of it¨o type, in particular, parabolic equations. the main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

We introduce a multitree-based adaptive wavelet Galerkin algorithm for space-time discretized linear parabolic partial differential equations, focusing on time-periodic problems. It is shown that the method converges with the best possible rate in linear complexity and can be applied for a wide range of wavelet bases. We discuss the implementational challenges arising from the Petrov-Galerkin n...

Recently, parabolic equations in infinite dimensions have received much attention in literature (see, for example, Pa Prato [DP] and Da Prato Zabczyk [DZ]). In particular, Cannarsa and Da Prato [CD1] showed that the Laplacian (with a certain weight) generates a semigroup on BUC(H), the space of all bounded uniformly continuous functions on a separable Hilbert space H, which is called the heat s...

We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equati...