Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Dedicated to Professor Vidar Thomée on the occasion of his 65 birthday, August 20, 1998

In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...

Parabolic equation methods for fluid and elastic media are extended to layered poro-elastic media, including some shallow-water sediments. A previous parabolic equation solution for one model of range-independent poro-elastic media [Collins et al., J. Acoust. Soc. Am. 98, 1645-1656 (1995)] does not produce accurate solutions for environments with multiple poro-elastic layers. First, a dependent...

The nonlocal boundary value problem for a hyperbolic-parabolic equation in a Hilbert space H is considered. The difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. In applications, the stability estimates for the solutions of the difference schemes of the mixed type boundary val...

In 2002, J.M.Rassias (Uniqueness of quasi-regular solutions for bi-parabolic elliptic bi-hyperbolic Tricomi problem, Complex Variables, 47 (8) (2002), 707-718) imposed and investigated the biparabolic elliptic bi-hyperbolic mixed type partial differential equation of second order. In the present paper some boundary-value problems with non-local initial condition for model and degenerate parabol...

A new velocity profile equation for the description of microcirculatory blood flow in vivo was proposed in 2009. However various recently published papers still use the assumption of parabolic velocity profile (Poiseuille flow). The purpose of this work was to evaluate the performance of 3 different fitting cases: 1) best parabolic fit, 2) axial fit with the proposed equation and 3) best fit wi...

A classical result of Aleksandrov allows us to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if ∫ Ω detD 2u dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov’s result that provides a similar estimate, in som...

We consider parabolic equations of the form ut = ∆u + f(u) + h(x, t), (x, t) ∈ R × (0,∞) , where f is a C1 function with f(0) = 0, f ′(0) < 0, and h is a suitable function on RN × [0,∞) which decays to zero as t → ∞ (hence the equation is asymptotically autonomous). We show that, as t → ∞, each bounded localized solution u ≥ 0 approaches a set of steady states of the limit autonomous equation u...

Abstract: We consider three level difference replacements of parabolic equations focusing on the heat equation in two space dimensions. Through a judicious splitting of the approximation, the scheme qualifies as an alternating direction implicit (ADI) method. Using the well known fact of the parabolic-elliptic correspondence, we shall derive a two stage iterative procedure employing a fractiona...

This paper will discuss the heat equation or as known as parabolic equation by Jacobi, Gauss Seidel and Alternating Direct Implicit (ADI) methods with the implementation of parallel computing on it. The numerical method is emphasized as platform to discretize the one dimensional heat equation. The result of three types of numerical methods will be presented graphically. The parallel algorithm i...