Quasilinear Parabolic Initial Boundary Value Research Articles

Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz–Minkowski plane R12

In this paper, we investigate the evolution of spacelike curves in the Lorentz–Minkowski plane R12 along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike grim reaper curve as time tends to infinity.

Weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation with nonstandard growth

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part II: Convergence

In this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial boundary value problems is pursued. Compared to other types of exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility of constructing schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time step sizes is incorporated. The convergence analysis is based upon two fundamental ingredients. The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time step sizes, have been deduced in the recent work [C. Gonzalez and M. Thalhammer, SIAM J. Numer. Anal., 53 (2015), pp. 701--719]. The core of the present work is devoted to the derivation of suitable local and global error representations. In conjunction with the stability bound...

Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

The numerical solution of fourth order mildly quasi-linear parabolic initial boundary value problem of second kind

In this article, we report on three-level implicit stable finite difference formulas of O(k 2 + h 2) and O(k 2 + h 4) for the numerical integration of certain mildly quasi-liner fourth order parabolic partial differential equations in one-space dimension. The numerical solution of u xx is obtained as a by-product of the method. In all cases, we use only (3 + 3 + 3)-grid points and a single computational cell. Difference schemes for the fourth order linear parabolic equation in polar coordinates are also discussed. The stability analysis for the model linear problem is given as a representative example. Numerical results are presented to demonstrate the order and accuracy of the proposed methods.

High Accuracy Difference Formulae For A Fourth Order Quasi-Linear Parabolic Initial Boundary Value Problem Of First Kind

In this paper, new three level implicit finite difference methods of O(k^2+h^2) and O(k^2+h^4) are proposed for the numerical solution of fourth order quasi-linear parabolic partial differential equations in one space variable, where k\\gt 0 and h\\gt 0 are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \\partial u/\\partial x is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper, we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper, we consider a nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.

Difference Methods for Quasilinear 2D-diffusion Problems in Toroidal Configurations

Implicit one-step difference methods for quasilinear parabolic initial boundary value problems in two space variables and in polar coordinates are considered. Problems of this kind are suitable for the description of diffusion processes in plasma physics. Therefore, because of its special physical interest the dependence of the coefficients both on the solution and on its space derivatives (the fluxes) is included. The first part of this paper deals with proving convergence of the discrete approximations for vanishing step sizes. For this purpose, bounds for the inverse difference operators have to be derived previously. The nonlinear systems arising from the discretizations have exactly one solution for all step sizes being sufficiently small. In the second part of the article numerical tests are performed. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

On degenerate quasilinear parabolic equations of higher order

In the paper existence results for degenerate quasilinear parabolic initial boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space using the generalized degree theory.