Large solutions for nonlinear parabolic equations without absorption terms

Variational iteration method is a semi analytic solution used to solve the parabolic differential equation both of homogen or nonhomogen. In the process of determining an approximation solution, this method did not use a linearization and a small pertubation. In this paper, the variational iteration method is implemented in the parabolic differential equation in the form of ut = uxx + f(u) + g(x, t) with appropriate intial condition. Furthermore, some examples of special parabolic differential equations are given to test the reliability and convergence of the method. Based on the result of study shows that the variational iteration method is able to solve the parabolic differential equation with a good accuration.

Theorems on the existence and uniqueness of the statistical solutions and for the existence of stationary solutions for nonlinear stochastic equations are proved. Markov families of statistical solutions are constructed. The ergodic properties of monotone systems are investigated. A Girsanov type theorem on the absolute continuity of the statistical solution of a nonlinear stochastic parabolic equation with respect to the statistical solution of the monotone equation is proved.

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Nonlinear Parabolic Equations with Measure Data

Boundedness of global solutions of nonlinear parabolic equations

. . , ae

We propose to enhance the stability of IMEX schemes applied to the numerical solution of nonlinear parabolic equations by using proper smoothing procedures on bands of frequency components of the solution. The separation of those different components is realized by considering a multigrid (MG) approach. The time scheme on the coarsest grid is implicit and energy diminishing, while time marching is done by a weighted damped IMEX scheme on the finer grids. This approach allows to concentrate most computational efforts on a small set of components (the coarsest grid) and to fine-tune the damping parameters so that there is limited consistency perturbation while enhancing stability. We provide stability results and numerical illustrations on pattern dynamics, Swift-Hohenberg, and crystal-phase fields equations, in both 2D and 3D cases. The results obtained with MG are comparable with those given by a one-grid simulation.

Consider the first initial-boundary value problem $$u = 0\quad on\quad \partial \Omega \times \left( {0,\infty } \right)$$ (1.1) , $$u = 0\quad on\quad \partial \Omega \times \left( {0,\infty } \right)$$ (1.2) , $$u\left( {x,0} \right) = \phi (x)\quad on\quad \Omega$$ (1.3) When Ω is a bounded domain in ℝ N with C 2 boundary, $$Lu = \sum\limits_{i,j = 1}^N {\frac{\partial }{{\partial {x_i}}}} \left( {{a_{ij}}\left( x \right){u_{{x_j}}}} \right) + \sum\limits_{i = 1}^N {{b_i}\left( x \right)} {u_i} + c\left( x \right)u$$ (aij) is uniformly positive definite matrix, the coefficients of L are sufficiently smooth in (math), say in C 1,ϕ is Holder continuous in Ω;, and F(u, w) is Lipschitz continuous in ℝ N +1.

In this paper we prove the existence of a renormalized solution for nonlinear parabolic equations of the type:$$\displaystyle{\partial b(x,u)\over\partial t} - {\rm div}\Big(a(x,t,\nabla u)\Big)=\mu\qquad \text{in}\ \Omega\times (0,T),$$ where the right handside is a general measure, $b(x,u)$ is anunbounded function of $u$ and $- {\rm div}(a(x,t,\nabla u))$is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in$\nabla u$.

We show that a mild solution of the initial value problem for u' = Au + F(u) is a strict solution, under suitable assumptions for A and F. Two examples show an application and the limitation of the result.

Existence of Renormalized Solutions for Nonlinear Parabolic Equations

Chapter 6. Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations was published in Singular Solutions of Nonlinear Elliptic and Parabolic Equations on page 283.