Solutions Of Nonlinear Parabolic Equations Research Articles

Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

Numerical simulation for European and American option of risks in climate change of Three Gorges Reservoir Area

Abstract With the climate change processes over times, all professions and trades in Three Gorges Reservoir Area will be influenced. One of the biggest challenges is the risk of rising sea level. In this situation, a large number of uncertainties for climate changes will be faced in Three Gorges Reservoir Area. Therefore, it is of importance to investigate the complexity of decision making on investing in the long term rising sea level risk related projects in Three Gorges Reservoir Area. This paper investigates the sea level and the temperature as the underlying assets in Three Gorges Reservoir Area. A real option model is constructed to evaluate potential sea level rising risk. We formulate European and American real option models into a linear parabolic variational inequalities and propose a power penalty approach to solve it. Then we obtain a nonlinear parabolic equation. It shows that the nonlinear parabolic equation is unique and solvable. Also, the solutions of the nonlinear parabolic equation converge to the solutions of the parabolic variational inequalities at the rate of order O(λ −k/2). Since the analytic solution of nonlinear parabolic equation is difficult to obtain, a fitted finite volume method is developed to solve it in case of European and American options, and the convergence of the nonlinear parabolic equation is obtained. An empirical analysis is presented to illustrate our theoretical results.

Existence of a renormalized solution of nonlinear parabolic equations with general measure data

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$\nabla u$.

Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density

In this paper the properties of solutions of nonlinear parabolic equation not in divergence form are studied. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic equation not in divergence form, depending on the value in the equation of the numerical parameters necessary and sufficient signs of their existence. The compactly supported solution of the Cauchy problem for a cross-diffusion parabolic equation not in divergence form with a source and a variable density is obtained.

Entropy Solutions for Nonlinear Parabolic Equations with Nonstandard Growth in Non-reflexive Orlicz Spaces

We prove in this paper an existence result of entropy solutions for nonlinear parabolic equations of the form: $$\begin{aligned} \displaystyle \frac{\partial u}{\partial t}-{\text {div}}\, a(x,t,u,\nabla u)-\mathrm{div}\Phi (x,t,u)= f \quad \text {in }{Q_T=\Omega \times (0,T)}, \end{aligned}$$ where the lower order term $$\Phi $$ satisfies only a natural growth condition prescribed by the N-function M defining the Orlicz spaces framework and the data f are an element of $$L^1(Q_T)$$ . We do not assume any restriction neither on M nor on its complementary $$\overline{M}$$ . No particular growth is considered on $$\Phi $$ .

L^{infty } decay estimates of solutions of nonlinear parabolic equation

In this paper, we are interested in L^{infty } decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain L^{infty } decay estimates of weak solutiona.

Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data

We give an existence result of a renormalized solution for a classof nonlinear parabolic equations@b(u)/@t div(a(x; t;grad(u))+ H(x; t;ru) = ,where the right side is a general measure, b is a strictly increasing C1-function,div(a(x; t;grad(u)) is a Leray{Lions type operator with growth in grad(u)and H(x; t;grad(u) is a nonlinear lower order term which satisfy the growth condition with respect to grad(u).

Entropy solutions for nonlinear parabolic equations in Musielak Orlicz spaces without Delta_2-conditions

In this paper, we are interested in results concerning entropy solutions for nonlinear parabolic equations in Musielak Orlicz spaces without Δ2-condition.

The solution of nonlinear parabolic equation using variational iteration method

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.

Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations

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.

On The Stabilization of Solutions of Nonlinear Parabolic Equations with Lower-Order Derivatives

For parabolic equations of the form $$ \frac{\partial u}{\partial t}-\sum \limits_{i,j=1}^n{a}_{ij}\left(x,u\right)\frac{\partial^2u}{\partial {x}_i\partial {x}_j}+f\left(x,u, Du\right)=0\kern0.5em \mathrm{in}\kern0.5em {\mathbb{R}}_{+}^{n+1}, $$ where $$ {\displaystyle \begin{array}{cc}{\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^n\times \left(0,\infty \right),& n\ge 1,D=\Big(\partial \end{array}}/\partial {x}_1,...,\partial /\partial {x}_n\Big), $$ and f satisfies some constraints, we obtain conditions that ensure the convergence of any its solution to zero as t→∞.

Constructive error estimates for full discrete approximation of periodic solution for heat equation

We consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation with time-periodic condition. Our numerical scheme is based on the finite element semidiscretization in space direction combined with an interpolation in time using the fundamental matrix for the semidiscretized problem. We derive the optimal order H1 and L2 error estimates that are critical in the numerical verification method of exact solutions for nonlinear parabolic equations. Several numerical examples that confirm the optimal rate of convergence are presented.

The Lambert Function and Exact Solutions of Nonlinear Parabolic Equations

In this paper, we consider diffusion equations with a power-type coefficient and nonlinear source. We pay the main attention to constructing exact solutions in terms of the Lambert function. We subject the source function to certain conditions that guarantee the existence of exact solutions of a given type. For illustrating the obtained results, we give several examples of exact solutions to nonlinear diffusion equations, including those with polynomial and fractional-rational source functions.

An Efficient Algorithm Based on Extrapolation for the Solution of Nonlinear Parabolic Equations

Abstract Two numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite difference discretization is used to approximate the solution at each time level. The convergence analysis is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equation with applications in fluid mechanics as well as convection–diffusion problems with applications in transport are solved to show the applicability and good performance of the procedures.

Asymptotic behavior of solutions for a free boundary problem with a nonlinear gradient absorption

This paper deals with the free boundary problem for a parabolic equation, $$u_t-u_{xx}=u^{p}-\lambda |u_x|^{q}$$ , $$t>0$$ , $$0<x<s(t)$$ , with $$p, q>1$$ . It is well known that global existence or blowup of solutions of nonlinear parabolic equations depends on which one dominating the model, the source or absorption, and on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of p, q, initial data and free boundary on the asymptotic behavior of solutions. At first, the ecological meaning of this model is explained by deriving the equation and the free boundary condition. Then, local existence and uniqueness are discussed, and the continuous dependence on initial data and comparison principle are proved. Furthermore, the finite time blowup and global solution are given by constructing sub- and super-solutions. In the case of $$ p>q>1 $$ , our results show that the solution blows up in finite time for the initial value sufficiently large; but if the initial data are small, there exist global fast solutions which decay exponentially. Also, with a suitable initial value, the existence of global slow solution with at most polynomial decaying is established. While, in the case of $$q\ge p>1$$ , all solutions with nonnegative initial data of exponential decay exist globally. Finally, the problem with double free boundaries, $$u_t-u_{xx}=u^p-\lambda |u_x|^q, t>0, g(t)<x<h(t)$$ , is also discussed and the similar conclusions are obtained.

Функция Ламберта и точные решения нелинейных параболических уравнений

Функция Ламберта и точные решения нелинейных параболических уравнений

Renormalized solution of nonlinear parabolic equations without sign condition and general measure data

We give an existence result of a renormalized solution for a class of nonlinear parabolic equations (∂b(u) ∕ ∂ t) - div (a(x, t, ∇ u)) + h(u) |∇u|p = μ, where the right side is a general measure and b(u) is a strictly increasing C1-function with b(0)=0, - div(a(x, t, ∇ u)) is a Leray--Lions type operator with growth |∇u|p-1 in ∇ u and h: ℝ → ℝ+ is a continuous positive function that belongs to L1(ℝ).

Об одном решении нелинейного параболического уравнения с нестационарным показателем нелинейности

Об одном решении нелинейного параболического уравнения с нестационарным показателем нелинейности

An Efficient Algorithm Based on Extrapolation for the Solution of Nonlinear Parabolic Equations

AbstractTwo numerical procedures are developed to approximate the solution of one-dimensional parabolic equations using extrapolated collocation method. By defining two different end conditions and forcing cubic spline to satisfy the interpolation conditions along with one of the end conditions, we obtain fourth- (CBS4) and sixth-order (CBS6) approximations to the solution in spatial direction. Also in time direction, a weighted finite-difference discretization is used to approximate the solution at each time level. The convergence analysis of the methods is discussed in detail and some error bounds are obtained theoretically. Finally, some different examples of Burgers’ equations with applications in fluid mechanics as well as convection–diffusion problems with applications in transport problems are solved to show the applicability and good performance of the procedures.

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

Abstract In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods.