共查询到20条相似文献，搜索用时 15 毫秒
1.
This paper concerns the study of the numerical approximation for the following initialboundary value problem
|
$
\left\{ \begin{gathered}
u_t - u_{xx} = f\left( u \right), x \in \left( {0,1} \right), t \in \left( {0,T} \right), \hfill \\
u\left( {0,t} \right) = 0, u_x \left( {1,t} \right) = 0, t \in \left( {0,T} \right), \hfill \\
u\left( {x,0} \right) = u_0 \left( x \right), x \in \left[ {0,1} \right], \hfill \\
\end{gathered} \right.
$
\left\{ \begin{gathered}
u_t - u_{xx} = f\left( u \right), x \in \left( {0,1} \right), t \in \left( {0,T} \right), \hfill \\
u\left( {0,t} \right) = 0, u_x \left( {1,t} \right) = 0, t \in \left( {0,T} \right), \hfill \\
u\left( {x,0} \right) = u_0 \left( x \right), x \in \left[ {0,1} \right], \hfill \\
\end{gathered} \right.
相似文献
2.
Miglena Koleva Lubin Vulkov 《Numerical Methods for Partial Differential Equations》2007,23(2):379-399
The numerical solution of the heat equation in unbounded domains (for a 1D problem‐semi‐infinite line and for a 2D one semi‐infinite strip) is considered. The artificial boundaries are introduced and the exact artificial boundary conditions are derived. The original problems are transformed into problems on finite domains. The space semi‐discretization by finite element method and the full approximation by the implicit‐explicit Euler's method are presented. The solvability of the full discretization schemes is analyzed. Computational examples demonstrate the accuracy and the efficiency of the algorithms. Also, the behavior of blowing up solutions is examined numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 379–399, 2007 相似文献
3.
Ronald E. Mickens P. M. Jordan 《Numerical Methods for Partial Differential Equations》2004,20(5):639-649
A positivity‐preserving nonstandard finite difference scheme is constructed to solve an initial‐boundary value problem involving heat transfer described by the Maxwell‐Cattaneo thermal conduction law, i.e., a modified form of the classical Fourier flux relation. The resulting heat transport equation is the damped wave equation, a PDE of hyperbolic type. In addition, exact analytical solutions are given, special cases are mentioned, and it is noted that the positivity condition is equivalent to the usual linear stability criteria. Finally, solution profiles are plotted and possible extensions to a delayed diffusion equation and nonlinear reaction‐diffusion systems are discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004. 相似文献
4.
This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. 相似文献
5.
We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u0(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools. 相似文献
6.
Construction of nonstandard finite difference schemes for $1{1\over 2}$ space‐dimension‐coupled PDEs
Ronald E. Mickens P.M. Jordan 《Numerical Methods for Partial Differential Equations》2007,23(1):211-219
Two coupled PDEs, where one has a diffusion term and the other does not, are defined to be space‐dimension systems. We show how to construct nonstandard finite difference schemes for such systems and demonstrate that they are positivity‐preserving. These methods also allow the calculation of an explicit functional relationships between the time and space step‐sizes. The case of water flowing through fractured bedrock is used to illustrate our procedure. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
7.
Ron Buckmire 《Numerical Methods for Partial Differential Equations》2004,20(3):327-337
The boundary value problem Δu + λeu = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λc, no solutions for λ > λc and a unique solution when λ = λc. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λc = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004 相似文献
8.
Marin Slodi
ka 《Numerical Methods for Partial Differential Equations》2005,21(2):191-212
We study a nonlinear degenerate parabolic equation of the type accompanied by an initial datum and mixed boundary conditions. The symbol [ · ]+ denotes the usual cutoff function. The problem represents a model of a reactive solute transport in porous media. The exponent p fulfills p ∈ (0, 1). This limits the regularity of a solution and leads to inconveniences in the error analysis. We design a new robust linear numerical scheme for the time discretization. This is based on a suitable combination of the backward Euler method and a linear relaxation scheme. We prove the convergence of relaxation iterations on each time point ti. We derive the error estimates in suitable function spaces for all values of p ∈ (0, 1). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005. 相似文献
9.
Thierry Cazenave Flvio Dickstein Fred B. Weissler 《Journal of Mathematical Analysis and Applications》2009,360(2):537-547
In this paper, we consider the nonlinear heat equation ut−Δu=|u|αu, | in the unit ball Ω of with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set of initial values in C0(Ω) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of , which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that is not convex. This contrasts with the case of nonnegative initial values, where the analogous set is known to be convex. 相似文献
