共查询到20条相似文献,搜索用时 15 毫秒
1.
《Mathematical and Computer Modelling》1999,29(2):1-18
This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) ut)t = Q(t)uxx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oRr×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied. 相似文献
2.
Dinh Nho Ho 《Mathematical Methods in the Applied Sciences》1992,15(8):537-545
The non-characteristic Cauchy problem for the heat equation uxx(x,t) = u1(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), ux(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate. 相似文献
3.
For the Cauchy problem, ut = uxx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, ux(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υt = υxx, 0 < x, 0 < t ? T, υx(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented. 相似文献
4.
Nathaniel Chafee 《Journal of Differential Equations》1974,15(3):522-540
We consider a parabolic partial differential equation ut = uxx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis. 相似文献
5.
Richard E. Ewing 《Journal of Mathematical Analysis and Applications》1979,71(1):167-186
Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)ux)x ? q(x)u = p(x)ut, 0 < x < 1,0 < t? T; ; ; p(0) ux(0, t) = g(t), 0 < t0 ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?1, ?2 and g are known only approximately. 相似文献
6.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):291-296
The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form ut - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in QT =]0,T[×Ω, Ω ⊂ RN, with an initial condition u(0) = u0, where u0 is not bounded, |H(t,x, u, ξ)⩽ β|ξ|p + f(t,x) + βeλ1|u|f, |g|p/(p-1) ∈ Lr(QT) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator. 相似文献
7.
Joo-Paulo Dias Mrio Figueira Luis Sanchez 《Mathematical Methods in the Applied Sciences》1998,21(12):1107-1113
In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
8.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
9.
Alkis S. Tersenov 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):437-452
The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u t + g(t, u)u x + f(t, u) = εu xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well. 相似文献
10.
《Mathematical and Computer Modelling》1997,25(11):31-44
This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation utt + c(t)ut + b(t)u = a(t)uxx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed. 相似文献
11.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
12.
Fred B Weissler 《Journal of Differential Equations》1984,55(2):204-224
The initial value problem on [?R, R] is considered: ut(t, x) = uxx(t, x) + u(t, x)γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψxx + ψγ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then limt → Tu(t, x) and limt → Tux(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0. 相似文献
13.
Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations 总被引:1,自引:0,他引:1
In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques. 相似文献
14.
Claire Chainais‐Hillairet 《Mathematical Methods in the Applied Sciences》2000,23(5):467-490
We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation ut(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u0. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u0 ∈ L∞ ∩ BVloc(ℝN), we get an error estimate of order h1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
15.
We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with +(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u0∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u0(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L1-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L∞‖ for (2+mn)/(n+1)<p<2. 相似文献
16.
Fahuai YiYuqing Liu 《Journal of Differential Equations》2002,183(1):189-207
Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule uε(Rε(t),t)=εRε′(t) at the moving boundary are considered. We prove, when ε approaches zero, Rε(t) converges to R(t) in C1+δ/2[0,T] for any finite T>0, 0<δ<1. 相似文献
17.
Jerry L Bona Vassilios A Dougalis 《Journal of Mathematical Analysis and Applications》1980,75(2):503-522
An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form ut + ux + uux ? vuxx ? α2uxxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes. 相似文献
18.
This paper studies the propagation of pulse-like solutions of semilinear hyperbolic equations in the limit of short wavelength. The pulses are located at a wavefront Σ?{φ=0} where φ satisfies the eikonal equation and dφ lies on a regular sheet of the characteristic variety. The approximate solutions are uεapprox=U (t, x, φ(t, x)/ε) where U(t, x, r) is a smooth function with compact support in r. When U satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions uεexact such that uεapprox has relative error O(ε) as ε→0. While the transport equation is familiar, the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope. 相似文献
19.
Müjdat Kaya 《Mediterranean Journal of Mathematics》2013,10(1):227-239
The problem of determining the initial value u(x, 0) = μ 0(x) in the parabolic equation u t = (k(x)u x (x, t)) x + F(x, t) from the final overdetermination μ T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. 相似文献
20.
Marcelo Montenegro 《Journal of Mathematical Analysis and Applications》2011,384(2):591-596
We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u0(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t0>0 such that the solution is identically equal to zero. 相似文献