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1.
We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f
1(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞.
__________
Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007. 相似文献
2.
A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent
moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) 01(t)=0
exp(−λt), (ii) 02(t) =0(t/t
*)exp(−λt), and 03(t)=0[1+a
cos(ωt)], where λ and ω are real parameters and t
* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized
representation of an incomplete gamma function Γ(α,x;b) and its decomposition C
Γ and S
Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present
analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations.
Received on 13 June 1997 相似文献
3.
The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy
force effects, for the situation in which the surface temperature T
′
w(x) and C
′
w(x) are subjected to the power-law surface heat and mass flux as K(T
′/r) = −ax
n
and D(C
′/r) = −bx
m
. The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are
obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are
shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a
good agreement is found to exist.
Received on 7 July 1998 相似文献
4.
Yoshihisa Morita Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861
We deal with a reaction–diffusion equation u
t
= u
xx
+ f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c
1
t) (c
1 < 0) and ψ2(x + c
2
t) (c
2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all
. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c
1
t) and ψ2(x + c
2
t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c
1, we show the existence of an entire solution which behaves as ψ1( − x + c
1
t) in
and φ(x + ct) in
for t≈ − ∞. 相似文献
5.
D.V. Bel’skii 《Nonlinear Oscillations》2006,9(2):166-172
We establish new properties of C
1[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞.
__________
Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006. 相似文献
6.
Matteo Bonforte Gabriele Grillo Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680
We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for
x ? \mathbb Rd{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular
to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted
here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold
(\mathbb Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard
\mathbb Rd{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated
with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics,
which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear
Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known
exponential decay of such representation for all other values of m. 相似文献
7.
The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water
at 4°C (maximum density) when the surface heat flux varies as x
m
and the velocity outside the boundary layer varies as x
(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions
of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions
when the mixed convection parameter λ is greater than some critical value λ
c
(dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ
c
is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and
the limits as m → ∞ and as m → −1. 相似文献
8.
Wehavediscussedconceptofequationwithn_turningpointsinmypaper[1],i.e.,asecondorderlinearordinarydifferentialequationd2ydx2+[λ2q1(x)+λq2(x,λ)]y=0,whereq1(x)=(x-μ1)(x-μ2)…(x-μn)f(x),f(x)≠0,andλisalargeparameter.Althroughthefirsttermoftheasymptoticexpan… 相似文献
9.
We investigate the time periodic solutions to the viscous Burgers equation ut − μuxx + uux = f for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function
spaces.
相似文献
10.
M. Banerjee 《Transport in Porous Media》2008,73(3):333-347
The field measurements and numerical results for intermittent flow regime in a sandy soil show that the time distributions
of the soil water flux q(z, t), and the soil water content θ(z, t)at various depths are periodic in nature, where t is time and z is the depth (i.e., at the surface z = 0 and at depths z = − 5, − 10, − 15 cm, etc). The period of q(z, t) and θ(z, t) variations are generally determined by the sum of the duration of pulse and the duration between the initiation of two consecutive
pulses of water at the soil surface. Fourier series models have been given for q(z, t) and θ(z, t) variations. The predicted Fourier results for these variations have been compared with the experimentally verified numerical
results—designated as observed values. The results show that the amplitudes of these variations were damped exponentially with depth, and the phase shift
increased linearly with depth. 相似文献
11.
Luis Barreira Meng Fan Claudia Valls Jimin Zhang 《Journal of Dynamics and Differential Equations》2012,24(1):101-118
We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x
t
+ f (t, x
t
, λ), assuming that the linear equation x′ = L(t)x
t
admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the
parameter λ. We also consider the general case of nonuniform polynomial dichotomies. 相似文献
12.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410
Let
be the exterior of the closed unit ball. Consider the self-similar Euler system
Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary
data on ∂Ω, we prove that this system has a unique solution
, vanishing at infinity, precisely
The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L
2 − norm of curl v. 相似文献
13.
Xiao-Biao Lin 《Journal of Dynamics and Differential Equations》2007,19(4):1037-1074
We study the spectral and linear stability of Riemann solutions with multiple Lax shocks for systems of conservation laws.
Using a self-similar change of variables, Riemann solutions become stationary solutions for the system u
t
+ (Df(u) − x
I)u
x
= 0. In the space of O((1 + |x|)−η) functions, we show that if , then λ is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of
a transcendental matrix. On some vertical lines in the complex plane, called resonance lines, the determinant can be arbitrarily small but nonzero. A C
0 semigroup is constructed. Using the Gearhart–Prüss Theorem, we show that the solutions are O(e
γ t
) if γ is greater than the real parts of the eigenvalues and the coordinates of resonance lines. We study examples where Riemann
solutions have two or three Lax-shocks.
Dedicated to Professor Pavol Brunovsky on his 70th birthday. 相似文献
14.
The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution
T
w(x) = T
∞+A·x
λ and lateral mass flux v
w(x) = a·x
(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions
are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection
parameter in detail. The results are compared with the numerical findings of previous authors.
Received on 8 March 1999 相似文献
15.
We consider the class of wave equations u
tt−u
xx=f(u, u
t, u
x). By using the differential invariants, with respect to the equivalence transformation algebra of this class, we characterize
subclasses of linearizable equations. Wide classes of general solutions for some nonlinear forms of f(u, u
t, u
x) are found. 相似文献
16.
We consider a time-dependent free boundary problem with radially symmetric initial data: σt − Δσ + σ = 0 if and σ(r,0)=σ0(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σS(r), RS), where RS depends only on the number . We prove that there exists a number μ*, such that if μ < μ* . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ* then the stationary solution is unstable. 相似文献
17.
Kenneth R. Meyer Patrick McSwiggen Xiaojie Hou 《Journal of Dynamics and Differential Equations》2010,22(3):367-380
The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search
for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ
1 as t → −∞ and u(t) → γ
2 as t → ∞ where γ
1, γ
2 are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that
f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions
we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied. 相似文献
18.
Thierry Cazenave Flávio Dickstein Fred B. Weissler 《Journal of Dynamics and Differential Equations》2007,19(3):789-818
In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.
相似文献
19.
Hermano Frid 《Archive for Rational Mechanics and Analysis》2006,181(1):177-199
We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems
of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems
whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which
are hyperplanes. In particular, we obtain the uniqueness of the self-similar L∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is
in the sense of the convergence as t→∞ in Lloc1 of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in Lloc1(ℝn), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem. 相似文献
20.
The present paper focuses on the analysis of unsteady flow and heat transfer regarding an axisymmetric impinging synthetic
jet on a constant heat flux disc. Synthetic jet is a zero net mass flux jet that provides an unsteady flow without any external
source of fluid. Present results are validated against the available experimental data showing that the SST/k − ω turbulence model is more accurate and reliable than the standard and low-Re k − ε models for predicting heat transfer from an impinging synthetic jet. It is found that the time-averaged Nusselt number
enhances as the nozzle-to-plate distance is increased. As the oscillation frequency in the range of 16–400 Hz is increased,
the heat transfer is enhanced. It is shown that the instantaneous Nu distribution along the wall is influenced mainly by the interaction of produced vortex ring and wall boundary layer. Also,
the fluctuation level of Nu decreases as the frequency is raised. 相似文献