A Cauchy problem for the heat equation

Summary Let u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | u_x(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then, , where M₁ and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and u_x(0, t)=g(t) is discussed. Work performed under the auspices of the U. S. Atomic Energy Commission.     

A nonstandard Cauchy problem for the heat equation

Mathematical Notes - General necessary conditions on the real parameters α, β, C, D for the function $${e^{ - alpha rho left( x right)}}left( {Ccos beta rho left( x right) +...     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Regularization of a non-characteristic Cauchy problem for the heat equation

The non-characteristic Cauchy problem for the heat equation u_xx(x,t) = u₁(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), u_x(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.     

An iterative method for a Cauchy problem for the heat equation

^** Email: tomjo{at}itn.liu.se An iterative method for reconstruction of the solution to aparabolic initial boundary value problem of second order fromCauchy data is presented. The data are given on a part of theboundary. At each iteration step, a series of well-posed mixedboundary value problems are solved for the parabolic operatorand its adjoint. The convergence proof of this method in a weightedL²-space is included.     

A direct numerical procedure for the Cauchy problem for the heat equation

For the Cauchy problem, u_t = u_xx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, u_x(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υ_t = υ_xx, 0 < x, 0 < t ? T, υ_x(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.     

Cauchy problem for a degenerate heat equation in Banach spaces

We study the solvability of a degenerate heat equation with closed linear operators B multiplying the time derivative and A multiplying the Laplace operator in the class of generalized functions in Banach spaces. Under various assumptions on the operator pencil λB-A (it can be Fredholm of index zero, Fredholm, spectrally bounded, sectorial, or radial), we construct the fundamental operator function for the differential operator Bδ′(t) × δ ( $ \bar x $ ) ? Aδ(t) × Δδ ( $ \bar x $ ) and use it for the closed-form construction of the desired generalized solution of the Cauchy problem for the equation in question. We single out uniqueness classes for these solutions and analyze the relationship between continuous and generalized solutions.     

Cauchy problem for the generalized Kadomtsev-Petviashvili equation

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 160–172, January–February, 1992.     

The Cauchy problem for the Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces ${H^s(\mathfrak{R})}$ with ${s < \frac{3}{2}}$ in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in ${H^{s}(\mathfrak{R})}$ with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in ${H^{s}(\mathfrak{R})}$ with s > 3/2.     

Cauchy problem for the beam vibration equation

We study a problem on the vibrations of an infinite beam at an arbitrary time after an initial perturbation. We obtain sufficient conditions for the existence of a solution, which is constructed in explicit form.     

Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation

In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.     

Resolvent metric and the heat kernel estimate for random walks

In this paper we introduce the resolvent metric, the generalization of the resistance metric used for strongly recurrent walks. By using the properties of the resolvent metric we show heat kernel estimates for recurrent and transient random walks.     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ²u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either , s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown.