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) +...     

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.     

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.     

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.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

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 generalized Kadomtsev-Petviashvili equation

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

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.     

An asymptotic Cauchy problem for the Laplace equation

The Cauchy problem for the Laplace operator $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $$\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $$ with a given majoranth, satisfyingh(+0)=0. Thisasymptotic Cauchy problem only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy dataf, g, and this smoothness is strictly controlled byh. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

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.