Remarks on polynomial expansions of solutions of the heat equation in two space variables

Results on polynomial expansions of analytic solutions of the heat equation can be used for the discussion of the continuation of analytic solutions. A system of polynomial solutions introduced by Col ton and Wimp [3] is found good for such investigations in the two dimensional case. A Banach scales approach is the base for the results of the present paper     

The generalized Cayley-Hamilton theorem inn dimensions

Résumé Pour tout nombre entier positifn, cette étude tire la forme explicite de l'identité enn matrices, chacunen×n, qu'on obtient par la polarisation complète du théorème familier deCayley-Hamilton, et qu'on emploie à plusieurs reprises, avecn=2 ou 3, pour déterminer une base d'intégrité des tenseurs symétriques.     

Uniform asymptotic expansions of solutions of the Mathieu equation and the modified Mathieu equation

By the method of the model equation, uniform asymptotic expansions of the Floquet solutions of the Mathieu equation and two linearly independent solutions of the modified Mathieu equation are obtained for any real values of the separation parameter contained in these equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 60–91, 1976.     

Power expansions for the self-similar solutions of the modified Sawada-Kotera equation

The fourth-order ordinary differential equation that defines the self-similar solutions of the Kaup—Kupershmidt and Sawada—Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points z = 0, z = ∞ and near an arbitrary point z = z ₀ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.      

Exponential expansions of solutions to an ordinary differential equation

Doklady Mathematics -     

Power-elliptic expansions of solutions to an ordinary differential equation

A rather general ordinary differential equation is considered that can be represented as a polynomial in variables and derivatives. For this equation, the concept of power-elliptic expansions of its solutions is introduced and a method for computing them is described. It is shown that such expansions of solutions exist for the first and second Painlevé equations.     

Discretization of the solutions of the heat equation

We obtain the exact order of discretization (reconstruction) errors, given linear information on the solutions of the heat equation.     

Series solutions of a semilinear wave equation

We are concerned with the reconstruction of series solutions of a semilinear wave equation with a quadratic nonlinearity. The solution which may blow up in finite time is sought as a sum of exponential functions and is shown to be a classical one. The constructed solutions can be used to benchmark numerical methods used to approximate solutions of nonlinear equations.     

Random generalized solutions to the heat equation

A solution for the heat conduction problem with random source term and random initial and boundary conditions is defined. Existence, uniqueness, properties, and asymptotic behavior of such a solution are investigated. Applications to one-dimensional problems are presented.     

Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.     

Almost automorphic solutions of non-homogeneous heat equation

Summary In this paper we consider the non-homogeneous heat equation with almost-automorphic right-hand side. Some results concerning bounded and almost-automorphic solutions are given.

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Riassunto In questo lavoro si considera l'equazione del calore non-omogenes con termine noto quasi-automorfico. Sono dimostrati alcuni risultati concernenti soluzioni limitate e quasiautomorfiche.

     

Additional solutions of a weighted heat equation

We obtain a simple algorithm for computing additional solutions of a weighted heat equation.     

Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)