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This paper deals with the numerical solution of the random Cauchy one-dimensional heat model. We propose a random finite difference numerical scheme to construct numerical approximations to the solution stochastic process. We establish sufficient conditions in order to guarantee the consistency and stability of the proposed random numerical scheme. The theoretical results are illustrated by means of an example where reliable approximations of the mean and standard deviation to the solution stochastic process are given.

The heat is the energy which flows from the higher to the lower temperature and the transport coefficient depends on the specific mode transfer. The transfer modes are the diffusive transport of thermal energy (the conduction mode), the exchange of heat between a moving fluid and an adjoining wall (the convection mode), and the radiation mode where all bodies can emit thermal radiation [

When dealing with a partial differential equation together with the initial and boundary conditions, it is crucial to obtain a well-posed problem. The extent of the spatial domain is another division for the partial differential equation that makes one method of solution preferable over another. Spatial domain may be a finite interval or an infinite interval, such as the whole real line. If the spatial domain is unbounded, the boundary conditions are not an important issue and in that case the problem is called initial value problem (IVP). In mathematics, a pure IVP is usually referred to as a Cauchy problem [

Mathematical models described by means of partial differential equations (PDEs) appear often in many areas of science and engineering and also in medicine and finance, for example, [

In the deterministic scenario, the heat equation on unbounded domains has been studied by different authors [

In this paper, we propose a random finite numerical scheme to approximate the solution s.p. of the Cauchy problem (

This paper is organized as follows. In Section

This section is devoted to introducing the numerical technique that will be considered later in order to approximate the solution s.p. to the random IVP (

The next step is to approximate the solution s.p. to the IVP (

Substituting the approximations (

As it is well known from the deterministic case, the study of the consistency and stability is a main issue when dealing with numerical schemes. This motivates the analysis of consistency and stability of the random numerical scheme (

According to the definition of the consistency of a finite difference numerical scheme in the deterministic case, below we extend this definition to the random scenario taking into account the norm (

The random finite difference scheme

In the context of Definition

Next, we shall prove that the RFDS (

Let us consider the random IVP (

Let us denote

Following the same idea we have used for introducing the concept of random consistency, below we extend the deterministic definition of stability of a finite numerical scheme to the random scenario using the norm (

The random finite difference scheme (

Below, we establish conditions under which the RFDS (

Let us consider the random IVP (

Taking into account the definition of the norm (

Under hypotheses (

It is important to point out that the hypothesis of boundedness on r.v.

This section is devoted to illustrating the theoretical results previously established by means of a test example where reliable approximations for the mean and the standard deviation (or equivalently the variance) of the solution s.p. of IVP (

Let us consider the random Cauchy problem (

We will approximate the mean and standard deviation of the solution s.p.,

In order to compute approximations of the mean and the standard deviation of the solution s.p.

In Figure

Expectation of the exact solution s.p. and the approximations at the time instant

An analogous comparison for the standard deviation at the time instant

Standard deviation of the exact solution s.p. and the approximations at the time instant

To complete the numerical analysis, in Figures

Relative errors at the time instant

Relative errors at the time instant

In this paper we have studied the randomized Cauchy heat model by assuming that the diffusion coefficient is a random variable and considering a deterministic initial condition over an unbounded domain. Thus, boundary conditions have not been required. We have proposed a random finite difference scheme for solving this model. The mean square consistency of the random finite difference scheme has been studied. Sufficient conditions for the mean square stability of the random finite difference scheme have been provided. The numerical experiments show that the proposed random finite difference scheme gives reliable approximations for the mean and the standard deviation of the solution stochastic process.

The authors declare that there is no conflict of interests regarding the publication of this article.

This work has been partially supported by the Ministerio de Economía y Competitividad Grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. M. A. Sohaly is also indebted to Egypt Ministry of Higher Education, Cultural Affairs, for its financial support [mohe-casem (2016)].