Applied Mathematics
Vol.5 No.15(2014), Article ID:48882,16 pages DOI:10.4236/am.2014.515226

The Cauchy Problem for the Heat Equation with a Random Right Part from the Space

Yuriy Kozachenko, Anna Slyvka-Tylyshchak

Department of Probability Theory, Statistics and Actuarial Mathematics, The Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Email: yvk@univ.kiev.ua, aslyvka@ukr.net

Received 16 May 2014; revised 24 June 2014; accepted 7 July 2014

ABSTRACT

The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random factors is a classical problem of the parabolic type of mathematical physics. In this paper, the heat equation with random right side is examined. In particular, we give conditions of existence with probability, one classical solutions in the case when the right side is a random field, sample continuous with probability one from the space. Estimation for the distribution of the supremum of solutions of such equations is founded.

Keywords:Cauchy Problem, Heat Equation, Stochastic Process

1. Introduction

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes.

The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet [1] . In the works [2] and [3] , a new approach studying the solutions of partial differential equations with random initial conditions was proposed. The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem. The mentioned approach was used in the works [4] -[7] . In the paper [3] , the application of the Fourier method for the homogeneous hyperbolic equation with Gaussian initial conditions is justified. The conditions of the existence of the classical solution of this equation in terms of correlation functions are also studied. Homogeneous hyperbolic equation with random initial conditions from the space is considered in [8] -[11] . The model of a solution of a hyperbolic type equation with random initial conditions was investigated in the papers [12] [13] . There is a study on a boundary-value problem of mathematical physics for the inhomogeneous hyperbolic equation with -subgaussian in right part [8] [14] . The parabolic type equations of Mathematical Physics with random factors of Orlicz spaces have been studied in the papers [15] [16] . Further references can be found in [8] [17] -[21] .

We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space. The Gaussian stochastic process with zero mean belongs to [22] . The conditions of existence with probability one of the classical solution of this problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

The paper consists of the introduction and three parts. Section 2 contains necessary definitions and results of the theory of the space. In Section 3, we consider heat equations with random right-hand side. For such problem conditions of existence, with probability one, of classical solution with random right-hand side from the space are found. The estimation for distribution of supremum of this problem has been got in Section 4.

2. Random Processes from Space

Definition 1. [23] An even continuous convex function, such that and

for and is called an function.

Definition 2. [21] We say an function satisfies the condition if there exist constants, , such that for all,.

Lemma 1. [21] Let be an function. Then 1. for and;

2. for and;

3. for;

4. The function is non decreasing for.

Lemma 2. [21] Let be the inverse to an function for. Then is a convex increasing function such that 1. for and;

2. for and;

3. for;

4. the function is nonincreasing for.

Definition 3. [23] Let be an function. The function is called the YoungFenchel transform of the function. The function is an function as well.

Let be a standard probability space.

Definition 4. [21] Let be an function for which there exist constants and such that for. The set of random variables, , is called the space generated by the function if and there exists a constant such that for all.

The space is a Banach space with respect to the norm [21] .

Definition 5. [23] The stochastic process belongs to space, if for all.

Remark 1. [24] The Gaussian stochastic process with zero mean belongs to, where and.

A Family of Strongly Random Variables and a Family Strongly Stochastic Processes

Lemma 3. [21] If, then there exists a constant such that.

Definition 6. [21] The random variable is called strongly, random variable if.

Properties and applications of random variables and stochastic processes from can be found in [21] .

Definition 7. [7] A family of random variables of the space is called family if

for all, where is at most countable and,.

Theorem 1. [7] Let be a strongly family of random variables. Then the linear closure of the family in the space and in the mean square sense is a strongly family.

Definition 8. [21] The stochastic process is called an process if the family of random variables is an.

Theorem 2. [7] Let be a family of jointly strongly stochastic processes. Then is a measurable space. If is a family of measurable functions in and the integral is well defined in the mean square sense, than the family of random variables is an family.

Theorem 3. [9] Let be the -dimensional space,

, ,.. Assume that the process is separable and

where is a monotone increasing continuous function such that as. We also assume that,where and is the inverse function to. If the processes converge in probability to the process for all, then converge in probability in the space.

Theorem 4. [9] Let and let, , be a separable random field such that. Put and assume that the partial derivatives

and

exist. Let there exist a monotone increasing continuous function, , such that as for, , and,. Assume that

If for all and for sufficiently small where, then with probability one the partial derivatives, , , exist and are continuous.

3. The Heat Equations with Random Right Part

We consider the Cauchy problem for the heat equation

(1)

, subject to the initial condition

(2)

Let the function is a random field sample continuity with probability one from the space, such that,. Let us denote. Let be a continuous function. Problem when the function nonrandom has been seen in [25] .

Lemma 4. Let is a random field, sample continuity for each with probability one, there is a continuous derivative for and satisfy condition

(3)

Then for the function for each the integral Fourier transform

exist and.

Proof. Since, by Fubini’s theorem, , we deduce that the integral exist with probability one, and therefore the integral, and therefore it implies from [26] that the integral Fourier transform

exist, and the inverse integral Fourier transform

exist.

Theorem 5. Let the conditions of Lemma 4 be satisfied and

and

(4)

If the following integrals exist

, ,

and for all and there exists a sequence, for, such that the sequence of integrals

(5)

(6)

converges in probability, uniformly for, , then is the classical solution to the problem (1) and (2).

Proof. Since the integrals (5) and (6) converges in probability uniformly for, , there exists a subsequence, as, such that

converges with probability one to

uniformly for, , Let

(7)

By deriving (7) with respect to and, we easily see that

Since for converges to, and con­verges to uniformly for, with probability one, we conclude that satisfies Equation (1).

Indeed,

Lemma 5. [9] Let be a random field, sample continuity from the space. Let be the correlation function of the field. For all assume that:

1. The derivatives exist;

2

3., at or.

Then Lebesgue integrals

exist with probability one.

Proof. We shall prove the existence of the integral

For existence of this integral with probability one it is enough to prove that there exists following integral

There is an inequality

Consider

Integrating by parts and using the conditions of the lemma, we obtain for y ≠ 0

Then

Therefore

for. The latter integral converges under. The existence of integrals, can be proved similarly.

Lemma 6. [15] Let a function, and be such that:

1.

2. for all,. Let, be a continuous increasing function such that for all, and the function is increasing for, and for some constant.

Then

for all and.

Corollary 1. Let in the conditions of Lemma 6 the function, ,. Then

(8)

for all.

Proof. Indeed, it is easy to show that the function

increases with. Therefore in Lemma 6 taking function we obtain the inequality 8.

Corollary 2.

. (9)

. (10)

for some.

Remark 2. If in the conditions of Corollary 2, , then for sufficiently small inequality (9) and (10) will have the form

Let

Theorem 6. Let be a random field, sample continuous with probability one from the and the conditions of Lemma 4 and Lemma 5 hold,

For, where is a monotone increasing continuous function such that as, moreover,

, (11)

where, and is the inverse function to. Then the function which is represented in the form (4) is a classical solution to the problems (1) and (2).

Proof. This theorem follows from Theorems 5 and 3.

Example 1. Let be a function such that, for some and all. Then for and condition (11) holds for all

(12)

Condition (12) holds if, for,

,. In this case, the condition of Theorem 6 is satisfied if for there exist constants such that

, (13)

For all, and sufficiently small.

Theorem 7. Let be a random field, sample continuous with probability one from the space, where is a function such that for some and all and the conditions of Lemma 4 and Lemma 5 hold and , for some, ,. Then the function which is represented in the form (4) is classical solution to the problems (1) and (2).

Proof. It follows from Lemma 5 that there exist integrals with probability one

,.

According to Theorem 5 to make the function be the solution of problems (1) and (2) it is sufficient to prove that integrals (5) and (6) converge uniformly in probability in, to the integrals

, for any,. According to Theorem 6, using the Example (1), to make integral (5) and (6) converge in probability in the following conditions must hold

Using generalized Minkovskoho inequality we obtain

(14)

Let and for sufficiently small, using the inequality (10), we have

(15)

Consider

It follows from Lemma 4 that

Therefore

(16)

Let then

Let and for sufficiently small, using the inequality (9), we have

Therefore

(17)

Thus we obtain from (14), (15), (16) and (17) that

Consider

Since, we have

Using that, , then the and for sufficiently small, we have

So we have

Therefore Then for, we havewhere

, are some constants.

Consider

From Lemma 4, we obtain

Similarly

where, are some constants.

Consider

From Lemma 4 we obtain

Then

where are some constants.

4. Estimates of the Distribution of the Supremum of a Solution

Theorem 8. [9] Let be the -dimensional space, , ,. Assume that is separable and. Ifwhere is a monotone increasing continuous function such that as, and

, where and is the inverse function to. Then, for all and, where

Theorem 9. Let the conditions of Theorem 6 hold

.

where

. Then

for all and, where

where is a monotone increasing continuous function such that as, and is the inverse function to.

Proof. This theorem follows from Theorem 8.

References

1. de Feriet, K. (1962) Statistical Mechanics of Continuous Media. Proceedings of Symposia in Applied Mathematics, American Mathematical Society, Providence, 165-198.
2. Beisenbaev, E. and Kozachenko, Yu.V. (1979) Uniform Convergence in Probability of Random Series, and Solutions of Boundary Value Problems with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 21, 9-23.
3. Buldygin, V.V. and Kozachenko, Yu.V. (1979) On a Question of the Applicability of the Fourier Method for Solving Problems with Random Boundary Conditions. Random Processes in Problems Mathematical Physics, Academy of Sciences of Ukrain.SSR, Institute of Mathematics, Kuiv, 4-35.
4. de La Krus, E.B. and Kozachenko, Yu.V. (1995) Boundary-Value Problems for Equations of Mathematical Physics with Strictly Orlics Random Initial Conditios. Random Operators and Stochastic Equations, 3, 201-220.http://dx.doi.org/10.1515/rose.1995.3.3.201
5. Kozachenko, Yu.V. and Endzhyrgly (1994) Justification of Applicability of the Fourier Method to the Boundary-Value Problems with Random Initial Conditions I. Theory of Probability and Mathematical Statistics, 51, 78-89.
6. Kozachenko, Yu.V. and Endzhyrgly (1994) Justification of Applicability of the Fourier Method to the Boundary-Value Problems with Random Initial Conditions II. Theory of Probability and Mathematical Statistics, 53, 58-68.
7. Kozachenko, Yu.V. and Kovalchuk, Ya.A. (1998) Boundary Value Problems with Random Initial Conditions and Series of Functions of Subφ(Ω). Ukrainian Mathematical Journal, 50, 504-515.
8. Dovgay, B.V., Kozachenko, Yu.V. and Slyvka-Tylyshchak, G.I. (2008) The Boundary-Value Problems of Mathematical Physics with Random Factors. Kyiv University, Kyiv, 173 p. (Ukrainian)
9. Kozachenko, Yu.V. and Slyvka, G.I. (2004) Justification of the Fourier Method for Hyperbolic Equations with Random Initial Conditions. Theory of Probability and Mathematical Statistics, 69, 67-83.http://dx.doi.org/10.1090/S0094-9000-05-00615-0
10. Slyvka, A.I. (2002) A Boundary-Value Problem of the Mathematical Physics with Random Initials Conditions. Bulletin of University of Kyiv. Series: Physics & Mathematics, 5, 172-178.
11. Slyvka-Tylyshchak, A.I. (2012) Justification of the Fourier Method for Equations of Homogeneous String Vibration with Random Initial Conditions. Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 38, 211-232.
12. Kozachenko, Y.V. and Slyvka, G.I. (2007) Modelling a Solution of a Hyperbolic Equation with Random Initial Conditions. Theory Probability and Mathematical Statistics, 74, 59-75.
13. Tylyshchak, A.I.S. (2012) Simulation of Vibrations of a Rectangular Membrane with Random Initial Conditions. Annales Mathematicae and Informaticae, 39, 325-338.
14. Dovgay, B.V. and Kozachenko, Y.V. (2005) The Condition for Application of Fourie Method to the Solution of Nongomogeneous String Oscillation Equation with φ-Subgaussianright Hand Side. Random Operators and Stochastic Equations, 13, 281-296.
15. Kozachenko, Y.V. and Veresh, K.J. (2010) The Heat Equation with Random Initial Conditions from Orlicz Space. Theory of Probability and Mathematical Statistics, 80, 71-84. http://dx.doi.org/10.1090/S0094-9000-2010-00795-2
16. Kozachenko, Y.V. and Veresh, K.J. (2010) Boundary-Value Problems for a Nonhomogeneous Parabolic Equation with Orlicz Right Side. Random Operators and Stochastic Equations, 18, 97-119. http://dx.doi.org/10.1515/rose.2010.005
17. Angulo, J.M., Ruiz-Medina, M.D., Anh, V.V. and Grecksch, W. (2000) Fractional Diffusion and Fractional Heat Equation. Advances in Applied Probability, 32, 1077-1099. http://dx.doi.org/10.1239/aap/1013540349
18. Kozachenko, Y.V. and Leonenko, G.M. (2006) Extremal Behavior of the Heat Random Field. Extremes, 8, 191-205.http://dx.doi.org/10.1007/s10687-006-7967-8
19. Beghin, L., Kozachenko, Y., Orsingher, E. and Sakhno, L. (2007) On the Solution of Linear Odd-Order Heat-Type Equations with Random Initial. Journal of Statistical Physics, 127, 721-739.http://dx.doi.org/10.1007/s10955-007-9309-x
20. Ratanov N.E., Shuhov, A.G. and Suhov, Y.M. (1991) Stabilization of the Statistical Solution of the Parabolic Equation. Acta Applicandae Mathematicae, 22, 103-115.
21. Buldygin, V.V. and Kozachenko, Y.V. (2000) Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Rhode.
22. Antonini, R.G., Kozachenko, Y. and Nikitina, T. (2003) Spaces of φ-Subgaussian Random Variables. Memorie di Matematica e Applicazioni, Accademia Nazionale delle Scinze detta dei XL, Vol. 27, 95-124.
23. Krasnoselsky, M.A. and Rutitcky, Y.B. (1961) Convex Functions and Orlicz Spaces. Noordhof, Gröningen.
24. Kozachenko, Y.V. and Ostrovskij, E.V. (1986) Banach Spaces of Random Variables of Sub-Gaussian Type. Theory of Probability and Mathematical Statistics, 532, 42-53.
25. Markovich, B.M. (2010) Equations of Mathematical Physics. Lviv Polytechnic Publishing House, Lviv, 384 p. (Ukrainian).
26. Budylin, A.M. (2002) Fourier Series and Integrals. Saint Petersburg, 137 p.