Abstract
This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.
1. Introduction
Temperature and heat flow are two important quantities in the problem of heat conduction. Temperature at any point of a solid is completely determined by its numerical value because it is a scalar quantity, whereas heat flow is defined by its value and direction, and it depends on the properties of the material that are neither uniform nor pure [1]. These uncertainties about material apart from measurement error deserve a random approach. Random heat transfer in a finite medium has been treated in [2] by developing a random perturbation method, in [3] by using finite element method, and in [4, 5] using finite difference methods.
It is well known that the integral transform technique to solve deterministic heat problems is very powerful and efficient because, by the combined use of the direct and inverse transform, the problem is simplified, by transforming partial differential equations into ordinary differential ones. In this paper, we introduce the random Fourier sine and cosine transforms and their mean square operational calculus to solve random temperature distribution in a semi-infinite bar with random temperature or random heat flow at one end. Using results about random ordinary differential equations of [6] allows us to find closed-form solutions stochastic process (s.p.) of random heat problems as in the deterministic case.
Although some authors deal with the uncertainty by using the Brownian motion and the Itô calculus [7, 8], we use a random mean square approach (m.s.) for two main reasons. The first one is that the m.s. solution coincides with the deterministic solution when data are deterministic. The second one is that if is an m.s. approximation of , then the expectation and the variance of converge to the expectation and variance of , respectively; see [9, chapter 4].
This paper is organized as follows. In Section 2, some preliminaries about m.s. calculus definitions, properties, and results are included. In Section 3, the random Fourier sine and cosine transforms are introduced and important m.s. operational rules related to the random Fourier sine (cosine) transform of a s.p. and of its m.s. derivatives, that is, the key to solve partial differential equations in terms of ordinary differential equations, are given. Sections 4 and 5 deal with random boundary value problems associated with the random heat equation where is a positive random variable (r.v.) whose properties will be specified later. For the deterministic case, the boundary conditions of these problems are called of third and second kind, respectively [1, page 52]. In Section 6, some illustrative examples are studied and in Section 7 a set of conclusions are given.
2. Preliminaries about Random Mean Square Calculus
In this section, we review some important concepts, definitions, and results related to the random calculus, mainly focusing on the mean square (m.s.) and mean fourth (m.f.) calculus, which correspond to and , respectively (see [10] for further details). After that, a relevant class of r.v.’s that will play an important role in the development of next sections is studied.
Let be a probabilistic space. Let be a real number. A real r.v. defined on is called of order , if where denotes the expectation operator. The space of all the real r.v.’s of order , endowed with the norm is a Banach space, [11, page 9].
Let be a sequence of r.v.’s of order . We say that it is convergent in the th mean to the real r.v. , if
This type of convergence is often expressed by . The symbol . denotes the limit in the th mean. If , then . In addition, if is th mean convergent to , then is also th mean convergent to [11, page 13]. Convergences in and are usually referred to as m.s. and m.f. convergence, respectively. If is a sequence of 2-r.v.’s in m.s. convergent to , then from Theorem 4.3.1 of [9, page 88] one gets where denotes the variance operator.
Let be a subset of the real line. A family of real r.v.’s of order is said to be a s.p. of order or, in short, a -s.p. if
We say that is th mean continuous at , if
Furthermore, if there exists a s.p. of order , such that then we say that is th mean differentiable at and is the -derivative of .
In the particular cases that , aforementioned definitions lead to the corresponding concepts of mean square (m.s.) and mean fourth (m.f.) continuity and differentiability. Furthermore, it is easy to establish by the Schwarz’s inequality that (see [10]), which prove that m.f. continuity and differentiability entail m.s. continuity and differentiability, respectively.
In accordance with [9, page 99], [12], we say that a s.p. with for all , is locally integrable in if, for all finite interval , the integral exits in . We say that is absolutely integrable in , if
Now, we introduce an important type of r.v.’s, , that have played a significant role in the m.s. solution of random ordinary differential equations (see [13] and references therein), and which will be used later. We will assume that such r.v.’s have absolute moments with respect to the origin that increases at the most exponentially; that is, there exist a nonnegative integer and positive constants and such that
From (11) and definition (3) for , for each , one gets Thus,
Remark 1. The lack of explicit formulae for the absolute moments with respect to the origin of some standard r.v.’s as well as the aim of looking for a general approach to deal with the widest range of random inputs, we are going to take advantage of censuring method (see [14, chapter 5]) to show that truncated r.v.’s satisfy condition (11). Let us assume a r.v. that satisfies
Then,
where denotes the probability density function (p.d.f.) of r.v. and . Indeed, in the case that , one gets
Notice that, in the last step, we have applied that the integral of the right-hand side is just because is a p.d.f. The other cases can be analyzed analogously. Substituting the integral by a sum in (15), previous reasoning remains true when is a discrete r.v. As a consequence, important r.v.’s such as binomial, hypergeometric, uniform, or beta satisfy condition (11) related to the absolute moments of . Although many other unbounded r.v.’s can also verify condition (11), we do not need to check it each case; since censuring their codomain suitably, we are legitimated to approximate them. Hence, truncations of r.v.’s such as exponential or Gaussian satisfy condition (11). The larger the censured interval is, the better the approximations are. However, in practice, intervals relatively short provide very good approximations. For instance, as an illustrative example, notice that the truncated interval contains the 99.7% of the probability mass of a Gaussian r.v. with mean and standard deviation .
For the sake of clarity in the presentation, we state the m.s. differentiation of integrals whose proof is a direct consequence of the deterministic case [12, page 99] and the m.s. differentiation theorem for a sequence of 2-s.p. [15].
Lemma 2 (m.s. differentiation of infinite integrals). Let be a 2-s.p. m.s. continuous with m.s. continuous partial derivative . Assume the following hypotheses.(i) is m.s. pointwise convergent for each . (ii) is m.s. uniformly convergent in , for each . Then, the process is m.s. differentiable, and
3. Random Fourier Sine and Cosine Transforms’ Operational Calculus
We begin this section by introducing the definition of the random Fourier sine and cosine transforms of a 2-s.p. m.s. locally integrable, and m.s. absolutely integrable; that is, as the s.p.’s respectively. Note that from (18) both integrals appearing in (19) and (20) are convergent in and thus they are 2-s.p.’s well defined.
Following the ideas of the deterministic inverse Fourier sine and cosine transforms, [16, chapter 2], we define the random inverse Fourier sine and cosine transforms of a 2-s.p. m.s. locally and m.s. absolutely integrable by the formulae respectively.
The following result contains some m.s. operational rules that will be used in Sections 4 and 5 to solve random heat problems in a semi-infinite medium.
Theorem 3. Let be a 2-s.p. twice m.s. differentiable with m.s. locally integrable, and with , , and m.s. absolutely integrable in . Then,
Proof. We present the proof of each formula separately.(i)By the rule for the m.s. derivative of a product of a 2-s.p. by a deterministic function, [9, page 96], it follows that or From definitions (19) and (28), one gets From the fundamental theorem of the m.s. calculus [9, page 104], we have From (30), as is m.s. absolutely integrable, it follows that Hence, the limit is finite. Furthermore, as is m.s. absolutely integrable, by the Cauchy condition of the integral one gets that By the fundamental theorem of the m.s. calculus [9, page 104], we also have From (33) and (34) one gets that Finally, from (29) and (35) and definition (20), one gets the proof of (i).(ii)By the rule for the m.s. derivative of a product of a 2-s.p. by a deterministic function [9, page 96] we have or Now, by definition (20) applied to and (37), it follows that By the fundamental theorem of the m.s. calculus [9, page 104], As in the proof of part (i), , and thus from (39) one gets From (38) and (40), one gets (24).(iii)By applying part (i) to , it follows directly (25).(iv)It is a direct consequence of the application of part (ii) to .
4. Random Heat Problem with Third Kind Boundary Condition
In this section, we deal with the random heat problem for the temperature distribution in a semi-infinite bar with random temperature at the end and zero initial temperature where and both are independent positive 4-r.v.’s, satisfying properties to be specified later. Assume that problem (41)–(43) admits a solution 2-s.p. m.s. locally and m.s. absolutely integrable, and let us denote what means that is regarded as a process of the active variable , for fixed . By applying the random Fourier sine transform to both members of (41) and properties of Theorem 3, it follows that and from Lemma 2,
From condition (43), it follows that
Hence, the transformed problem becomes the following random initial value problem for the variable :
Let us assume that being the moment generating function of r.v. . Then, by Theorem 8 of [6], the solution s.p. of problem (48) is given by
By using the random inverse Fourier sine transform (21), one gets where
Putting the substitution , we have and dealing with , we have
We start paying attention to the derivative with respect to the variable of the s.p. . Note that, under condition (11), the property (13) holds: for some positive constants and . Hence, it is easy to check that is m.s. uniformly convergent in a neighbourhood ,, of each ,.
Note also that, from (55), the integral of the derivative with respect to of the integrand of (see (52)) is m.s. absolutely uniformly convergent in a neighbourhood ,, of each ,. Hence, for each ,, (56) defines a 2-r.v. By Lemma 2, it follows that
By [12, page 61], fixed ,, each realization of (56), denoted by , can be computed as follows:
Thus,
By (52) , the fundamental theorem of the m.s. calculus [9, page 104] yields
From (51), (53), and (60), one gets
Using the independence of r.v.’s and , one gets that the expectation and the variance function of the solution s.p. are, respectively, where
Summarizing, the following result has been established.
Theorem 4. Let us consider the random heat problem given by (41)–(43) where and are independent positive 4-r.v.’s. Assume that also satisfies conditions (11) and (49). Then, the solution stochastic process of this problem is given by (61). In addition, (62)–(64) are closed expressions for its expectation and variance.
5. Random Heat Problem with Second Kind Boundary Condition
This section deals with the random heat problem for the temperature distribution in a semi-infinite bar with zero initial temperature and where the heat flow at the end is given by s.p. : where and both are independent 4-r.v.’s and is positive and satisfies properties (11) and (49). We also assume that is m.f. continuous. Assuming that problem (65)–(67) admits a solution 2-s.p. locally and absolutely m.s. integrable, let us denote which means that is regarded as a process of the active variable , for fixed . By applying the random Fourier cosine transform to both members of (65) and properties of Theorem 3, one gets
In addition, from (68) and (67), one gets
By Lemma 2, we also have Thus, the transformed random ordinary initial value problem in the variable becomes
By using Theorem 8 of [6], the solution of (72) is given by Taking in (73) the random inverse Fourier cosine transform (22) and using Fubini theorem in normed spaces [17, page 175], it follows that
With the notation of (56), we have
For the sake of convenience, let us write Note that under condition (11), the random integral appearing in (76) is m.s. absolutely uniformly convergent in a neighbourhood , , of each , . Furthermore, from [12, page 61] each realization
Hence, from (76)-(77), one gets and from (75) it follows that
Using the independence of r.v.’s and , one computes the expectation of the solution s.p. :
Taking into account (63), the variance of is computed from (80) and
Summarizing, the following result has been established.
Theorem 5. Let one consider the random heat problem given by (65)–(67) where is a positive 4-r.v. satisfying conditions (11) and (49). Let one assume that is a mean fourth continuous process depending on r.v. such that, for each , the 4-r.v. is independent of . Then, the solution stochastic process of this problem is given by (79). In addition, (80) together with (63) and (81) are closed expressions for its expectation and variance.
6. Numerical Examples
Example 1. Let us consider problem (41)–(43) where the positive 4-r.v.’s and are assumed to follow a beta distribution of parameters and : and a gamma distribution of parameters and : , respectively. We will assume that both r.v.’s are independent. Note that satisfies condition (55) since it is bounded (see Remark 1 and (11)–(13)). Furthermore, it is easy to check that the moment generating function of r.v. , , satisfies hence, it is locally bounded about . Therefore, by Theorem 4 expression (61) is a solution s.p. of problem (41)–(43). In Figure 1, we have plotted the expectation, , and the standard deviation, , of the solution s.p. on the spatial domain for some selected values in the time interval . One observes the average of temperature pulls out of zero as time increases and, as a consequence, its variability, measured through standard deviation, behaves analogously.
(a)
(b)
Example 2. Let us consider problem (65)–(67) where is assumed to follow a beta distribution of parameters and : . Notice that is a positive 4-r.v. and satisfies condition (11) because it is bounded. In addition, it is straightforward to check that its moment generating function is given by and satisfies ; hence, it is locally bounded about . Let us consider the boundary condition , where is a Gaussian 4-r.v. of mean and standard deviation , that is, independent of r.v. . Since (see [9, page 26]), , is m.f. continuous. Hence, the hypotheses of Theorem 5 are satisfied and expression given by (79) is a solution s.p. of problem (65)–(67). In Figure 2, we show, by means of a surface, approximations on the spatial domain to the expectations and standard deviation according to (80), (63), and (81) at different instants in the time interval .
(a)
(b)
7. Conclusions
In this paper, we show that the well-known Fourier sine and cosine transforms’ technique used in the deterministic case can also be used to solve random heat problems with the same quality answer. This fact requires the proof of the m.s. operational rules for the random Fourier sine and cosine transforms, as well as results about random ordinary differential equations obtained by the authors.
Thus, the paper opens a fruitful research activity in the management of random partial differential problems not only with these random Fourier transforms but also with other random transforms.
Acknowledgments
This work has been partially supported by the Ministerio de Economía y Competitividad Grant no. DPI2010-20891-C02-01 and Universitat Politècnica de València Grant no. PAID06-11-2070.