Mathematical and Computer Modelling 50 (2009) 109–115

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

On the heat equation involving the Dirac δ distribution as a coefficientDarko MitrovićUniversity of Montenegro, Faculty of Mathematics, Cetinjski put bb, 81000 Podgorica, Montenegro

a r t i c l e i n f o

Article history:Received 5 March 2008Received in revised form 22 January 2009Accepted 6 February 2009

Keywords:Heat equationSingular coefficientsMultiplication of distributions

a b s t r a c t

We consider the equation

(aH(x+ st)+ bH(−x− st)+ δ(x+ st))ut(x, t) = uxx(x, t),

where (x, t) ∈ R × R+, a, b, s ∈ R are fixed constants, H is the Heaviside function, and δis the Dirac distribution. We augment the equation with appropriate initial and boundarydata. We give a physical model justifying such an equation, and introduce a new solutionconcept with the help of a distribution space defined on discontinuous test functions. Weprove the existence and uniqueness of a solution in the framework of the proposed solutionconcept.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

We are interested in the heat equation with the δ distribution as a coefficient. Formally posed, our problem looks like

(aH(x+ st)+ bH(−x− st)+ δ(x+ st)) ut(x, t) = uxx(x, t), x ∈ R, t ∈ R+, (1)u|t=0 = H(x) (2)u|x=−∞ = 0, u|x=+∞ = 1, (3)

where δ,H ∈ D ′(R) are, respectively, Dirac and Heaviside distributions and s is a constant, different from zero. The valuesof the temperature u at infinity are chosen to make the presentation simpler and they do not play any essential role in thepaper.But, the assumption s 6= 0 seems to be necessary in the current construction (see (16)). It is possible to formalize the

situation when s = 0 as well, but due to obvious physical analogy between the nonstationary case s 6= 0 and the stationaryone s = 0, we will just comment on the case s = 0 in the last section.The physical background of this problem is the following. The equation describes the temperature behavior in one-

dimensional space which is initially divided on two homogeneous parts (let us call them A and B). In the first part thetemperature is one degree and in the other part it is zero degrees. The space is divided by some infinitely thin super-isolatorwhose specific heat capacity (c) multiplied by the density (ρ) is extremely high (see [1], chapter 2, paragraph 2, for anexplanation). On the other hand, we assume that the free path length (λ) of that super-isolator is such that cρλ is finite.As time passes, this isolator moves along the x-axis with the velocity−s. It is obvious how the temperature will change;

on the left-hand side of the isolator the temperature will be one degree and on the right-hand side it will be zero degrees.Still, it appears that it is not so easy to find an appropriate functional space in which one will have a physically reasonablesolution of the problem.More precisely, it is obvious that the solution of (1), (2), (3) should be the function H(x + st), where H is the Heaviside

function, since the temperatureudepends only on the position of the isolator. Ifwe look back to the equation,we can see that,

E-mail addresses:matematika@t-com.me, matematika@cg.yu.

0895-7177/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2009.02.005

110 D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115

according to the predicted form of the solution, we have to find the functional space in which the ‘‘product’’ of two δ distri-butions gives an sδ′ distribution for the constant s. Theword ‘‘product’’ is in quotationmarks since it does notmean standardmultiplication. We will define an appropriate operation which we will call a product. As is well known, a standard productcannot be properly defined in the Schwartz distribution space; this is the so-called Schwartz impossibility result [2,3].Also, note that we have a rather similar situation in the case of the equation for the photon distribution function f [1]

written in the form appropriate for us:

∂t f + c∂x f = χ f , f (x, t) = f (x, t)− fp, (x, t) ∈ R× R+

where χ is proportional to the coefficient of absorption, c is the velocity of the photon and fp is the function of balanceddistribution (which is constant). If we assume that we have a situation such that at one point of the space the coefficient ofabsorption is equal to infinity it is clear that all the photons will be placed exactly at that point. So, the number of photonsf (up to probability one) will be infinity at that one point and there will be no particles anywhere else. Since the mass ofthe photons is finite the distributions function f will be of the delta function type. Also, the coefficient of absorption willbe of the delta function type and we have to find the functional space in which the product of delta distributions gives thederivative of delta multiplied by some constant.In order to solve the problem we introduce a distribution space on discontinuous test functions (Section 2). We call

it the ‘new distribution space’. We can explain the introduction of such a space of the distribution using the argumentsleading to the principle of indeterminacy in quantumphysics. Namely,we know thatwe can detectmicroparticles onlywhenthey interact with some substance or some field. If they do not interact we cannot speak even about their existence and,accordingly,we cannot notice any of their properties. Since all properties of themicroparticles are obtained as a consequenceof their interactions and not from themselves as independent natural objects, we cannot be sure how they will behave insome situation (even if we have completely the same conditions, two different particles will behave as we predict only upto some probability).We have a similar situation in distribution theory. Distributions can be ‘‘detected’’ only in the ‘‘interactions’’ with some

space of test functions. Therefore, their behavior will depend on the test function space that ‘‘interacts’’ with them. Wecan agree that the linear properties of the standard distribution space will remain unchanged independently of the testfunction space (Section 2) while the nonlinear phenomena will change depending on the test function space defining thedistributions.When we say nonlinear properties, we mean the operation of multiplication between distributions defined in

an appropriate manner. We will define multiplication between two distributions analogically with the Colombeauconstruction [4,3]. Roughly speaking, that means that we replace a new distribution f by a family of smooth functions (fε)εwhich weakly converges toward the new distribution (i.e. fε ⇀ f , ε → 0, where the weak convergence is understood inthe ‘new distributions’ sense; Definition 5). Then, we can define the product between the new distributions f and g as thefamily (fεgε)ε , where fε ⇀ f and gε ⇀ g , ε→ 0. Although the spaces of standard and new distributions are homomorphic(cf. Theorem 4), the approximating families constructed over the ‘new distribution space’ are substantially richer thansmooth families approximating the distribution in a standard weak sense. Roughly speaking, unlike the situation that wehave in the Colombeau algebra, the concept presented here allows us to embed δ2 into D ′ (see Example 6(b)). Therefore,the approach from Nedeljkov et al. [5], where the heat equation involving singularities is considered in the framework ofthe Colombeau algebra, does not give satisfactory results in the case of our problem.The idea of defining the distribution space not as Schwartz [2] does but using some other trial function space is indeed

not new. A successful attempt on this subject was made by Bredimas [6]. But, as far as we know, he did not offer anyphysical model which would approve the use of his more general µ′x0 space [6] instead of a standard distribution spaceD ′(R). Probably therefore his idea has remained, in some sense, forgotten until now.We conclude the introduction by summarizing the main results of the paper.First, we give an answer to the question whether it is possible to define ‘‘reasonably’’ the product of two Dirac

distributions with the same support. Second, we introduce a model suggesting what the product should be, and then wemake a structure allowing us to define the product between Dirac’s distributions in an appropriate manner.

2. Introduction to the new distribution space

Definition 1. Let φ ∈ L∞(R) and S = {x1, . . . , xn} be a nonempty subset of R such that xi < xj for i < j.By F −S (φ) and F +S (φ)we denote, respectively,

F −S (φ) =

−φ(x), x ∈ (−∞, x1)(−1)kφ(x), x ∈ [xk−1, xk), k = 2, . . . , n− 1(−1)n+1φ(x), x ∈ [xn,+∞),

F +S (φ) =

φ(x), x ∈ (−∞, x1)(−1)k−1φ(x), x ∈ [xk−1, xk), k = 2, . . . , n− 1(−1)nφ(x), x ∈ [xn,+∞).

D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115 111

In what follows, we shall construct the new distribution space using the mapping F +S . A completely analogicalconstruction can be accomplished using the mapping F −S . We shall provide remarks on this issue when we find itappropriate.We denote:

DS(R) ={φ|φ ∈ L∞(R)such that F +S (φ) ∈ D(R)

}where D(R) is the Schwartz space of infinitely differentiable functions with compact support. The topology on the spaceDS(R) is projective topology with respect to the mapping

J : D(R)→ DS(R),

defined by J(φ) = F +S (φ). It is obvious that when the latter mapping is continuous then the mapping J(φ) = F −S (φ) isalso continuous. Therefore, if we define the mapping J by J(φ) = F −S (φ)we will get the same topology onDS(R).

Definition 2. For every S ⊂ Fin(R) we denote by D ′S+(R) the set of all linear functionals over DS(R) such that for everyf ∈ D ′S+(R) there exists f ∈ D ′(R) (standard distribution space) for which we have

〈f , φ〉 = 〈f ,F +S (φ)〉,

for every φ ∈ DS(R).We write f = c+S (f ).We call the spaceD ′S+(R) the space of new distributions.

The following lemma immediately follows from the definition of the mapping F +:

Lemma 3. For any two functions ψ, η ∈ L∞(R) we have

F +S (ψη) = ηF+

S (ψ) = ψF +S (η).

The last lemma shows us that we can define the product of new distributions with a smooth function in a usual manner. So,we can formulate the following theorem:

Theorem 4. For any S ∈ Fin(R), the modulus over the ring of smooth functions(D ′S+(R),+, · ; C

∞(R))is linearly embedded

into the Schwartz space of distributions.

Proof. We define for f ∈ D ′S+(R) the mapping

i(f ) = f

where f ∈ D ′(R) is such that f = s+S (f ).Let f , g ∈ D ′S+(R) and ψ ∈ C

∞(R). It is clear that we have i(f + g) = i(f ) + i(g) and from the last lemmai(ψ f ) = ψ i(f ). �

We shall need the notion of weak convergence in the new space of distributions (we will also call it the new weakconvergence).Wewill define the notion only on families of C∞(R) functions sincewewillworkwith smooth approximationsof new distributions. This notion will actually provide the mentioned rich structure of the new distribution space.

Definition 5. We say that the family (fε)ε ⊂ C∞(R) tends to h ∈ D ′S+(R) for S = {x1, . . . , xn} ∈ Fin(R), in the new weaksense along the set Sε = {xε1, . . . , x

εn} where x

εi → xi as ε → 0, if, for every family (φε)ε ∈ DSε (R) satisfying F +Sε (φε) = φ

for every ε > 0 and some φ ∈ D(R), we have

limε→0〈fε, φε〉D ′

S+ε= 〈h, φ〉D ′

S+,

where limε→0 φε = φ ∈ DS(R), and 〈h, φ〉D ′S+is action of h ∈ D ′S+(R) on φ ∈ DS(R).

By OD ′S+ε(εβ), α ∈ R, we denote the family (fε)ε ∈ C∞(R) such that, for every family (φε)ε ∈ DSε (R) satisfying

F +Sε (φε) = φ for every ε > 0 and some φ ∈ D(R), we have

〈fε, φε〉 = O(εβ), ε→ 0,

where limε→0 φε = φ ∈ DS(R), and O is standard Landau symbol.

112 D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115

Example 6. (a) Let the function w(z) be such that, for some real constants βl < α < βr , we have∫F +{α}(ω(z))dz = 1,

∫ω(z)dz = 0,

ω(z) ≥ 0 for z ≤ α, ω(z) ≤ 0 for z ≥ α,supp ω(z) ⊂ (βl, α) ∪ (α, βr).

Take the following family:

(ρε(x))ε =(1εω( xε

))ε

, ε > 0.

We will show that the family (ρε)ε tends to δ(x) ∈ D ′{0}+(R) in the new weak sense along the set {αε}, where δ = c

+

{0}(δ);δ is the Dirac distribution.Indeed, for arbitrary φ ∈ D(R) put

φε(x) = F +{αε}(φ(x)) =

{φ(x), x < αε−φ(x), x ≥ αε ∈ D{αε}(R), (4)

and consider∫1εω( xε

)φε(x)dx =

∫ αε

−∞

1εω( xε

)φ(x)dx−

∫+∞

αε

1εω( xε

)φ(x)dx

=

( xε= z H⇒ dx = εdz

)=

∫ α

βl

ω(z)φ(εz)dz −∫ βr

α

ω(z)φ(εz)dz

=

∫ α

βl

ω(z)(φ(0)+ εzφ(z)

)dz −

∫ βr

α

ω(z)(φ(0)+ εzφ(z)

)dz

=

∫F +{α}(ω(z))φ(0)dz + O(ε) = φ(0)+ O(ε)

→ φ(0) = 〈δ,F +{0}(φ)〉 := 〈c

+

{0}(δ), φ〉, as ε→ 0,

where φ = F{0}(φ).(b) It is well known that for an arbitrary standard weak approximation (ρε)ε of the Dirac distribution the family (ρ2ε )ε

does not have a weak limit. This will not be the case in the new distribution space. Indeed, take the family (ρε)ε from part(a) with an additional assumption∫ α

−∞

ω2(z)dz =∫∞

α

ω2(z)dz. (5)

Taking (4) into account, and using the same procedure as in (a), we get⟨1ε2ω2( xε

), φε(x)

⟩=

∫1ε2ω2( xε

)φε(x)dx

=

∫ α

−∞

1εω2(z)

(φ(0)+ εzφ′(0)+ O(z2)ε2

)dz −

∫+∞

α

1εω2(z)(φ(0)+ εzφ′(0)+ O(z2)ε2)dz

(5)= −φ′(0)

(∫+∞

α

zω2(z)dz −∫ α

−∞

zω2(z)dz)+ O(ε).

So, in the new weak sense along {αε}, we have

ρ2ε ⇀ K δ′ = Kc+{0}(δ

′), ε→ 0,

where δ′ is the distributional derivative of the Dirac δ distribution and

K =∫+∞

α

zω2(z)dz −∫ α

−∞

zω2(z)dz. (6)

D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115 113

Notice that we can make K to be an arbitrary number from the interval (0,∞) by choosing different ω and α > 0. Indeed,without loss of generality we can assume that ω is of box type [7]:

ω(x) =

1

2(α − βl), βl − γ < x < α − γ

−12(βr − α)

, α + γ < x < βr + γ

0, otherwise.

From this and the definition of K , we have

K =∫ βr

α

z4(βr − α)2

dz −∫ α

βl

z4(α − βl)2

dz. (7)

It is enough to prove that it is possible to make K arbitrarily large and arbitrarily small. So, take α = −1 and γ = 0.We have

−

∫−1

βl

z4(−1− βl)2

dz =β2l − 1

2(−1− βl)2→18, as βl →∞. (8)

Similarly, by taking βr = 0, we have∫ 0

−1

z4dz = −

18. (9)

Comparing (7) with (8) and (9), we see that K can be made arbitrarily close to zero.Similarly, by taking α = 0, βl = −1, βr = 1 and letting γ →∞, we conclude that K can be arbitrarily large.(c) Notice that

(∫ x−∞

1εω( xε

)dx)εtends to zero in the new weak sense along {αε}. Indeed, the family is bounded and it

tends pointwisely to zero. Then, using the Lebesgue dominated convergence theorem we conclude that the family tendsto zero strongly in L1loc(R). From the strong convergence, the (new) weak convergence to zero follows immediately. So, inthe new distribution space we have an element converging to the c+

{0}(δ) distribution (homomorphic image of the Dirac δdistribution), in the new weak sense along {αε}, while at the same time the integral over (−∞, x) of the family tends tozero. We recall that, in the classical setting, the integral over (−∞, x) of a family approximating the δ distribution tends tothe Heaviside function. This example demonstrates how rich a structure the new distribution space has. It is also importantto notice that the family

(∫ x−∞

F +{αε}

( 1εω( xε

))dx)εweakly converges to the Heaviside function. This last remark shows that

we can still use the rich structure of the new distribution space without loss of classical facts.

3. A solution concept for the heat equation with a singularity

3.1. The nonstationary case

In this section, we consider problem (1), (2), (3) when s 6= 0. It is posed using the elements from the (standard) D ′(R)distribution space where the multiplication is not defined. One of the ways to overcome this problem is to regularize thedistribution appearing in the equation and then to find an approximate solution of the problem. If we decided to use suchapproach we would have to multiply two regularizations (weak approximations) of the δ distribution. We would have tochoose them to give the weak approximation of the δ′ distribution (see Section 1). Even if it were possible, the constructionwould be rough and unnatural since wewould have to choose different regularizations for the same distribution. Therefore,the concept of solving problem (1), (2), (3) is the following.We choose s > 0 (if s < 0 we would useD ′

{−st}−(R) instead ofD′

{−st}+(R) below). Then, we modify the approach frome.g. [8] (weak asymptotic solution concept) where the variable t is regarded as a parameter. First, we will pose the problemusing elements from D ′

{−st}+(R+) (we recall that we regard t ∈ R+ as a parameter). But, in the realm of L∞(R) functions

we do not have any substantial enrichment of the structure after passing to the new distribution space. Therefore, we willrewrite (‘singularize’) problem (1), (2), (3) not only by passing toD ′

{−st}+(R), but also by taking the derivative of the initialand boundary data. In such a setting, the problem looks like the following:(

aH(x+ st)+ bH(−x− st)+ δ(x+ st))ut = uxx, (10)

∂xu|t=0 = δ(x), (11)

∂xu|x=−∞ = 0, ∂xu|x=+∞ = 0, (12)

where δ = c+{0}(δ) for the Dirac distribution δ, and H = c

+

{0}(H) for the Heaviside function H . Clearly, H(x) := c+

{0}(H)(x) =H(−x), x ∈ R.

114 D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115

Then, we will regularize the coefficients and the initial condition of problem (10), (11), (12) in the sense of the newweakconvergence. So, for an unknown family (uε) ∈ C∞0 (R

+× R) our problem becomes the following:(

aHε(x+ st)+ bHε(−x− st)+1εφ

(x+ stε

))uεt(x, t) = uεxx(x, t), (13)

∂xuε|t=0 =1εφ( xε

)(14)

∂xuε|x=−∞ = 0, ∂xuε|x=+∞ = 0. (15)

Eq. (13) is understood up to OD ′{−st+αε}+

(εβ) for a β > 0 and α defined in (16). We call the family (uε) solving the problem

in this sense (i.e. up to OD ′{−st+αε}+

(εβ)) the approximate solution.

The function φ = φ(z), z ∈ R, appearing in (13) and (14), satisfies the same conditions as the function ω fromExample 6(a). In other words, the new weak limit along the set {st + αε} of the family

(1εφ( x+st

ε

))is δ(x + st). We also

assume that φ satisfies (see (6))

1s=

∫+∞

α

zφ2(z)dz −∫ α

−∞

zφ2(z)dz, (16)

which is always possible for s > 0 according to Example 6(b).The function Hε is such that

Hε ⇀ H := c+{0}(H)

in the new weak sense along the set {st + αε}.Furthermore, we assume that, for every ε > 0 and every t ∈ R+,

Hε(x)φ(x+ stε

)= Hε(−x)φ

(x+ stε

)= 0, x ∈ R, (17)

i.e. the supports of the function Hε and φ(·+stε

)are disjoint for every ε > 0 and every t ∈ R+. Such an assumption is

rather natural since in that way we separate the investigation of behavior of the temperature in the neighborhood of thesuper-isolator and outside that neighborhood (see also Definition 7).Now, we return to Example 6(c). There, we have noticed that the new distribution space admits families which do not

behave as expected from the viewpoint of the classical settings. We have also shown how to avoid this anomaly (see the lastsentences of the example). In that spirit and in the spirit of assumption (17) we introduce the following definition:

Definition 7. We say that u ∈ D ′(R+ × R) is a new weak solution to (1), (2), (3) if(a) it satisfies the initial and boundary conditions (2) and (3), respectively;(b) for every (t0, x0) 6∈ {(t, x)| x = st} the function u locally represents a classical solution to Eq. (1);(c) there exists a family (uε) approximately solving (13), (14), (15) such that, for almost every t ∈ R+ and an α ∈ R,∫ x

−∞

F +{−st+αε}(∂xuε(x

′, t))dx′ ⇀ u(x, t) as ε→ 0

in the standard distribution spaceD ′(R) over R.

Notice that in the previous definition we first ‘singularize’ the possible solution by finding its derivative, and then wego back to the standard situation (again see the last sentences in Example 6(c)). This actually means that in the first stepwe enlarge the space of possible solutions (‘singularization’ of the problem), and in the second step we recover informationwhich is in accordance with the standard results.The following two assertions follow from Example 6 and Definition 7:

Theorem 8. The family

(uε(x, t))ε =(∫ x+st

−∞

1εφ( zε

)dz)ε

∈ C∞(R+ × R), (18)

represents an approximate solution of problem (13), (14), (15) for the constant α ∈ R determined in (16).

Proof. Notice that

∂x

∫ x+st

−∞

1εφ( zε

)dz =

1εφ

(x+ stε

)⇀ δ(x+ st) ∈ D ′

{−st}+(R), (19)

D. Mitrović / Mathematical and Computer Modelling 50 (2009) 109–115 115

in the newweak sense along the set {−st+αε}. From this, after substituting (uε(x, t))ε from (18) in (13) and having inmind(17), it follows that on the left-hand side of (13) we have product of two new weak approximations of the δ, while on theright-hand side we have a new weak approximation of δ′ (the notations are taken from Example 6):

sε2φ2(x+ stε

)=1ε2φ′(x+ stε

)where the equality is understood up to OD ′

{−stαε}+(εβ) for β > 0.

Therefore, repeating the procedure from Example 6(b) and using (16), we prove the assertion. �

Corollary 9. The distribution H(x+ st) represents a unique new weak solution to problem (1), (2), (3).

Proof. It is easy to check that (see also Example 6(c))∫ x+st

−∞

1εF +{αε}φ

( zε

)dz ⇀ H(x+ st), ε→ 0.

According to Definition 7, from (19) and Theorem 8 it follows that H(x + st) represents the new weak solution to (1), (2),(3).Uniqueness follows from the fact that, for (t, x) 6∈ {(t, x)|x = st}, Eq. (1) reduces to uxx = 0. Then, from the boundary

conditions (2), it easily follows that

u(x, t) ={0, x < st1, x > st,

implying the uniqueness of H(x+ st) as the solution to the considered problem. �

3.2. The stationary case

In this section we shall comment on problem (1), (2), (3) in the case when s = 0. In this case one of the possible conceptsis the introduction of two parameter families. Accordingly, we introduce the following definition:

Definition 10. We say that the distribution u ∈ D ′(R+ × R) is the new weak solution to (1), (2), (3) with s = 0 if thereexists a positive sequence (sn)n tending to zero such that the sequence (un)n of newweak solutions to (1), (2), (3) with s = sntends to u in the standard distributional sense.

Using the latter definition, it is clear that we can repeat the considerations from the previous section to obtain theexistence and uniqueness to (1), (2), (3) with s = 0.

Acknowledgement

The work of the author is supported in part by the local government of the municipality of Budva.

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