On solutions of a discretized Heat equation

in discrete Clifford analysis

F. Baaske∗, S. Bernstein†, H. De Ridder‡, F. Sommen§

Abstract

The main purpose of this paper is to study solutions of the Heat equation in thesetting of discrete Clifford Analysis. More precisely we consider this equation withdiscrete space and continuous time. Thereby we focus on representations of solutions bymeans of dual Taylor series expansions. Furthermore we develop a discrete convolutiontheory, apply it to the inhomogeneous Heat equation and construct solutions for therelated Cauchy problem by means of Heat polynomials.

Keywords: heat equation, heat kernel, discrete Clifford analysis

MSC2010 Classification: 15A66,35K05,35K08,39A12

1 Introduction

Clifford analysis is a natural higher-dimensional generalization of complex functiontheory and based on the study of properties of null solutions of the so-called Diracoperator which generalizes the ∂ operator. The null solutions of the Dirac operatorare called monogenic functions. Because the Dirac operator factorizes the LaplacianClifford analysis is also a refinement of classical harmonic analysis.A very new branch of Clifford analysis is discrete Clifford analysis. The interest in adiscrete theory is driven by computational physics and numerical computations.Because Clifford analysis is a generalization of complex function theory we look firstat complex discrete functions.Discrete monodiffric functions are defined on square lattices where introduced by J.Ferrand [10] and also [9]. They are based on the discretization of the Cauchy-Riemannequations

fm,n+1 − fm+1,n = i(fm+1.n+1 − fm,n).

A discrete function theory should mirror the properties of the continuous one, i.e.discrete holomorphic functions should be discrete harmonic and one needs a discrete

∗Friedrich-Schiller-University Jena, Department of Mathematics, Germany, franka.baaske@uni-jena.de†Institut fur Angewandte Analysis, TU Bergakademie Freiberg, swanhild.bernstein@math.tu-freiberg.de‡Ghent University, Department of Mathematical Analysis, hdr@cage.ugent.be§Ghent University, Department of Mathematical Analysis, fs@cage.ugent.be

1

version of the Cauchy integral. A major problem consists in the fact that discreteholomorphic functions do not form an algebra.There are already a lot of contributions regarding the construction of discrete Diracoperators [11, 12, 14, 23, 24, 15, 25] but these constructions are based on potential-theoretic arguments [18, 13, 11, 20, 21]. Classical function theory is also built up bypolynomials. All the theories mentioned above don’t give basic polynomials, a Fischerdecomposition or a Taylor series.An alternative approach is based on one type of difference operator [22], more preciselyeither a forward or backward difference operator, and leads to basic polynomials anda Fischer decompositions. But here it is not possible to factorize the star Laplacian,because such a factorization needs both types of difference.In [2] the problem was considered from a more abstract point of view. Differentiatingmeans lowering the power of a polynomial and multiplication with a variable raisesthe power. This gives a way to apply umbral calculus [19]. The connection betweenlowering operator and raising operator is given by the so-called “Weyl relations”

∂jxj − xj∂j = 1 or, applied to a function f, ∂j(xjf(x))− xj∂jf(x) = f(x).

Then the corresponding theory is based on a duality argument, the so-called Fischerduality.Even though this looks like a possible approach there is a major obstacle. While forwardand backward differences are mutually commuting with each other, this is no longertrue for the corresponding vector variables. The idea is to split the basis elements ekinto two new basis elements e+

k and e−k and construct an appropriate discrete Diracoperator involving both forward and backward difference operators. Because the raisingoperators do not commute with each other, the standard Weyl relations had beenreplaced by a type of “skew” Weyl relations.The success of this theory is proven by a couple of papers where it was possible toconstruct an Euler operator, discrete homogeneous polynomials, a discrete Cauchy-Kovalevskaya extension theorem [3], Fueter polynomials [6], discrete Taylor series [4],discrete Fourier transform and distributions [5] in a discrete Clifford setting.In this paper we study the Cauchy problem for the Heat equation

∂

∂tu(x, t)−∆xu(x, t) = 0, t > 0, x ∈ Rn

u(x, 0) = u0(x), x ∈ Rn

as a first step towards applications of the discrete function theory. A standard methodto solve the continuous problem is to use a fundamental solution [8]. The function

Φ(x, t) =1

(4πt)n/2e−|x|

2/4t

with x ∈ Rn and t > 0 is the fundamental solution of the Heat equation, which isalso denoted as Heat kernel or Gaussian distribution. For all fixed t > 0 it has theproperties of a probability distribution. For an initial function u0 that belongs to theSchwartz space S = S(Rn) of rapidly decreasing infinitely differentiable functions onRn, the solution u(x, t) is found by convolution of u0 with the fundamental solution.

2

Drawing inspiration from the continuous setting, we study the discrete Gaussian andthe discrete Gaussian distribution as fundamental solutions of the discrete heat equa-tion. A function of the form e−a|x|

2

, a > 0, is usually called a “Gaussian.” It is easilyseen that such a function is bell-shaped and has a maximum at 0. Another impor-

tant property is the function f(x) = e−|x|22 is equal to their Fourier transform, i.e.

f(ξ) = e−|ξ|2

2 . Therefore such functions have a rich history within mathematics andappear in a wide variety of settings.It is obvious that a discrete analog of the Gaussian should have similar properties. Asimple answer is to sample the continuous Gaussian. Unfortunately, this is not a gooddiscrete analog. A better choice is the discrete Gaussian kernel:

Gt = e−2t BesselI(|x|, 2t),

where BesselI denotes the modified Bessel functions of the first kind of integer order.With the aid of the discrete Gaussian kernel we construct a discrete fundamentalsolution allowing us to solve the Cauchy problem.The paper is organized as follows: we start with an introduction. To keep the paperself-explanatory, Section 2 contains the preliminaries about the discrete framework.In Section 3 we introduce the discrete Gaussian distribution and compare its densityfunction with the continuous counterpart. It turns out that the density function showsthe well-known bell-shape with a maximum at the origin. In Section 4 we calculatethe fundamental solution for the discrete Heat equation, based on a dual Taylor seriesexpansion, showing an intrinsic connection with the discrete Gaussian. In Section 5we give the basic properties of discrete convolution theory. In Section 6 we apply thisfor the solutions of the discrete Heat equation and we consider some special solutionswhich could be considered as basic building blocks. In Section 7, we construct thesolutions of the Heat equation by a series expansion. The discrete Heat polynomialsare the basic building blocks for this expansion.In this paper, we considered a discrete version of the Heat equation at which space isdiscrete and time continuous. The fundamental solution is given by the discrete Gaussdistribution, allowing us to construct solutions of the inhomogeneous Heat equation.We furthermore constructed solutions of the Cauchy problem for the Heat equation bymeans of a series expansion in the Heat polynomials.

2 Preliminaries

Let Rn be the n-dimensional Euclidian space with orthonormal basis ej , j = 1, . . . , nand consider the Clifford algebra R0,n over Rn. Passing to the so called discreteHermitian setting, we imbed the Clifford algebra R0,n into the bigger complex oneC2n, i.e. the underlying vector space is of twice the dimension, and introduce forwardand backward basis elements e±j satisfying the following anti-commutator rules, cf. [1]:

e−j e−` + e−` e−j = e+j e+

` + e+` e+

j = 0 and e+j e−` + e−j e+

` = δj,`.

The connection to the original basis ej is given by e+j + e−j = ej . Observe that this

implies e2j = 1 in contrast to the usual Clifford setting where traditionally e2

j = −1 ischosen.

3

This approach is motivated due to the aim of constructing a discrete Dirac operatorfactorizing the discrete Laplacian. Consider therefore the standard equidistant latticeZn, i.e. the coordinates of a Clifford vector x take only integer values. The partialderivatives ∂xj used in Euclidian Clifford analysis are replaced by forward and backward

differences ∆±j , j = 1, . . . , n, acting on Clifford valued functions f :

∆+j [f ] = f(·+ ej)− f(·) and ∆−j f = f(·)− f(· − ej).

These operators are seen as lowering operators. With respect to the Zn -neighbourhoodof x the usual definition of the discrete Laplacian reads

∆∗[f ](x) =

n∑j=1

∆+j ∆−j [f ] =

n∑j=1

(f(x+ ej)− f(x− ej))− 2nf(x).

This operator is also known as “Star Laplacian”. An appropriate definition of a discreteDirac operator ∂ factorizing ∆∗ is obtained by means of the forward and backward basiselements, more precisely by

∂ =

n∑j=1

(e+j ∆+

j + e−j ∆−j ).

In order to receive an analogue of the classical Weyl relations ∂xjxk − xk∂xj = δjk,

we introduce raising operators X±j being characterized by their interaction with the

corresponding lowering operators ∆±j according to the skew Weyl relations, cf. [2]

∆+j X

+j −X

−j ∆−j = ∆−j X

−j −X

+j ∆+

j = 1.

In the discrete vector variable ξ =n∑j=1

(e+j X−j + e−j X

+j ), the operator components

X±j occur intertwined with the forward and backward basis elements e±j . Observe that

the co-ordinate variables ξj = e+j X−j + e−j X

+j and ∂j = e+

j ∆+j + e−j ∆−j of ξ =

n∑j=1

ξj

and ∂ =n∑j=1

∂j thus satisfy the following anti-commutator relations

∂j ξj − ξj ∂j = 1 and ∂` ξj + ξj ∂` = 0, ` 6= j, j, ` = 1, . . . , n

which imply

∂j ξkj [1] = k ξk−1

j [1], ∂` ξkj [1] = 0 and ∂j ξ

k1j ξk2` [1] = k1 ξ

k1−1j ξk2` [1], j 6= `.

The ξj mutually anti-commute (ξjξk = −ξkξj for j 6= k) as do the differences ∂j .

The natural powers ξkj [1] of the operator ξj acting on the ground state 1 are the basicdiscrete homogeneous polynomials of degree k in the variable xj , replacing the basic

4

powers xkj in the continuous setting and constituting a basis for all discrete polynomials,

cf. [3]. The odd and even powers are explicitly given by ξj [1](xj) = xj(e+j + e−j

)and

ξ2nj [1](xj) =

(x2j + nxj(e

+j e−j − e−j e+

j )) n−1∏i=1

(x2j − i2

)ξ2n+1j [1](xj) = xj

n∏i=1

(x2j − i2)

(e+j + e−j

), n > 1.

Observe thatξkj [1](xj) = 0 if k > 2 |xj |+ 1. (1)

By means of these operators a discrete function with compact support Ω can beexpanded in a Taylor series about the point zero

f(x) =∞∑k=0

∑|α|=k

1

α1! · · ·αn!ξα11 . . . ξαnn [1](x) ∂αnn . . . ∂α1

1 f(0).

Therefor we assume that Ω contains the origin and for all x ∈ Ω the inclusionyj : |yj | 6 |xj | ⊂ Ω holds, since each difference of order αj requires

⌈αj2

⌉extra

points in forward and backward direction where f needs to be defined, cf. [4]. Due to(1) such a series for functions on bounded domains always converges. If f is additionallyleft discrete monogenic in Ω and defined on the set

⋃x∈Ω

y : |yj | 6 |xj | + h, then its

Taylor series can be expressed in terms of discrete Fueter polynomials Vα1,...,αn whichform a basis for the space of discrete sperical monogenics with degree |α| = α1+· · ·+αn

f(x) =

∞∑k=0

∑|α|=k

Vα1,...,αn∂αnn . . . ∂α1

1 f(0)

.

For the converse assertion and further details cf. [4].

We consider now the space of polynomials and restrict ourselves for the moment tothe one-dimensional case. A discrete distribution is defined as a Clifford-valued linearfunctional over this space, cf. [5]. We distinguish between two types of distributions.In analogy to regular distributions we associate with a discrete function f a discretedistribution F acting on a polynomial V in the following way:

〈F, V 〉 :=∑x∈Z

V (x)f(x)

under the condition that the sum converges, f is the so-called density function. InRemark 1 we explain more detailed the chosen distribution space.

Associating the discrete delta distribution δδδj with the discrete delta function δjapplied to a homogeneous polynomial leads to⟨

δδδj , ξ`[1]⟩

=∑x∈Z

ξ`[1](x) δj(x) = ξ`[1](j), j ∈ Z.

5

Thus, distributions which posses a density function can be expressed by means ofdiscrete delta distributions: F =

∑n∈Z

δδδnf(n). Since δj [V ] = V (j) we obtain

∑n∈Z

δδδnf(n) =∑n∈Z

V (n)f(n) = F [V ].

Singular distributions, i.e. distributions which cannot be assigned to a discrete functionare represented using a sequence of its moments, see below.

For both types of distributions the action of the lowering and raising operators ∆±

and X± are given by 〈∆±F, V 〉 = −〈F, V∆∓〉 and 〈X±F, V 〉 = 〈F, V X∓〉 respectively,such that the skew Weyl relations are satisfied. Thus, the actions of the co-ordinatedifference operator ∂ and the co-ordinate vector variable ξ are also determined fordiscrete distributions:

〈∂F, V 〉 = −⟨F, V

(∆−e+ + ∆+e−

)⟩= −〈F, V ∂†〉

〈ξF, V 〉 =⟨F, V

(X−e− +X+e+

)⟩= 〈F, V ξ†〉

where ∂† = e−∆+ + e+∆− and ξ† = e+X+ + e−X−. Applying powers of differenceoperators ∂k to δδδj

⟨∂kδδδj , ξ

`[1]⟩

= (−1)k⟨δδδj , (ξ

`[1])(∂†)k⟩

=

(−1)k `!

(`−k)! ξ`−k[1](j), k 6 `

0, k > `

leads to an equivalent representation of arbitrary discrete distributions by its dualTaylor series

F =

∞∑k=0

(−1)k

k!∂kδδδ0

⟨F, ξk[1]

⟩where

⟨F, ξk[1]

⟩k is the sequence of the moments of F .

In higher dimensions the dual Taylor decomposition of F is given by, cf. [7]:

F =

∞∑αn=0

· · ·∞∑

α1=0

(−1)α1+···+αn

α1! · · ·αn!∂αnn . . . ∂α1

1 δδδ0 〈F, ξα11 . . . ξαnn [1]〉 .

Remark 1. In general distributions can be defined as linear functionals, that is ascontinuous, Clifford-valued linear forms on spaces of functions on Z. In particular thetopological dual of compactly supported functions on Z corresponds to the space of allfunctions on Z. More general one can consider spaces of functions with weight condi-tions of any type, e.g. rapidly decreasing sequences leading to tempered distributions, cf.[5]. However, we define distributions as linear functionals on the space of polynomials.The reason therefor is, that this space contains not only regular distributions. But eachdistribution admits a dual Taylor series expansion in terms of the derivatives of thediscrete delta distribution ∂kδδδ what we will use to represent the Gaussian distribution.

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3 The Gaussian distribution

As we will show later on, there is an intrinsic connection between the fundamentalsolution of the discrete Heat equation and the discrete Gaussian distribution. There-fore, we first discuss this discrete Gaussian distribution in detail in the present section.We begin to introduce the one-dimensional discrete Gaussian distribution and continuewith the generalization to higher dimensions.

3.1 The one-dimensional discrete Gaussian distribution

Similar to the continuous setting where the Gaussian distribution acts on the classicalhomogeneous polynomials x` according to∫

x∈Rx` exp

(−x2

2

)dx =

0, ` odd

(`− 1)!!√

2π, ` even,

we introduce a discrete Gaussian distribution G by defining its action on the discretehomogeneous polynomials ξ`[1]:

G[ξ`[1]

]:=

√

2π, ` = 0

0, ` odd

(`− 1)!!√

2π, ` even.

Since the sequence of moments G[ξ`[1]

]` of the distribution G is known, we can

express G in its dual Taylor series:

G =

∞∑`=0

(−1)`

`!∂`δδδ0 G

[ξ`[1]

].

Using (2k − 1)!! = (2k−1)!2k−1 (k−1)!

, the representation of G by its dual Taylor series then is

given by

G =

∞∑k=0

√2π

2k k!∂2kδδδ0 =

√2π exp

(∂2

2

)δδδ0.

One can recognize the exponential which also appears in classical Gaussian distribu-tions. However, now the exponent is a lowering operator ∂ instead of the usual raisingoperator.

Remark 2. Given the apparent similarity to the continuous setting, one could con-

sider the discrete function exp(− ξ

2

2

)[1] and examine whether it can be used as a den-

sity function. Note that the order of operations in this expression is important, since(ξ2[1]

)kdoes not equal ξ2k[1]. This function is the formal analogue of the continuous

density function of the continuous Gaussian distribution, as may be seen also from thefundamental property

∂ exp

(−ξ

2

2

)[1] = −ξ exp

(−ξ

2

2

)[1].

7

However, it is not scalar-valued because the discrete homogeneous powers ξ`[1] are notscalar-valued, and even more importantly, for this function to be a density function ofa (regular) discrete distribution, it should have compact support or at least vanish atinfinity. However, neither of these requirements is fulfilled, hence it is not suitable asdensity function of a discrete Gaussian distribution.

The distribution G has a discrete density function since one can rewrite it as follows

G =

∞∑k=0

√2π

2k k!

k∑i=−k

(−1)i+k(

2k

i+ k

)δδδi =

√2π

∞∑p∈Z

∞∑k=|p|

(−1)p+k

2k k!

(2k

p+ k

)δδδp=

∞∑p∈Z

√2π

eBesselI(p, 1)δδδp

with BesselI the modified Bessel function of the first kind. This density function isdepicted in Figure 1.

Lemma 1. The discrete Gaussian distribution satisfies in distributional sense: ξ G =−∂ G.

Proof. The co-ordinate difference ∂ acting on the Gaussian distribution G gives

∂G =

∞∑k=0

√2π

(2k)!(2k − 1)!! ∂2k+1δδδ0.

The skew Weyl relation ξ ∂ = ∂ ξ − 1 then implies that ξ ∂2k = ∂2kξ − 2k ∂2k−1,combined with ξδδδ0 = 0 leads to

ξ G =

∞∑k=1

√2π

(2k)!(2k − 1)!!

(−2k ∂2k−1

)δδδ0 = −

∞∑k=1

√2π

(2k − 2)!(2k − 3)!! ∂2k−1 δδδ0

= −∞∑k=0

√2π

(2k)!(2k − 1)!! ∂2k+1 δδδ0 = −∂G

proving the lemma.

Remark 3. The previous property only holds in distributional sense. Since the actionof ξ on the delta function δ0 is not zero (unlike its action on the delta distributionδδδ0), this property does not hold for the density function associated with the discreteGaussian distribution.

3.2 The higher dimensional discrete Gaussian distribution

Following the strategy developed in the one–dimensional case, we now define a discreteGaussian distribution G in general dimension m, as

G =

∞∑k1,...,km=0

(√

2π)m

2k1+...+km k1! . . . km!∂2kmm . . . ∂2k1

1 δδδ0 = (2π)m2 exp

(∂2

2

)δδδ0

8

with discrete density function

g (x1, . . . , xm) =(√

2π)m

emBesselI(x1, 1) . . . BesselI(xm, 1).

The action of the co-ordinate vector variable ξj on this distribution follows the tradi-tional rule ξj G = −∂j G, from which it follows that

ξ G = −∂ G and EG =

m∑j=1

ξj ∂j G = −m∑j=1

ξ2j G = −ξ2G.

Note that here the order of the co-ordinate difference operators ∂2kmm , . . ., ∂2k1

1 doesn’tmatter since the even powers ensure that they are scalar-valued.

4 A fundamental solution of the discrete Heat equa-tion

Consider a discrete version of the Heat equation, given by

(∆∗ − ∂t)u(x, t) = 0, x ∈ Zm, t ∈ R+,

with ∆∗ the discrete Laplacian, i.e. we consider the case where space is discrete andtime continuous. The method to determining solutions of the Heat equation with agiven initial temperature, is based on the notion of a fundamental solution Gt of theHeat equation, satisfying in distributional sense

(∆∗ − ∂t)Gt(x, t) = δ(t)δδδ0

where δ(t) is the continuous delta distribution in the time variable and δδδ0 the discretedelta distribution in the space variable.

Any discrete distribution can be written as a dual Taylor series. Based on theformal equivalence to the continuous space-time setting, we consider a distributionwhich only involves even powers of differences acting on the discrete delta distribution:∂2kδδδ0, i.e. we propose the following form for the fundamental solution:

Gt =

∞∑k=0

ck(t) ∂2kδδδ0

with the continuous functions ck(t) yet to be determined, in such a way that

(∂t −∆∗)Gt = δ(t)δδδ0. (2)

Therefore, we determine both ∂tGt and ∆∗Gt:

∆∗Gt =

∞∑k=0

ck(t) ∂2k+2δδδ0

∂tGt = c′0(t)δδδ0 +

∞∑k=1

c′k(t) ∂2kδδδ0 = c′0(t)δδδ0 +

∞∑s=0

c′s+1(t) ∂2s+2δδδ0.

9

In order for (2) to be fulfilled, it must hold that

c′0(t)δδδ0 +

∞∑k=0

(c′k+1(t)− ck(t)

)∂2k+2δδδ0 = δ(t)δδδ0

or equivalently c′0(t) = δ(t) and c′k+1(t) = ck(t). We thus see that putting c0(t) = H(t),

c1(t) = tH(t), . . . , ck(t) = tk

k! H(t) with H(t) the continuous Heaviside, ensures that

Gt = H(t)

∞∑k=0

tk

k!∂2kδδδ0 = H(t) exp

(t ∂2)δδδ0

is a fundamental solution of the discrete Heat equation. It consists of continuousdistributions in t combined with discrete distributions in x.

Remark 4. This definition of the discrete fundamental solution resembles (up to aconstant) the discrete Gaussian distribution, and is in this way formally equivalent tothe fundamental solution of the continuous Heat equation.

4.1 A density function of the fundamental solution Gt

The fundamental solution Gt can be rewritten in function of the discrete delta distri-butions δδδn, n ∈ Z:

Gt = H(t)

∞∑k=0

tk

k!∂2kδδδ0 = H(t)

∞∑k=0

tk

k!

[k∑

i=−k

(−1)i+k(

2k

i+ k

)δδδi

]

= H(t)∑n∈Z

(−1)n

∞∑k=|n|

(−1)k tk

k!

(2k

n+ k

)δδδn.If

∞∑k=|n|

(−1)k tk

k!

(2k

n+ k

)converges for all n ∈ Z, the discrete function

g := (x, t)→ (−1)n∞∑

k=|n|

(−1)k tk

k!

(2k

n+ k

)H(t)

is said to be the density function of the distribution Gt. The infinite sum convergesfor all n ∈ Z. Indeed,

∞∑k=|n|

(−1)n+k tk

k!

(2k

n+ k

)=

BesselI(|x| , 2t)e2t

with BesselI the modified Bessel function of the first kind. This density function isdepicted in Figure 2 for several values of t. Note that for t = 0, this density functionis the discrete delta function δ0.

10

5 Discrete Convolution Theory

Definition 1. Consider two discrete functions f and g and define their convolutionf ∗ g as the discrete function

(f ∗ g) (n) =∑x∈Z

f(x) g(n− x) =∑m∈Z

f(n−m) g(m).

For this convolution to exist, it suffices that the intersection of the support of f withthe support of g(n−·) is compact, which is certainly the case if one of the two discretefunctions has a compact support. For example, consider the case where supp(f) =supp(g) = Z+. The intersection of supp(f) with supp (g(n− ·)) is always compact forevery n ∈ Z.

Remark 5. Note that this convolution is only symmetric if f and g commute, forexample if either of the two functions f or g is scalar. This is not necessarily true forall discrete functions, as one can see from the example f = e+δ0 and g = e−δ0 whichgives:

(f ∗ g) (n) =∑x∈Z

f(x) g(n− x) = e+e−δ0(n)

(g ∗ f) (n) =∑x∈Z

g(x) f(n− x) = e−e+δ0(n).

Lemma 2. Given two discrete functions f and g and a Clifford constant a, it holdsthat

(a f) ∗ g = a (f ∗ g) , f ∗ (a g) = (f a) ∗ g, f ∗ (g a) = (f ∗ g) a

Lemma 3. Given two discrete functions f and g, the convolution has the followingproperty:

(f∂) ∗ g = f ∗ (∂g) .

Proof.((f∂) ∗ g

)(n) =

∑x∈Z

(f∂) (x) g(n− x)

=∑x∈Z

(f(x+ 1) e+ − f(x)

(e+ − e−

)− f(x− 1) e−

)g(n− x)

=∑y∈Z

f(y)

(e+ g(n− (y − 1))−

(e+ − e−

)g(n− y)− e− g(n− (y + 1)

)=∑y∈Z

f(y) (∂g) (n− y) = f ∗ (∂g) (n).

11

Example 1. The convolutions of discrete functions ∂kδ with compact support with adiscrete function f are given by(

∂kδ ∗ f)

(n) =(∂kf

)(n).

Indeed, (δ ∗ f) (n) =∑x∈Z δ(x) f(n− x) = f(n) and((

∂2g)∗ h)

(n) =

((g∂2)∗ h)

(n) =

(g ∗(∂2h

))(n)

and for g scalar (in this case: g = ∂2`δ for some `),

(∂g ∗ h) (n) = (g∂ ∗ h) (n) =

(g ∗ ∂h

)(n).

On the other hand, by using lemma 3, one can determine that(f ∗ ∂kδ

)(n) =

((f∂k

)∗ δ)

(n) =(f∂k

)(n).

Remark 6. For a discrete function f , the convolution with ∂kδ0 is not necessarilysymmetric. For example, choose f = e+δ0 then

∂δ0 ∗ f = (∂δ0 ∗ δ0) e+ = ∂δ0 e+ = e−e+ (δ0 − δ1)

f ∗ ∂δ0 =(δ0 e+

)∗ ∂δ0 = e+ ∂δ0 = e+e− (δ0 − δ1) .

However, the convolution with ∂kδ0 is symmetric if k is even or f scalar valued.

Definition 2. Consider two regular distributions Tf and Tg with discrete density func-tions f , respectively g, at least one of them having compact support. The convolutionof Tf and Tg is the distribution denoted Tf ∗ Tg which acts as follows on discretepolynomials:

〈Tf ∗ Tg, V 〉 =

⟨Tg(y),

⟨Tf (x), V (x+ y)

⟩⟩=∑y∈Z

(∑x∈Z

V (x+ y) f(x)

)g(y).

Remark 7. Let f and g be discrete density functions of the distributions Tf and Tg,at least one of them having compact support. Since

〈Tf ∗ Tg, V 〉 =∑y∈Z

(∑x∈Z

V (x+ y) f(x)

)g(y) =

∑x∈Z

V (x)

∑y∈Z

f(y) g(x− y)

=∑x∈Z

V (x) (f ∗ g) (x) = 〈Tf∗g, V 〉 ,

we see that Tf ∗ Tg = Tf∗g.

Example 2. The convolution of the discrete distributions ∂kδδδ0 with a regular distri-bution Tf with density function f satisfies

∂kδδδ ∗ Tf = ∂kTf and Tf ∗ ∂kδδδ = Tf∂k.

12

Indeed⟨∂kδδδ ∗ Tf , V

⟩=

⟨Tf (x),

⟨∂kδδδ(y), V (x+ y)

⟩⟩=∑x∈Z

(∑y∈Z

V (x+ y) ∂kyδδδ(y)

)f(x)

= (−1)k∑x∈Z

(∑y∈Z

(V (∂†y)k

)(x+ y) δδδ(y)

)f(x)

= (−1)k∑x∈Z

(V (∂†x)k

)(x) f(x) = (−1)k

⟨Tf , V (∂†)k

⟩=⟨∂kTf , V

⟩.

On the other hand, consider Tf ∗ ∂kδδδ:⟨Tf ∗ ∂kδδδ, V

⟩=

⟨∂kδδδ(x),

⟨Tf (y), V (x+ y)

⟩⟩=∑x∈Z

(∑y∈Z

V (x+ y) f(y)

)∂kxδδδ(x)

= (−1)k∑x∈Z

(∑y∈Z

V (x+ y) f(y)

)(∂†x)kδδδ(x).

Note that(∑y∈Z

V (x+ y) f(y)

)∂†x

=∑y∈Z

(V (x+ y + 1) f(y) e− − V (x+ y) f(y)

(e− − e+

)− V (x+ y − 1) f(y) e+

)

=∑y∈Z

V (x+ y)

(f(y − 1) e− − f(y)

(e− − e+

)− f(y + 1) e+

)= −

∑y∈Z

V (x+ y) (f∂y) (y)

and hence⟨Tf ∗ ∂kδδδ, V

⟩=∑x∈Z

(∑y∈Z

V (x+ y)(f∂ky

)(y)

)δδδ(x) =

∑y∈Z

V (y)(f∂ky

)(y) =

⟨Tf∂

k, V⟩.

To define the convolution of the distribution ∂`δδδ0 and a general distribution G, wewrite down G as its dual Taylor series

G =

∞∑k=0

(−1)k

k!∂kδδδ0 ck, with ck = G

[ξk[1]

].

The distribution ∂`δδδ0 ∗G is then given by

∂`δδδ0 ∗G =

∞∑k=0

(−1)k

k!

(∂`δδδ0 ∗ ∂kδδδ0

)ck =

∞∑k=0

(−1)k

k!∂`+kδδδ0 ck = ∂`G.

13

If G is regular with density function g, i.e. G = Tg, the result indeed corresponds withthe first definition.

Lemma 4. The derivative ∂ of a convolution of two regular distributions Tf and Tgis the convolution of the distribution ∂Tf and Tg:

∂ (Tf ∗ Tg) = (∂Tf ) ∗ Tg.

Proof. When Tf is a derivative of the delta distribution, i.e. Tf = ∂kδδδ0, one easily seesthat

∂(∂kδδδ0 ∗ Tg

)= ∂

(∂kTg

)= ∂k+1Tg = ∂k+1δδδ0 ∗ Tg.

For a general distribution Tf we see that

〈∂ (Tf ∗ Tg) , V 〉 = −⟨Tf ∗ Tg, V ∂†

⟩= −

⟨Tg,⟨Tf , V ∂

†⟩⟩ = 〈Tg, 〈∂Tf , V 〉〉= 〈(∂Tf ) ∗ Tg, V 〉 .

Example 3. Consider the convolution G of the (discrete part of the) fundamentalsolution Gt with a regular distribution Tf with density function f :

G :=

( ∞∑k=0

tk

k!∂2kδδδ0

)∗ Tf =

∞∑k=0

tk

k!

(∂2kTf

).

This is a regular distribution with density function g(x) =

∞∑k=0

tk

k!∂2kf(x).

6 Solutions of the discrete Heat equation

Consider the problem (∂t −∆∗)u(x, t) = f(x, t) where f(x, t) is a given function. Thesolution u(x, t) can be found in the following way:

1. Determine the convolution FS ∗ f =: g(x, t) of the (density function of the)fundamental solution FS and the function f . If f is a density function of a regulardistribution Tf , the function g(x) is then the density function of the distributionFS ∗Tf =: G, which will satisfy in distributional sense (∂t −∆∗)G = Tf . Indeed,lemma 4 combined with the classical property of the derivative of a continuousconvolution shows that

(∂t −∆∗)G = (∂t −∆∗) (FS ∗ Tf ) = ((∂t −∆∗)FS) ∗ Tf = δ ∗ Tf = Tf .

hence g(x, t), which is the density function of G is the suitable solution of theconsidered problem.

2. Note that the convolution FS ∗ f is a combination of a discrete and a continuousconvolution, i.e. if f is a function in the discrete variable x and the continuous

14

time variable s, then FS ∗ f is a distribution in the discrete variable y and thecontinuous variable s given by

(FS ∗ f) (y, s) =

∞∑k=0

1

k!

∫ ∞−∞

H(t) tk(∂2kδ0 ∗ f

)(y, s− t) dt

=

∞∑k=0

1

k!

∫ ∞−∞

H(t) tk(∂2kf

)(y, s− t) dt

=

∞∑k=0

1

k!

(H(t) tk ∗t ∂2kf(y, t)

)(y, s)

where we first considered the discrete convolution and second the continuousconvolution with respect to the time-variable. Although the discrete convolution∂2kδδδ0 ∗ f will always exist, one still has to check the convergence of the series int:

∞∑k=0

1

k!

∫ ∞−∞

H(t) tk(∂2kf

)(y, s− t) dt.

Example 4. Consider the function f(x, t) = χ[ 0,+∞[ (t) δ(x), which is the example of‘polution in one point’. The convolution

FS ∗ f =

∞∑k=0

1

k!

(∫ ∞−∞

H(t) tk χ[ 0,+∞[ (s− t) dt)(

∂2kδ ∗ δ)

=

∞∑k=0

1

k!

(χ[ 0,+∞[ (s)

∫ s

0

tk dt

)∂2kδ = χ[ 0,+∞[ (s)

∞∑k=0

1

k!

[tk+1

k + 1

]s0

∂2kδ

= χ[ 0,+∞[ (s)

∞∑k=0

sk+1

(k + 1)!∂2kδ.

We now consider the convergence of this function g(x, s):

χ[ 0,+∞[ (s)

∞∑k=0

sk+1

(k + 1)!∂2kδ = χ[ 0,+∞[ (s)

∞∑k=0

sk+1

(k + 1)!

k∑j=−k

(−1)k+j

(2k

k + j

)δj

= χ[ 0,+∞[ (s)∑p∈Z

∞∑k=|p|

(−1)k+p sk+1

(k + 1)!

(2k

k + p

) δpwhence we see that

g(x, s) = χ[ 0,+∞[ (s)

∞∑k=|x|

(−1)k+x sk+1

(k + 1)!

(2k

k + x

).

One can check, for example with Maple, that

g(x, s) =s1+|x|

(1 + |x|)!hypergeom

([1 + |x| , |x|+ 1

2

],[

3

2|x|+ 3

2− 1

2||x| − 1| , 3

2|x|+ 3

2+

1

2||x| − 1|

],−4s

).

15

We compare this solution to the continuous setting and we consider the problemu(x, 0) = 0, x ∈ R(∂t − ∂2

x

)u(x, t) = δ(x), x ∈ R, t > 0.

There, the solution u(x, t) is given by the convolution of f(x, t) = δ(x) with the con-

tinuous fundamental solution Φ(x, t) =1√4πt

e−x24t , in this case resulting in

u(y, s) =

∫ s

0

∫ ∞−∞

1√4πt

e−x24t f(y − x, s− t) dx dt

=

∫ s

0

∫ ∞−∞

1√4πt

e−x24t δ(y − x) dx dt

=

∫ s

0

1√4πt

e−y24t dt.

Both the discrete and continuous solution are depicted in Figure 3.

Example 5. When considering the Heat problem (∂t −∆∗)u(x, t) = f(x, t) withf(x, t) = δ(x) δ(t), for example a so-called ‘smoke bomb’, we end up with our funda-mental solution FS. Indeed, the convolution of the fundamental solution with δ(x) δ(t)results again in our fundamental solution:

FS ∗ f =

(H(t)

∞∑k=0

tk

k!∂2kδ

)∗ δ(x)δ(t) =

∞∑k=0

1

k!

(∫ ∞−∞

H(t) tk δ(s− t) dt)(

∂2kδ ∗ δ)

= H(s)

∞∑k=0

1

k!sk ∂2kδ = FS.

7 Heat polynomials

The continuous Heat polynomials pβ(x, t) are polynomial solutions to the Heat equation

(∂t −∆)u(x, t) = 0, x ∈ R, t > 0

with initial condition pβ(x, 0) = xβ . For integer values of β, we get the followingsolutions:

pn(x, t) = n!

bn2 c∑k=0

xn−2k tk

(n− 2k)! k!.

The Heat polynomials are suitable to determine in general solutions of the Heat equa-tion with a given initial condition.

16

7.1 Construction

In our setting of one discrete space variable x and a continuous time variable t, we searchfor analogous discrete polynomial solutions pn(x, t) of the discrete Heat equation

(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

with initial condition pn(x, 0) = ξn[1].

To this end, we first let the Fourier transform Fx in the variable x, act on the Heatequation. We can decompose the function u(x, t) in its Taylor series

u(x, t) =

∞∑k=0

ξk[1] ck(t)

with ck(t) functions of the continuous variable t. The Fourier transform of u(x, t) isthen the discrete distribution Fx [u] = u given by the dual Taylor series

u =

∞∑k=0

∂kδδδ0 ck(t).

Since for any discrete function f , the Fourier transform of the derivative of f is minusthe operator ξ acting on the Fourier transform of f :

Fx [∂f ] = −ξFx [f ] ,

we see that Fx [u] = u must satisfy (in distributional sense):(∂t − ξ2

)u = 0

or thus also∂t

(e−ξ

2 t u)

= 0.

The boundary condition then implies that e−ξ2 t u = F [ξn[1]] or thus u = eξ

2 t F [ξn[1]].Since F

[ξk[1]

]= ∂kδδδ0, we can hence find the appropriate solution u(x, t).

Example 6. As a first trivial example, consider the problem(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = φ(x), φ(x) = 1.

Then φ = δδδ0 and u = eξ2 t δδδ0 = δδδ0 since ξδδδ0 = 0. We’re thus looking for a discrete

function u(x, t) such that u = δδδ0. We may conclude that u(x, t) = 1.

Example 7. A more illustrative example is given by considering the problem(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = φ(x), φ(x) = ξ2k[1].

17

Then φ = ∂2kδδδ0 and

u = eξ2 t ∂2kδδδ0 =

∞∑`=0

t`

`!ξ2` ∂2kδδδ0 =

k∑`=0

t`

`!

(2k)!

(2k − 2`)!∂2k−2`δδδ0

since ξ∂kδδδ0 = −k ∂k−1δδδ0. We are thus looking for a discrete function u(x, t) such that

u =

k∑`=0

t`

`!

(2k)!

(2k − 2`)!∂2k−2`δδδ0.

We may conclude that

u(x, t) =

k∑`=0

t`

`!

(2k)!

(2k − 2`)!ξ2k−2`[1],

which formally resembles the continuous solutions.

Example 8. On the other hand, for the discrete Heat polynomials of odd degree, i.e.discrete polynomial solutions to the problem

(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = φ(x), φ(x) = ξ2k+1[1],

our discrete solution u(x, t) is given by

u(x, t) =

k∑`=0

t`

`!

(2k + 1)!

(2k − 2`+ 1)!ξ2k−2`+1[1].

We conclude that the discrete Heat polynomials pn(x, t) are given by

p2k(x, t) =

k∑`=0

t`

`!

(2k)!

(2k − 2`)!ξ2k−2`[1],

p2k+1(x, t) =

k∑`=0

t`

`!

(2k + 1)!

(2k − 2`+ 1)!ξ2k−2`+1[1]

or equivalently

pn(x, t) =

bn2 c∑`=0

t`

`!

n!

(n− 2`)!ξn−2`[1].

The first few discrete Heat polynomials are then given by

p0(x, t) = 1,

p1(x, t) = ξ[1],

p2(x, t) = ξ2[1] + 2 t,

p3(x, t) = ξ3[1] + 6 t ξ[1],

p4(x, t) = ξ4[1] + 12 t ξ2[1] + 12 t2,

p5(x, t) = ξ5[1] + 20 t ξ3[1] + 60 t2 ξ[1].

They are depicted in Figure 4.

18

Remark 8. When we determine the solution to the problem

(∂t −∆∗)u(x, t) = ξ2k[1] δ(t)

with the usual method of convoluting with the fundamental solution, we get

FS ∗(ξ2`[1] δ(t)

)=

∞∑k=0

((H(t)

tk

k!

)∗t δ(t)

)∂2kx ξ

2`[1]

=∑k=0

(∫ ∞−∞

H(s)sk

k!δ(t− s)ds

)(2`)!

(2`− 2k)!ξ2`−2k[1]

=∑k=0

(∫ ∞0

sk

k!δ(t− s)ds

)(2`)!

(2`− 2k)!ξ2`−2k[1]

= χ[ 0,+∞[ (t)∑k=0

tk

k!

(2`)!

(2`− 2k)!ξ2`−2k[1]

= χ[ 0,+∞[ (t) p2`(x, t).

In this way, we also see the appearance of the Heat polynomials.

7.2 Solutions of the Heat equation with given initial condition

By means of the discrete Heat polynomials, we can determine solutions u(x, t) of thefollowing Heat problem:

(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = φ(x).

Indeed, every discrete function φ(x) can be developed into a discrete Taylor series. Wegive the example where the initial function φ(x) is the discrete delta function δ0.

The discrete delta function δ0 can be expressed in terms of the discrete homogeneouspolynomials ξk[1] as follows:

δ0 =

∞∑`=0

(−1)`

(`!)2 ξ2` [1] +

∞∑`=0

(−1)`+1

`! (`+ 1)!ξ2`+1 [1]

(e+ − e−

).

Substituting every discrete polynomial ξk[1] by its corresponding Heat polynomialpn(x, t), gives us a solution u(x, t), satisfying the Heat equation, with initial condition

19

u(x, 0) = δ(x):

u(x, t) =

∞∑`=0

(−1)`

(`!)2

[∑s=0

ts

s!

(2`)!

(2`− 2s)!ξ2`−2s[1]

]

+

∞∑`=0

(−1)`+1

`! (`+ 1)!

[∑s=0

ts

s!

(2`+ 1)!

(2`− 2s+ 1)!ξ2`−2s+1[1]

] (e+ − e−

)=

∞∑s=0

[ ∞∑`=s

(−1)`

(2s)!

t`−s

(`− s)!

(2`

`

)]ξ2s[1]

+

∞∑s=0

[ ∞∑`=s

(−1)`+1

(2s+ 1)!

t`−s

(`− s)!

(2`+ 1

`

)]ξ2s+1[1]

(e+ − e−

).

One can easily check that for t = 0:

u(x, 0) =

∞∑s=0

(−1)s

s! s!ξ2s[1] +

∞∑s=0

(−1)s+1

s! (s+ 1)!ξ2s+1[1]

(e+ − e−

)= δ(x)

and that (∂t −∆∗)u(x, t) = 0. The function u(x, t) is depicted in Figure 5 for −5 6x 6 5 and t 6 10.

One can show that u(x, t) is the function e−2t BesselI(x, 2t), which is nothing elsethan the density function of the fundamental solution of the Heat equation. We may

thus conclude that the fundamental solutionBesselI(x, 2t)

e2tis a solution to the problem

(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = δ(x).

The following example will show that convergence in t of the solution u(x, t) is notalways obvious.

Example 9. We consider the problem(∂t −∆∗)u(x, t) = 0, x ∈ Z, t > 0

u(x, 0) = eξ2

[1].

The initial function φ(x) = eξ2

[1] =

∞∑`=0

1

`!ξ2`[1] converges for each x ∈ Z. Following

the usual method, we replace each ξ2`[1] by the corresponding Heat polynomial

p2`(x, t) =∑s=0

ts

s!

(2`)!

(2`− 2s)!ξ2`−2s[1],

20

and get the following solution u(x, t) of the given problem:

u(x, t) =

∞∑`=0

1

`!p2`(x, t) =

∞∑`=0

1

`!

[∑s=0

ts

s!

(2`)!

(2`− 2s)!ξ2`−2s[1]

]

=

∞∑p=0

∞∑`=p

1

`!

t`−p

(`− p)!(2`)!

(2p)!

ξ2p[1]

The series

∞∑`=p

1

`!

t`−p

(`− p)!(2`)!

(2p)!=

∞∑s=0

ts

s!

(2s+ 2p)!

(s+ p)! (2p)!converges only for t < 1

4 for

which it equals1

p! (1− 4t)p+ 1

2

We thus see that

u(x, t) =

∞∑p=0

1

p! (1− 4t)p+ 1

2

ξ2p[1]

This series converges for all x ∈ Z, but only for t < 14 .

8 Conclusion and further research

In this contribution, we have successfully applied the discrete function theory to theHeat equation, yielding a discretized Heat equation where space is discrete and timecontinuous. It turned out that a fundamental solution is given by the discrete Gaussiandistribution:

Gt = H(t)

∞∑k=0

tk

k!∂2kδδδ0.

We investigated some properties and exposed differences to the continuous Gaussiandistribution. After introducing the discrete convolution we applied this theory on theinhomogeneous Heat equation. Finally we constructed solutions of the Cauchy problemfor the Heat equation by means of Heat polynomials illustrated by several examples.

A topic for future research encompasses a further discretization of the Heat equationby considering the time variable to be discrete and studying

(∆−∆+

t

)f(x, t) = 0

where ∆+t denotes the forward difference with respect to the time variable. Both

fundamental solution and heat polynomials will be considered for this totally discreteheat equation and their connection with the corresponding heat polynomials in thispaper (discrete space - continuous time).

Acknowledgements

The third author acknowledges support by the institutional grant no. B/10675/02 ofGhent University (BOF).

21

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