# Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

Authors: M. N. Qureshi; A. Q. Khan; M. Ayub; Q. Din
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

One-dimensional optimal systems for nonhomogeneous discrete heat equation with different source terms are calculated. By utilizing these optimal systems invariant solutions are found. Also generating solutions are calculated, using the elements of the symmetry algebra.


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## Body

## 1. Introduction

Mathematics provides models, for real life phenomena, to precisely understand the underlying laws governing them. Dynamic changes in a physical phenomenon are usually modeled by differential equations. This means that it is supposed that the changes are continuous and these are taking place in a continuous domain. But there are many real life situations where either the domain or the phenomenon itself or the both are not continuous. These situations occur but are not limited to the fields of biology, physics (classical and quantum), geometry, mathematical design, finance and so forth. Discrete situations are better modeled by difference equations, in contrast with the differential equations. Therefore, difference equations make their appearance in almost every branch of mathematics and their importance cannot be overemphasized.Differences and their calculus are as old as is the differential calculus, but apart from its importance and usefulness the theory and methods of solving difference equations are not as developed as those of differential equations. Since differential equation is a limiting case of difference equation; therefore, it is natural to extend methods available for solving differential equations to the respective difference equations, and the same is done most of the times [1–3].Solving differential equations by exploiting symmetry properties of their solution spaces is one of the standard methods, introduced by the Norwegian mathematician Sophus Lie (1842–1899). In contrast with adhoc techniques this method provides an algorithmic and unified procedure to solve almost all types of differential equations, linear or nonlinear [4, 5].At present, the theory of difference equations is at the same stage as was the theory of differential equations at the time ofS. Lie. A number of adhoc approaches are available to solve difference equations, for example, substitution, nonlinear functional relation, Schroder’s generation function, Maeda’s method, or the theory of integrable maps. But a number of authors have attempted, following Lie’s method for differential equations, to develop a unified integration procedure based on invariance properties of difference equation [6, 7]. It is worth noting that the symmetry properties are not only helpful to find solution but play an important role to understand the physical phenomenon more deeply. In this direction, a number of researchers have attempted to apply symmetries to analyze different discrete physical phenomena [6–19].The discrete diffusion equation is widely used in many contexts [20, 21]. For instance, it has been applied to the area of population growth where one wishes to model geographic spread in addition to growth in number. In the area of physics it is used to model ionic diffusion on a lattice. It has also been used in digital filtering in the form of diffusion filtering [10]. Due to its wide utility, a number of researchers have analyzed different aspects of the discrete diffusion process. Recently, in [6], Levi et al. have attempted to generalize the Lie infinitesimal formulism for calculation of symmetries of difference equations and calculated generalized symmetries of diffusion type difference equations. The symmetry group, which they have found for the discrete equation, is a minimal extension of the Lie point symmetries of the corresponding differential equation. They also found that in the case of linear discrete equation the symmetry algebra is isomorphic to the continuous limit.Symmetries are used not only to integrate but also to analyze the solution space for more insight into the physical system at hand. So, once symmetries are given it is natural to ask for further analysis and find solutions of the equation. There is limited literature on the analysis of solutions of diffusion type difference equations via symmetries. In this paper one-dimensional optimal systems are obtained for the symmetry algebra of diffusion difference equations with different source terms found in [6]. We have also calculated invariant solutions corresponding to each representative of the one-dimensional optimal system, wherever they exist. Using elements of the symmetry algebra generating solutions are also obtained. Moreover, all those cases in which no group invariant solution exist are mentioned.The rest of the paper is arranged as follows. In the next two sections we give some preliminaries and summarize the results of [6], for completeness. In Section 3 one-dimensional optimal systems are obtained for all cases and corresponding group invariant solutions are given. Section 4 is devoted to obtain generating solutions corresponding to each symmetry generator. A brief conclusion is given in Section 5.


## 2. Preliminaries

Letu(x) be a continuous scalar function of p independent variables x=(x1,x2,…,xi,…,xp) evaluated at finitely many points on a uniformly distributed lattice with spacing σi>0. Then partial finite differences are defined by
(1)Δxiu(x)=1σi{u(x1,x2,…,xi-1,xi+σi,xi+1,…,xp)-u(x1,x2,…,xi-1,xi,xi+1,…,xp)}.
It is sometimes convenient to express finite difference operator in terms of shift operator defined as follows:
(2)Txiu(x)=u(x1,x2,…,xi-1,xi+σi,xi+1,…,xp).
Consequently,
(3)Δxiu(x)=1σi(Txi-1)u(x).
For example, for the factorial function x(n)=x(x-h)(x-2h)(x-3h)⋯[x-(n-1)h] (2) gives
(4)Txix(n)=σinx(n-1)+x(n).
Also the action of Txi-1 on x(n) is given by
(5)Txi-1x(n)=-σinx(n-1)+x(n).Consider a partial difference equation in the following form:(6)E(x,Tαu(x),TβiΔxiu(x),TγijΔxiΔxju(x),…)=0,
where the operator Tα operates on a function u(x) as follows:
(7)Tαu(x)=Tx1α1Tx2α2⋯Txpαpu(x),
with mi≤αi≤ni, i=1,2,…,p, and mi, ni are some fixed integers. The shift operators Tβi, Tγij are defined likewise.A discrete vector field in its evolutionary form,XQ=Q(∂/∂u) with characteristic Q given by
(8)Q=∑iξi(x,Tαu)TbΔxiu-Φ(x,Tcu)
is an infinitesimal generalized symmetry generator of the difference equation (6) if it satisfies the infinitesimal symmetry criterion:(9)XQ(n)E(x,Tαu(x),TβiΔxiu(x),TγijΔxiΔxju(x),…)|E(x,Tαu(x),TβiΔxiu(x),TγijΔxiΔxju(x),…)=0=0.Here,XQ(n) is the discrete analog of the nth prolongation of the vector field XQ. Prolonged vector field acts on the nth extended lattice and therefore has the following form:
(10)XQ(n)=∑αTαQ∂∂Tαu+∑βiTβiQxi∂∂TβiΔxiu+∑γijTγijQxixj∂∂TγijΔxiΔxju+⋯.
Here, the coefficient functions Qxi, Qxixj,… are the discrete total variations of Q given by
(11)Qxi=ΔxiTQ,Qxixj=ΔxiTΔxjTQ,
where the total variation ΔxT acts on functions of x,u,Δxu,… as follows:
(12)ΔxiTf(x,u(x),Δxu(x),…)=1σi{f((x1,x2,…,xi+σi,…,xp),mmmmmmu(x1,x2,…,xi+σi,…,xp),mmmmmmm(Δxu)(x1,x2,…,xi+σi,…,xp),…)mmmmmm-f((x1,x2,…,xp),mmmmmmmmm,,u(x1,x2,…,xp),mmmmmmmmmm(Δxu)(x1,x2,…,xp),…)}.


## 3. Symmetry Algebra of Nonhomogeneous Discrete Heat Equation

In this section we summarize the results of [6]. The difference equation
(13)Δtu-Δxxu+g(x,t,Tx,Tt)u=0
is a discrete heat or diffusion equation. Here u is a function of space variable x, time t and partial shifts Tx and Tt with respect to x and t, respectively. Let the following be the generalized symmetry generator in the evolutionary form:
(14)XQ=(ξΔxu+τΔtu+fu)∂∂u=Q∂∂u.
If (14) is admitted by (13) then the infinitesimal symmetry criterion gives
(15)XQ(2)(Δtu-Δxxu+g(x,t,Tx,Tt)u)|Δxxu=Δtu+g(x,t,Tx,Tt)u=0,
where
(16)XQ(2)=Q∂∂u+Qx∂∂ux+Qt∂∂ut+Qxx∂∂uxx+Qxt∂∂uxt+Qtt∂∂utt
is the second prolongation of XQ. Using (16) in (15) one gets the following set of determining equations:
(17)Δx(τ)=0,(18)-Δt(τ)Tt+2Δx(ξ)Tx+[τ,g]=0,(19)-Δt(ξ)Tt+Δxx(ξ)Tx2+2Δx(f)Tx+[ξ,g]=0,(20)-Δt(f)Tt+Δxx(f)Tx2+2Δx(ξ)Txg+ξΔx(g)Tx+τΔt(g)Tt+[f,g]=0.To obtain the symmetry generator (14) one needs to solve the set of equations (17) to (20) for unknown functions ξ, τ, and f. In [6] these functions have been found for three special source terms g(x,t,Tx,Tt). Here, we give the infinitesimal symmetry generators obtained in [6] and refer to the paper for details.


### 3.1. Free Heat Equation

In this caseg(x,t,Tx,Tt)=0 and (13) takes the form
(21)Δtu-Δxxu=0.The set of equations (17) to (20) are then solved to obtain the following symmetry algebra:
(22)X1e=Δtu∂∂u,X2e=Δxu∂∂u,X3e=u∂∂u,X4e=(2tΔtuTt-1+xTx-1Δxu)∂∂u,X5e=(2tΔxuTt-1+xTx-1u)∂∂u,X6e=(12t2Tt-2Δtu-tσtTt-2Δtu+xtTt-1Tx-1Δxu+14x2Tx-2u-14xσxTx-2u+12tTt-1u)∂∂u.


### 3.2. Nonzero Potential

In this case the potentialg is taken to be a nonzero function g=g(x,Tx). Plugging in this assumption on g in the set of equations (17) to (20) one gets
(23)ξ=12x(1)Δt(τ)TtTx-1+α(t,Tt),
where α(t,Tt) is an arbitrary function to be known. And
(24)f=18x(2)Δtt(τ)Tt2Tx-2+12x(1)ΔtαTtTx-1-14Δx-1[x(1)Δt(τ)TtTx-1,g]Tx-1+β(t,Tt),
where β(t,Tt) is an arbitrary function to be known. To find the unknown functions f and ξ one puts a condition on the potential function g such that the commutator [x(1)Δt(τ)TtTx-1,g] vanishes and (24) becomes
(25)f=18x(2)Δtt(τ)Tt2Tx-2+12x(1)ΔtαTtTx-1+β(t,Tt).
Using (25) in (20) and simplifying one gets
(26)-18x(2)Δttt(τ)Tt3Tx-2-12x(1)ΔttαTt2Tx-1-ΔtβTt+14Δtt(τ)Tt2+Δt(τ)Ttg+12x(1)Δt(τ)TtTx-1Δx(g)Tx+α(t,Tt)Δx(g)Tx=0.The authors then consider the following two particular physically important models that obey the assumptions imposed on the functiong.


#### 3.2.1. Discrete Harmonic Oscillator

The discrete harmonic oscillator follows the discrete heat equation with the following potential function:(27)g(x,Tx)=k2x(2)Tx-2,k∈R>0.
Solving the set of determining equations (17) to (20) for above value of g one gets the following symmetry algebra:
(28)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=E2k(t)(Δxu+x(1)kTx-1u)∂∂u,X4e=E-2k(t)(Δxu-x(1)kTx-1u)∂∂u,(29)X5e=E4k(t)×(Δtu+2x(2)k2Tx-2u+2x(1)kTx-1Δxu+ku)∂∂u,X6e=E-4k(t)×(Δtu+2x(2)k2Tx-2u-2x(1)kTx-1Δxu-ku)∂∂u.


#### 3.2.2. Discrete Centrifugal Barrier

Likewise, the discrete centrifugal barrier with potential function(30)g(x,Tx)=Txx(-2)Tx
has the following symmetry algebra:
(31)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=(t(1)Tt-1Δtu+12x(1)Tx-1Δxu)∂∂u,X4e=(x(1)t(1)Tx-1Tt-1Δxu+t(2)Tt-2Δtu+14x(2)Tx-2u+12Tt-1t(1)τ2u)∂∂u.We consider the following case where the potential function also depends on the variablest, Tt and yet satisfies the required condition.


#### 3.2.3. Time-Dependent Discrete Harmonic Oscillator

Now, if the potential function considered in (27) depends on all the four variables x, t, Tx, and Tt and is given by the following
(32)g(x,t,Tx,Tt)=k2x(2)Tx-2+k2t(2)Tt-2,then the values of coefficient functions of infinitesimal generator (14) are given by:
(33)τ=0,ξ=E2k(t)α1+E-2k(t)α2,f=x(1)kE2k(t)α1Tx-1-x(1)kE-2k(t)α2Tx-1+β0.
And one gets 3-dimensional subalgebra of the discrete harmonic oscillator obtained in ((28) and (29)). This is generated by the following infinitesimal generators:
(34)X1e=u∂∂u,X2e=(E2k(t)Δxu+x(1)kE2k(t)Tx-1u)∂∂u,X3e=(E-2k(t)Δxu-x(1)kE-2k(t)Tx-1u)∂∂u.


## 3.1. Free Heat Equation

In this caseg(x,t,Tx,Tt)=0 and (13) takes the form
(21)Δtu-Δxxu=0.The set of equations (17) to (20) are then solved to obtain the following symmetry algebra:
(22)X1e=Δtu∂∂u,X2e=Δxu∂∂u,X3e=u∂∂u,X4e=(2tΔtuTt-1+xTx-1Δxu)∂∂u,X5e=(2tΔxuTt-1+xTx-1u)∂∂u,X6e=(12t2Tt-2Δtu-tσtTt-2Δtu+xtTt-1Tx-1Δxu+14x2Tx-2u-14xσxTx-2u+12tTt-1u)∂∂u.


## 3.2. Nonzero Potential

In this case the potentialg is taken to be a nonzero function g=g(x,Tx). Plugging in this assumption on g in the set of equations (17) to (20) one gets
(23)ξ=12x(1)Δt(τ)TtTx-1+α(t,Tt),
where α(t,Tt) is an arbitrary function to be known. And
(24)f=18x(2)Δtt(τ)Tt2Tx-2+12x(1)ΔtαTtTx-1-14Δx-1[x(1)Δt(τ)TtTx-1,g]Tx-1+β(t,Tt),
where β(t,Tt) is an arbitrary function to be known. To find the unknown functions f and ξ one puts a condition on the potential function g such that the commutator [x(1)Δt(τ)TtTx-1,g] vanishes and (24) becomes
(25)f=18x(2)Δtt(τ)Tt2Tx-2+12x(1)ΔtαTtTx-1+β(t,Tt).
Using (25) in (20) and simplifying one gets
(26)-18x(2)Δttt(τ)Tt3Tx-2-12x(1)ΔttαTt2Tx-1-ΔtβTt+14Δtt(τ)Tt2+Δt(τ)Ttg+12x(1)Δt(τ)TtTx-1Δx(g)Tx+α(t,Tt)Δx(g)Tx=0.The authors then consider the following two particular physically important models that obey the assumptions imposed on the functiong.


### 3.2.1. Discrete Harmonic Oscillator

The discrete harmonic oscillator follows the discrete heat equation with the following potential function:(27)g(x,Tx)=k2x(2)Tx-2,k∈R>0.
Solving the set of determining equations (17) to (20) for above value of g one gets the following symmetry algebra:
(28)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=E2k(t)(Δxu+x(1)kTx-1u)∂∂u,X4e=E-2k(t)(Δxu-x(1)kTx-1u)∂∂u,(29)X5e=E4k(t)×(Δtu+2x(2)k2Tx-2u+2x(1)kTx-1Δxu+ku)∂∂u,X6e=E-4k(t)×(Δtu+2x(2)k2Tx-2u-2x(1)kTx-1Δxu-ku)∂∂u.


### 3.2.2. Discrete Centrifugal Barrier

Likewise, the discrete centrifugal barrier with potential function(30)g(x,Tx)=Txx(-2)Tx
has the following symmetry algebra:
(31)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=(t(1)Tt-1Δtu+12x(1)Tx-1Δxu)∂∂u,X4e=(x(1)t(1)Tx-1Tt-1Δxu+t(2)Tt-2Δtu+14x(2)Tx-2u+12Tt-1t(1)τ2u)∂∂u.We consider the following case where the potential function also depends on the variablest, Tt and yet satisfies the required condition.


### 3.2.3. Time-Dependent Discrete Harmonic Oscillator

Now, if the potential function considered in (27) depends on all the four variables x, t, Tx, and Tt and is given by the following
(32)g(x,t,Tx,Tt)=k2x(2)Tx-2+k2t(2)Tt-2,then the values of coefficient functions of infinitesimal generator (14) are given by:
(33)τ=0,ξ=E2k(t)α1+E-2k(t)α2,f=x(1)kE2k(t)α1Tx-1-x(1)kE-2k(t)α2Tx-1+β0.
And one gets 3-dimensional subalgebra of the discrete harmonic oscillator obtained in ((28) and (29)). This is generated by the following infinitesimal generators:
(34)X1e=u∂∂u,X2e=(E2k(t)Δxu+x(1)kE2k(t)Tx-1u)∂∂u,X3e=(E-2k(t)Δxu-x(1)kE-2k(t)Tx-1u)∂∂u.


## 3.2.1. Discrete Harmonic Oscillator

The discrete harmonic oscillator follows the discrete heat equation with the following potential function:(27)g(x,Tx)=k2x(2)Tx-2,k∈R>0.
Solving the set of determining equations (17) to (20) for above value of g one gets the following symmetry algebra:
(28)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=E2k(t)(Δxu+x(1)kTx-1u)∂∂u,X4e=E-2k(t)(Δxu-x(1)kTx-1u)∂∂u,(29)X5e=E4k(t)×(Δtu+2x(2)k2Tx-2u+2x(1)kTx-1Δxu+ku)∂∂u,X6e=E-4k(t)×(Δtu+2x(2)k2Tx-2u-2x(1)kTx-1Δxu-ku)∂∂u.


## 3.2.2. Discrete Centrifugal Barrier

Likewise, the discrete centrifugal barrier with potential function(30)g(x,Tx)=Txx(-2)Tx
has the following symmetry algebra:
(31)X1e=Δtu∂∂u,X2e=u∂∂u,X3e=(t(1)Tt-1Δtu+12x(1)Tx-1Δxu)∂∂u,X4e=(x(1)t(1)Tx-1Tt-1Δxu+t(2)Tt-2Δtu+14x(2)Tx-2u+12Tt-1t(1)τ2u)∂∂u.We consider the following case where the potential function also depends on the variablest, Tt and yet satisfies the required condition.


## 3.2.3. Time-Dependent Discrete Harmonic Oscillator

Now, if the potential function considered in (27) depends on all the four variables x, t, Tx, and Tt and is given by the following
(32)g(x,t,Tx,Tt)=k2x(2)Tx-2+k2t(2)Tt-2,then the values of coefficient functions of infinitesimal generator (14) are given by:
(33)τ=0,ξ=E2k(t)α1+E-2k(t)α2,f=x(1)kE2k(t)α1Tx-1-x(1)kE-2k(t)α2Tx-1+β0.
And one gets 3-dimensional subalgebra of the discrete harmonic oscillator obtained in ((28) and (29)). This is generated by the following infinitesimal generators:
(34)X1e=u∂∂u,X2e=(E2k(t)Δxu+x(1)kE2k(t)Tx-1u)∂∂u,X3e=(E-2k(t)Δxu-x(1)kE-2k(t)Tx-1u)∂∂u.


## 4. One-Dimensional Optimal System and Invariant Solution

In this section one-dimensional optimal system for each of the above cases of discrete diffusion equation is calculated. It is done in analogy with the procedure to calculate optimal systems for differential equation, laid down in [4]. These optimal systems are then used to calculate invariant solutions.


### 4.1. Free Heat Equation

A calculation gives eight equivalence classes of one-dimensional optimal systems represented by the following generators:〈X6e〉, 〈X5e〉, 〈X1e+X4e〉, 〈X4e〉, 〈X3e〉, 〈X2e-X1e〉, 〈X2e〉, and 〈X1e〉.


#### 4.1.1. Invariant Solution

Employing the method given in [4] for calculating invariant solution for differential equations, the invariant solution corresponding to the representatives of the optimal systems given above are obtained below.To obtain invariant solution corresponding to the subalgebra〈X2e-X1e〉 we find the group invariants u=h(x(1)+t(1)) and define new variables in terms of these invariants by y=x(1)+t(1), u=h(y). This change of variables reduces the partial difference equation (21) to the ordinary 2nd order difference equation Δyyh-Δyh=0. This equation is then solved to get the invariant solution u=c1+c22(x(1)+t(1)). Similarly, corresponding to X4e, u=h((x(2)-x(1)σx)t(-1)) is the group invariant. Substituting y=(x(2)-x(1)σ(x))t(-1), u=h(y) (21) reduces to the ordinary difference equation Δyyh+(1/2)yΔyh=0. A further change of variable Δyh=w(y) reduces the equation to the following first-order difference equation with variable coefficient:
(35)w(y+1)+12yw(y)=0.To solve (35) we use Laplace method given in [3]. Let w(y)=∫abty-1u(t)dt, yw(y)=tyu(t)|ab-∫abtyDu(t)dt, and w(y+1)=∫abtyu(t)dt. On substituting these values (35) becomes
(36)12tyu(t)|ab+∫abty(u(t)-12Du(t))dt=0.Now there are two possibilities eitheru(t)-(1/2)Du(t)=0 and this gives u(t)=ce2t or ctye2t=0; therefore, t→0 or t→-∞. So, w(y)=c∫-∞0ty-1e2tdt, and one gets the following invariant solution:
(37)u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(2x(1)-σx)t(-1)e2tdt.
Similar calculations give invariant solutions corresponding to other representatives of optimal classes. The results are summarized in Table 1.Table 1
Optimal systems for free heat equation and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
X
2
e
u
(
x
,
t
)
=
c
X
3
e
No invariant solution
X
2
e
-
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
2
x
-
t
X
4
e
w
(
y
+
1
)
+
1
2
y
w
(
y
)
=
0 (reduced form)  u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(1)(2x(1)-σx)t(-1)e2tdt
X
4
e
+
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
(
x
(
2
)
+
x
(
1
)
σ
x
)


### 4.2. Discrete Harmonic Oscillators and Centrifugal Barrier

For the discrete harmonic oscillator and the discrete centrifugal barrier the representatives of their one-dimensional optimal systems and the corresponding invariant solutions are summarized in Tables2 and 3.Table 2
The optimal systems for discrete harmonic oscillator and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
-
k
2
T
x
-
2
(
x
(
2
)
+
6
x
(
1
)
+
6
)
X
2
e
No invariant solution
X
3
e
u
(
x
,
t
)
=
c
(
k
+
1
)
2
-
k
2
T
x
-
2
k
+
1
(
x
(
2
)
+
2
x
(
1
)
K
+
1
+
2
(
k
+
1
)
2
)
X
4
e
u
(
x
,
t
)
=
c
+
k
2
T
x
-
2
(
x
(
2
)
+
2
x
(
1
)
-
1
)
X
5
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
+
4
t
(
1
)
+
(
2
+
k
2
T
x
-
2
)
x
(
2
)
+
(
8
+
4
k
2
T
x
-
2
)
x
(
1
)
+
(
2
σ
x
-
1
+
6
k
2
T
x
-
2
)Table 3
The optimal systems for discrete centrifugal barrier and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
+
T
x
2
(
4
x
(
-
1
)
-
x
(
-
2
)
-
6
)
X
2
e
No invariant solution
X
3
e
Δ
y
w
+
1
2
y
w
(
y
)
=
T
x
x
(
-
2
)
T
x (reduced form)  u(x,t)=c∫-∞0e((t(2)/x(2))-1)dt+2t(1)x(2)Tx-2x(-2)
X
2
e
+
X
3
e
Δ
y
v
-
H
(
x
,
t
)
v
=
-
T
x
2
x
(
-
2
)  Where  H(x,t)=12x(2)t(2)σt+12x(2)t(1)+1Remark 1.
The solution corresponding toX3e is obtained by applying Laplace method in reduced form.Since, in case of time-dependent harmonic oscillator, the symmetry algebra is a subalgebra of the time-independent case therefore one does not get new invariant solutions.


## 4.1. Free Heat Equation

A calculation gives eight equivalence classes of one-dimensional optimal systems represented by the following generators:〈X6e〉, 〈X5e〉, 〈X1e+X4e〉, 〈X4e〉, 〈X3e〉, 〈X2e-X1e〉, 〈X2e〉, and 〈X1e〉.


### 4.1.1. Invariant Solution

Employing the method given in [4] for calculating invariant solution for differential equations, the invariant solution corresponding to the representatives of the optimal systems given above are obtained below.To obtain invariant solution corresponding to the subalgebra〈X2e-X1e〉 we find the group invariants u=h(x(1)+t(1)) and define new variables in terms of these invariants by y=x(1)+t(1), u=h(y). This change of variables reduces the partial difference equation (21) to the ordinary 2nd order difference equation Δyyh-Δyh=0. This equation is then solved to get the invariant solution u=c1+c22(x(1)+t(1)). Similarly, corresponding to X4e, u=h((x(2)-x(1)σx)t(-1)) is the group invariant. Substituting y=(x(2)-x(1)σ(x))t(-1), u=h(y) (21) reduces to the ordinary difference equation Δyyh+(1/2)yΔyh=0. A further change of variable Δyh=w(y) reduces the equation to the following first-order difference equation with variable coefficient:
(35)w(y+1)+12yw(y)=0.To solve (35) we use Laplace method given in [3]. Let w(y)=∫abty-1u(t)dt, yw(y)=tyu(t)|ab-∫abtyDu(t)dt, and w(y+1)=∫abtyu(t)dt. On substituting these values (35) becomes
(36)12tyu(t)|ab+∫abty(u(t)-12Du(t))dt=0.Now there are two possibilities eitheru(t)-(1/2)Du(t)=0 and this gives u(t)=ce2t or ctye2t=0; therefore, t→0 or t→-∞. So, w(y)=c∫-∞0ty-1e2tdt, and one gets the following invariant solution:
(37)u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(2x(1)-σx)t(-1)e2tdt.
Similar calculations give invariant solutions corresponding to other representatives of optimal classes. The results are summarized in Table 1.Table 1
Optimal systems for free heat equation and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
X
2
e
u
(
x
,
t
)
=
c
X
3
e
No invariant solution
X
2
e
-
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
2
x
-
t
X
4
e
w
(
y
+
1
)
+
1
2
y
w
(
y
)
=
0 (reduced form)  u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(1)(2x(1)-σx)t(-1)e2tdt
X
4
e
+
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
(
x
(
2
)
+
x
(
1
)
σ
x
)


## 4.1.1. Invariant Solution

Employing the method given in [4] for calculating invariant solution for differential equations, the invariant solution corresponding to the representatives of the optimal systems given above are obtained below.To obtain invariant solution corresponding to the subalgebra〈X2e-X1e〉 we find the group invariants u=h(x(1)+t(1)) and define new variables in terms of these invariants by y=x(1)+t(1), u=h(y). This change of variables reduces the partial difference equation (21) to the ordinary 2nd order difference equation Δyyh-Δyh=0. This equation is then solved to get the invariant solution u=c1+c22(x(1)+t(1)). Similarly, corresponding to X4e, u=h((x(2)-x(1)σx)t(-1)) is the group invariant. Substituting y=(x(2)-x(1)σ(x))t(-1), u=h(y) (21) reduces to the ordinary difference equation Δyyh+(1/2)yΔyh=0. A further change of variable Δyh=w(y) reduces the equation to the following first-order difference equation with variable coefficient:
(35)w(y+1)+12yw(y)=0.To solve (35) we use Laplace method given in [3]. Let w(y)=∫abty-1u(t)dt, yw(y)=tyu(t)|ab-∫abtyDu(t)dt, and w(y+1)=∫abtyu(t)dt. On substituting these values (35) becomes
(36)12tyu(t)|ab+∫abty(u(t)-12Du(t))dt=0.Now there are two possibilities eitheru(t)-(1/2)Du(t)=0 and this gives u(t)=ce2t or ctye2t=0; therefore, t→0 or t→-∞. So, w(y)=c∫-∞0ty-1e2tdt, and one gets the following invariant solution:
(37)u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(2x(1)-σx)t(-1)e2tdt.
Similar calculations give invariant solutions corresponding to other representatives of optimal classes. The results are summarized in Table 1.Table 1
Optimal systems for free heat equation and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
X
2
e
u
(
x
,
t
)
=
c
X
3
e
No invariant solution
X
2
e
-
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
2
x
-
t
X
4
e
w
(
y
+
1
)
+
1
2
y
w
(
y
)
=
0 (reduced form)  u=c∫-∞0t((x(2)-x(1)σx)t(-1)-1)lnt(1)(2x(1)-σx)t(-1)e2tdt
X
4
e
+
X
1
e
u
(
x
,
t
)
=
c
1
+
c
2
(
x
(
2
)
+
x
(
1
)
σ
x
)


## 4.2. Discrete Harmonic Oscillators and Centrifugal Barrier

For the discrete harmonic oscillator and the discrete centrifugal barrier the representatives of their one-dimensional optimal systems and the corresponding invariant solutions are summarized in Tables2 and 3.Table 2
The optimal systems for discrete harmonic oscillator and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
-
k
2
T
x
-
2
(
x
(
2
)
+
6
x
(
1
)
+
6
)
X
2
e
No invariant solution
X
3
e
u
(
x
,
t
)
=
c
(
k
+
1
)
2
-
k
2
T
x
-
2
k
+
1
(
x
(
2
)
+
2
x
(
1
)
K
+
1
+
2
(
k
+
1
)
2
)
X
4
e
u
(
x
,
t
)
=
c
+
k
2
T
x
-
2
(
x
(
2
)
+
2
x
(
1
)
-
1
)
X
5
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
+
4
t
(
1
)
+
(
2
+
k
2
T
x
-
2
)
x
(
2
)
+
(
8
+
4
k
2
T
x
-
2
)
x
(
1
)
+
(
2
σ
x
-
1
+
6
k
2
T
x
-
2
)Table 3
The optimal systems for discrete centrifugal barrier and respective invariant solutions.
Optimal systems
Invariant solutions
X
1
e
u
(
x
,
t
)
=
c
1
+
x
(
1
)
c
2
+
T
x
2
(
4
x
(
-
1
)
-
x
(
-
2
)
-
6
)
X
2
e
No invariant solution
X
3
e
Δ
y
w
+
1
2
y
w
(
y
)
=
T
x
x
(
-
2
)
T
x (reduced form)  u(x,t)=c∫-∞0e((t(2)/x(2))-1)dt+2t(1)x(2)Tx-2x(-2)
X
2
e
+
X
3
e
Δ
y
v
-
H
(
x
,
t
)
v
=
-
T
x
2
x
(
-
2
)  Where  H(x,t)=12x(2)t(2)σt+12x(2)t(1)+1Remark 1.
The solution corresponding toX3e is obtained by applying Laplace method in reduced form.Since, in case of time-dependent harmonic oscillator, the symmetry algebra is a subalgebra of the time-independent case therefore one does not get new invariant solutions.


## 5. Generating Solutions

Since a symmetry of a given equation maps its solution space onto itself, so, one can find new solutions by applying symmetry transformations to a known solution of the equation. The solution thus obtained is called a generating solution. Here, these generating solutions are obtained by first obtaining the one parameter transformation corresponding to each infinitesimal generators obtained in Section2. These solutions are obtained in analogy with the method given in [4] for differential equations. The results are summarized in the form of Tables 4, 5, and 6.Table 4
Generating solutions for free heat equation.
Generators
Infinitesimal transformations
Generating solution
X
1
e
G
1
(
x
(
1
)
,
t
(
1
)
+
ε
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
,
t
(
1
)
-
ε
)
X
2
e
G
2
(
x
(
1
)
+
ε
,
t
(
1
)
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
-
ε
,
t
(
1
)
)
X
3
e
G
3
(
x
(
1
)
,
t
(
1
)
,
u
e
ε
)
u
(
x
,
t
)
=
e
ε
f
(
x
(
1
)
,
t
(
1
)
)
X
4
e
G
4
(
x
(
1
)
e
ε
,
t
(
1
)
e
2
ε
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
e
-
ε
,
t
(
1
)
e
-
2
ε
)
X
5
e
G
5
(
x
(
1
)
+
2
t
(
1
)
ε
,
t
(
1
)
,
u
exp
(
-
(
x
(
1
)
ε
+
ε
2
t
(
1
)
)
)
)
u
(
x
,
t
)
=
exp
(
ε
2
t
(
1
)
-
x
(
1
)
ε
)
f
(
x
(
1
)
-
2
t
(
1
)
e
ε
,
t
(
1
)
)
X
6
e
G
6
(
x
(
1
)
1
-
4
ε
t
(
1
)
,
t
(
1
)
1
-
4
ε
t
(
1
)
,
u
1
-
4
ε
t
(
1
)
exp
(
-
x
(
2
)
ε
+
x
(
1
)
σ
x
ε
1
-
4
ε
t
(
1
)
)
)
u
(
x
,
t
)
=
1
4
π
t
(
1
)
exp
(
(
-
x
(
2
)
-
x
(
1
)
σ
x
)
ε
4
t
(
1
)
)Table 5
Generating solutions for discrete harmonic oscillator.
Generators
Infinitesimal transformations
Generating solution
X
1
e
G
1
(
x
(
1
)
,
t
(
1
)
+
ε
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
,
t
(
1
)
-
ε
)
X
2
e
G
2
(
x
(
1
)
,
t
(
1
)
,
u
e
ε
)
u
(
x
,
t
)
=
e
ε
f
(
x
(
1
)
,
t
(
1
)
)
X
3
e
G
3
(
x
(
1
)
+
ε
,
t
(
1
)
,
u
exp
(
x
(
1
)
k
ε
+
k
ε
2
2
)
)
u
(
x
,
t
)
=
exp
(
x
(
1
)
k
ε
-
k
ε
2
2
)
f
(
x
(
1
)
-
ε
,
t
(
1
)
)
X
4
e
G
4
(
x
(
1
)
+
ε
,
t
(
1
)
,
u
exp
(
-
x
(
1
)
k
ε
-
k
ε
2
2
)
)
u
(
x
,
t
)
=
exp
(
-
x
(
1
)
k
ε
+
k
ε
2
2
)
f
(
x
(
1
)
-
ε
,
t
(
1
)
)
X
5
e
G
5
(
x
(
1
)
e
ε
,
t
(
1
)
+
ε
,
u
exp
(
exp
(
1
2
(
x
(
2
)
-
x
(
1
)
σ
x
)
e
4
ε
+
ε
)
)
)
,
k
=
1
u
(
x
,
t
)  =exp(exp(12(x(2)-x(1)σx)+ε))f(x(1)e2ε,t(1)-ε)
X
6
e
G
5
(
x
(
1
)
e
-
2
ε
,
t
(
1
)
+
ε
,
u
exp
(
exp
(
1
2
(
x
(
2
)
-
x
(
1
)
σ
x
)
e
4
ε
-
ε
)
)
)
,
k
=
1
u
(
x
,
t
)  =exp(exp-(12(x(2)-x(1)σx)e8ε-ε))f(x(1)e2ε,t(1)-ε)Table 6
Generating solutions for discrete centrifugal barrier.
Generators
Infinitesimal transformations
Generating solution
X
1
e
G
1
(
x
(
1
)
,
t
(
1
)
+
ε
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
,
t
(
1
)
-
ε
)
X
2
e
G
2
(
x
(
1
)
,
t
(
1
)
t
,
u
e
ε
)
u
(
x
,
t
)
=
e
ε
f
(
x
(
1
)
,
t
(
1
)
)
X
3
e
G
3
(
x
(
1
)
e
ε
,
t
(
1
)
e
2
ε
,
u
)
u
(
x
,
t
)
=
f
(
x
(
1
)
e
-
ε
,
t
(
1
)
e
-
2
ε
)
X
4
e
G
6
(
x
(
1
)
1
-
4
ε
t
(
1
)
,
t
(
1
)
1
-
4
ε
t
(
1
)
,
u
1
-
4
ε
t
(
1
)
exp
(
(
x
(
2
)
-
x
(
1
)
σ
x
)
ε
1
-
4
ε
t
(
1
)
)
)
u
(
x
,
t
)
=
1
+
4
ε
t
(
1
)
exp
(
(
x
(
2
)
-
x
(
1
)
σ
x
)
ε
1
+
4
ε
t
(
1
)
)
f
(
x
1
+
4
ε
t
(
1
)
,
t
1
+
4
ε
t
(
1
)
)


## 6. Conclusion

In this paper, in analogy with differential equations, the symmetry analysis of a nonhomogeneous discrete heat equation (13) has been carried out. The one-dimensional optimal systems are obtained for the equation with different source terms by using symmetry algebra given in [6]. These optimal systems are then used to obtain invariant solutions if they exist. Moreover, one parameter group of transformations corresponding to all infinitesimal generators of the symmetry algebra has been calculated to obtain generating solutions of the equation. Main results are summarized in Tables 1, 2, 3, 4, 5, and 6. Working on the same lines one can calculate conservation laws which we will report soon.



---

*Source: 146976-2013-09-19.xml*