Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

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.

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 of S. 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

Let 𝑢(𝑥) be a continuous scalar function of 𝑝 independent variables 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑖,…,𝑥_𝑝) evaluated at finitely many points on a uniformly distributed lattice with spacing 𝜎_𝑖>0. Then partial finite differences are defined by Δ_𝑥𝑖𝑢(𝑥)=1𝜎_𝑖𝑢𝑥₁,𝑥₂,…,𝑥_𝑖−1,𝑥_𝑖+𝜎_𝑖,𝑥_𝑖+1,…,𝑥_𝑝−𝑢𝑥₁,𝑥₂,…,𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑝. (1) It is sometimes convenient to express finite difference operator in terms of shift operator defined as follows: 𝑇_𝑥𝑖𝑢(𝑥)=𝑢𝑥₁,𝑥₂,…,𝑥_𝑖−1,𝑥_𝑖+𝜎_𝑖,𝑥_𝑖+1,…,𝑥_𝑝. (2) Consequently, Δ_𝑥𝑖𝑢(𝑥)=1𝜎_𝑖𝑇_𝑥𝑖−1𝑢(𝑥). (3) For example, for the factorial function 𝑥^(𝑛)=𝑥(𝑥−ℎ)(𝑥−2ℎ)(𝑥−3ℎ)⋯[𝑥−(𝑛−1)ℎ] (2) gives 𝑇_𝑥𝑖𝑥^(𝑛)=𝜎_𝑖𝑛𝑥^(𝑛−1)+𝑥^(𝑛). (4) Also the action of 𝑇_𝑥⁻¹_𝑖 on 𝑥^(𝑛) is given by 𝑇_𝑥⁻¹_𝑖𝑥^(𝑛)=−𝜎_𝑖𝑛𝑥^(𝑛−1)+𝑥^(𝑛). (5)

Consider a partial difference equation in the following form: 𝐸𝑥,𝑇^𝛼𝑢(𝑥),𝑇^𝛽𝑖Δ_𝑥𝑖𝑢(𝑥),𝑇^𝛾𝑖𝑗Δ_𝑥𝑖Δ_𝑥𝑗𝑢(𝑥),…=0, (6) where the operator 𝑇^𝛼 operates on a function 𝑢(𝑥) as follows: 𝑇^𝛼𝑢(𝑥)=𝑇^𝛼1_𝑥1𝑇^𝛼2_𝑥2⋯𝑇^𝛼𝑝_𝑥𝑝𝑢(𝑥), (7) with 𝑚_𝑖≤𝛼_𝑖≤𝑛_𝑖, 𝑖=1,2,…,𝑝, and 𝑚_𝑖, 𝑛_𝑖 are some fixed integers. The shift operators 𝑇^𝛽𝑖, 𝑇^𝛾𝑖𝑗 are defined likewise.

A discrete vector field in its evolutionary form, 𝑋_𝑄=𝑄(𝜕/𝜕𝑢) with characteristic 𝑄 given by 𝑄=_𝑖𝜉_𝑖𝑥,𝑇^𝛼𝑢𝑇^𝑏Δ_𝑥𝑖𝑢−Φ𝑥,𝑇^𝑐𝑢 (8) is an infinitesimal generalized symmetry generator of the difference equation (6) if it satisfies the infinitesimal symmetry criterion:𝑋_𝑄^(𝑛)𝐸𝑥,𝑇^𝛼𝑢(𝑥),𝑇^𝛽𝑖Δ_𝑥𝑖𝑢(𝑥),𝑇^𝛾𝑖𝑗Δ_𝑥𝑖Δ_𝑥𝑗𝑢(𝑥),…_{𝐸(𝑥,𝑇𝛼𝑢(𝑥),𝑇𝛽𝑖Δ𝑥𝑖𝑢(𝑥),𝑇𝛾𝑖𝑗Δ𝑥𝑖Δ𝑥𝑗𝑢(𝑥),…)=0}=0. (9)

Here, 𝑋_𝑄^(𝑛) is the discrete analog of the 𝑛𝑡ℎ prolongation of the vector field 𝑋_𝑄. Prolonged vector field acts on the 𝑛th extended lattice and therefore has the following form: 𝑋_𝑄^(𝑛)=_𝛼𝑇^𝛼𝑄𝜕𝜕𝑇^𝛼𝑢+_𝛽𝑖𝑇^𝛽𝑖𝑄^𝑥𝑖𝜕𝜕𝑇^𝛽𝑖Δ_𝑥𝑖𝑢+_𝛾𝑖𝑗𝑇^𝛾𝑖𝑗𝑄^{𝑥𝑖𝑥𝑗}𝜕𝜕𝑇^𝛾𝑖𝑗Δ_𝑥𝑖Δ_𝑥𝑗𝑢+⋯. (10) Here, the coefficient functions 𝑄^𝑥𝑖, 𝑄^{𝑥𝑖𝑥𝑗},… are the discrete total variations of 𝑄 given by 𝑄^𝑥𝑖=Δ^𝑇_𝑥𝑖𝑄,𝑄^{𝑥𝑖𝑥𝑗}=Δ^𝑇_𝑥𝑖Δ^𝑇_𝑥𝑗𝑄, (11) where the total variation Δ^𝑇_𝑥 acts on functions of 𝑥,𝑢,Δ_𝑥𝑢,… as follows: Δ^𝑇_𝑥𝑖𝑓𝑥,𝑢(𝑥),Δ_𝑥𝑢(𝑥)=1,…𝜎_𝑖𝑓𝑥₁,𝑥₂,…,𝑥_𝑖+𝜎_𝑖,…,𝑥_𝑝,𝑢𝑥₁,𝑥₂,…,𝑥_𝑖+𝜎_𝑖,…,𝑥_𝑝,Δ_𝑥𝑢𝑥₁,𝑥₂,…,𝑥_𝑖+𝜎_𝑖,…,𝑥_𝑝𝑥,…−𝑓₁,𝑥₂,…,𝑥_𝑝,𝑢𝑥₁,𝑥₂,…,𝑥_𝑝,Δ_𝑥𝑢𝑥₁,𝑥₂,…,𝑥_𝑝,…. (12)

3. Symmetry Algebra of Nonhomogeneous Discrete Heat Equation

In this section we summarize the results of [6]. The difference equation Δ_𝑡𝑢−Δ_𝑥𝑥𝑢+𝑔𝑥,𝑡,𝑇_𝑥,𝑇_𝑡𝑢=0 (13) is a discrete heat or diffusion equation. Here 𝑢 is a function of space variable 𝑥, time 𝑡 and partial shifts 𝑇_𝑥 and 𝑇_𝑡 with respect to 𝑥 and 𝑡, respectively. Let the following be the generalized symmetry generator in the evolutionary form: 𝑋_𝑄=𝜉Δ_𝑥𝑢+𝜏Δ_𝑡𝜕𝑢+𝑓𝑢𝜕𝜕𝑢=𝑄.𝜕𝑢 (14) If (14) is admitted by (13) then the infinitesimal symmetry criterion gives 𝑋_𝑄⁽²⁾Δ_𝑡𝑢−Δ_𝑥𝑥𝑢+𝑔𝑥,𝑡,𝑇_𝑥,𝑇_𝑡𝑢_{Δ𝑥𝑥𝑢=Δ𝑡𝑢+𝑔(𝑥,𝑡,𝑇𝑥,𝑇𝑡)𝑢}=0, (15) where 𝑋_𝑄⁽²⁾𝜕=𝑄𝜕𝑢+𝑄^𝑥𝜕𝜕𝑢_𝑥+𝑄^𝑡𝜕𝜕𝑢_𝑡+𝑄^𝑥𝑥𝜕𝜕𝑢_𝑥𝑥+𝑄^𝑥𝑡𝜕𝜕𝑢_𝑥𝑡+𝑄^𝑡𝑡𝜕𝜕𝑢_𝑡𝑡 (16) is the second prolongation of 𝑋_𝑄. Using (16) in (15) one gets the following set of determining equations: Δ_𝑥(𝜏)=0,−Δ_𝑡(𝜏)𝑇_𝑡+2Δ_𝑥(𝜉)𝑇_𝑥+𝜏,𝑔=0,−Δ_𝑡(𝜉)𝑇_𝑡+Δ_𝑥𝑥(𝜉)𝑇²_𝑥+2Δ_𝑥𝑓𝑇_𝑥+𝜉,𝑔=0,−Δ_𝑡𝑓𝑇_𝑡+Δ_𝑥𝑥𝑓𝑇²_𝑥+2Δ_𝑥(𝜉)𝑇_𝑥𝑔+𝜉Δ_𝑥𝑔𝑇_𝑥+𝜏Δ_𝑡𝑔𝑇_𝑡+𝑓,𝑔=0. (17)(18)(19)(20)

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,𝑇_𝑥,𝑇_𝑡). Here, we give the infinitesimal symmetry generators obtained in [6] and refer to the paper for details.

3.1. Free Heat Equation

In this case 𝑔(𝑥,𝑡,𝑇_𝑥,𝑇_𝑡)=0 and (13) takes the form Δ_𝑡𝑢−Δ_𝑥𝑥𝑢=0. (21)

The set of equations (17) to (20) are then solved to obtain the following symmetry algebra: 𝑋_1𝑒=Δ_𝑡𝑢𝜕𝜕𝑢,𝑋_2𝑒=Δ_𝑥𝑢𝜕,𝑋𝜕𝑢_3𝑒𝜕=𝑢,𝑋𝜕𝑢_4𝑒=2𝑡Δ_𝑡𝑢𝑇_𝑡⁻¹+𝑥𝑇_𝑥⁻¹Δ_𝑥𝑢𝜕,𝑋𝜕𝑢_5𝑒=2𝑡Δ_𝑥𝑢𝑇_𝑡⁻¹+𝑥𝑇_𝑥⁻¹𝑢𝜕,𝑋𝜕𝑢_6𝑒=𝑡²𝑇_𝑡⁻²Δ_𝑡𝑢−𝑡𝜎_𝑡𝑇_𝑡⁻²Δ_𝑡𝑢+𝑥𝑡𝑇_𝑡⁻¹𝑇_𝑥⁻¹Δ_𝑥𝑢+14𝑥²𝑇_𝑥⁻²1𝑢−4𝑥𝜎_𝑥𝑇_𝑥⁻²1𝑢+2𝑡𝑇_𝑡⁻¹𝑢𝜕.𝜕𝑢 (22)

3.2. Nonzero Potential

In this case the potential 𝑔 is taken to be a nonzero function 𝑔=𝑔(𝑥,𝑇_𝑥). Plugging in this assumption on 𝑔 in the set of equations (17) to (20) one gets 1𝜉=2𝑥⁽¹⁾Δ_𝑡(𝜏)𝑇_𝑡𝑇_𝑥⁻¹+𝛼𝑡,𝑇_𝑡, (23) where 𝛼(𝑡,𝑇_𝑡) is an arbitrary function to be known. And 1𝑓=8𝑥⁽²⁾Δ_𝑡𝑡(𝜏)𝑇²_𝑡𝑇_𝑥⁻²+12𝑥⁽¹⁾Δ_𝑡𝛼𝑇_𝑡𝑇_𝑥⁻¹−14Δ_𝑥⁻¹𝑥⁽¹⁾Δ_𝑡(𝜏)𝑇_𝑡𝑇_𝑥⁻¹,𝑔𝑇_𝑥⁻¹+𝛽𝑡,𝑇_𝑡, (24) where 𝛽(𝑡,𝑇_𝑡) 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⁾Δ_𝑡(𝜏)𝑇_𝑡𝑇_𝑥⁻¹,𝑔] vanishes and (24) becomes 1𝑓=8𝑥⁽²⁾Δ_𝑡𝑡(𝜏)𝑇²_𝑡𝑇_𝑥⁻²+12𝑥⁽¹⁾Δ_𝑡𝛼𝑇_𝑡𝑇_𝑥⁻¹+𝛽𝑡,𝑇_𝑡. (25) Using (25) in (20) and simplifying one gets −18𝑥⁽²⁾Δ_𝑡𝑡𝑡(𝜏)𝑇³_𝑡𝑇_𝑥⁻²−12𝑥⁽¹⁾Δ_𝑡𝑡𝛼𝑇²_𝑡𝑇_𝑥⁻¹−Δ_𝑡𝛽𝑇_𝑡+14Δ_𝑡𝑡(𝜏)𝑇²_𝑡+Δ_𝑡(𝜏)𝑇_𝑡1𝑔+2𝑥⁽¹⁾Δ_𝑡(𝜏)𝑇_𝑡𝑇_𝑥⁻¹Δ_𝑥𝑔𝑇_𝑥+𝛼𝑡,𝑇_𝑡Δ_𝑥𝑔𝑇_𝑥=0. (26)

The authors then consider the following two particular physically important models that obey the assumptions imposed on the function 𝑔.

3.2.1. Discrete Harmonic Oscillator

The discrete harmonic oscillator follows the discrete heat equation with the following potential function: 𝑔𝑥,𝑇_𝑥=𝑘²𝑥⁽²⁾𝑇_𝑥⁻²,𝑘∈𝑅^>0. (27) Solving the set of determining equations (17) to (20) for above value of 𝑔 one gets the following symmetry algebra: 𝑋_1𝑒=Δ_𝑡𝑢𝜕𝜕𝑢,𝑋_2𝑒𝜕=𝑢,𝑋𝜕𝑢_3𝑒=𝐸_2𝑘(𝑡)Δ_𝑥𝑢+𝑥⁽¹⁾𝑘𝑇_𝑥⁻¹𝑢𝜕,𝑋𝜕𝑢_4𝑒=𝐸_−2𝑘(𝑡)Δ_𝑥𝑢−𝑥⁽¹⁾𝑘𝑇_𝑥⁻¹𝑢𝜕,𝑋𝜕𝑢_5𝑒=𝐸_4𝑘(𝑡)×Δ_𝑡𝑢+2𝑥⁽²⁾𝑘²𝑇_𝑥⁻²𝑢+2𝑥⁽¹⁾𝑘𝑇_𝑥⁻¹Δ_𝑥𝜕𝑢+𝑘𝑢,𝑋𝜕𝑢_6𝑒=𝐸_−4𝑘(𝑡)×Δ_𝑡𝑢+2𝑥⁽²⁾𝑘²𝑇_𝑥⁻²𝑢−2𝑥⁽¹⁾𝑘𝑇_𝑥⁻¹Δ_𝑥𝜕𝑢−𝑘𝑢.𝜕𝑢 (28)(29)

3.2.2. Discrete Centrifugal Barrier

Likewise, the discrete centrifugal barrier with potential function 𝑔𝑥,𝑇_𝑥=𝑇_𝑥𝑥⁽⁻²⁾𝑇_𝑥 (30) has the following symmetry algebra: 𝑋_1𝑒=Δ_𝑡𝑢𝜕𝜕𝑢,𝑋_2𝑒𝜕=𝑢,𝑋𝜕𝑢_3𝑒=𝑡⁽¹⁾𝑇_𝑡⁻¹Δ_𝑡1𝑢+2𝑥⁽¹⁾𝑇_𝑥⁻¹Δ_𝑥𝑢𝜕,𝑋𝜕𝑢_4𝑒=𝑥⁽¹⁾𝑡⁽¹⁾𝑇_𝑥⁻¹𝑇_𝑡⁻¹Δ_𝑥𝑢+𝑡⁽²⁾𝑇_𝑡⁻²Δ_𝑡1𝑢+4𝑥⁽²⁾𝑇_𝑥⁻²𝑢+12𝑇_𝑡⁻¹𝑡⁽¹⁾𝜏₂𝑢𝜕.𝜕𝑢 (31)

We consider the following case where the potential function also depends on the variables 𝑡, 𝑇_𝑡 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, 𝑇_𝑥, and 𝑇_𝑡 and is given by the following 𝑔𝑥,𝑡,𝑇_𝑥,𝑇_𝑡=𝑘²𝑥⁽²⁾𝑇_𝑥⁻²+𝑘²𝑡⁽²⁾𝑇_𝑡⁻², (32)

then the values of coefficient functions of infinitesimal generator (14) are given by: 𝜏=0,𝜉=𝐸_2𝑘(𝑡)𝛼₁+𝐸_−2𝑘(𝑡)𝛼₂,𝑓=𝑥⁽¹⁾𝑘𝐸_2𝑘(𝑡)𝛼₁𝑇_𝑥⁻¹−𝑥⁽¹⁾𝑘𝐸_−2𝑘(𝑡)𝛼₂𝑇_𝑥⁻¹+𝛽₀. (33) And one gets 3-dimensional subalgebra of the discrete harmonic oscillator obtained in ((28) and (29)). This is generated by the following infinitesimal generators: 𝑋_1𝑒𝜕=𝑢,𝑋𝜕𝑢_2𝑒=𝐸_2𝑘(𝑡)Δ_𝑥𝑢+𝑥⁽¹⁾𝑘𝐸_2𝑘(𝑡)𝑇_𝑥⁻¹𝑢𝜕,𝑋𝜕𝑢_3𝑒=𝐸_−2𝑘(𝑡)Δ_𝑥𝑢−𝑥⁽¹⁾𝑘𝐸_−2𝑘(𝑡)𝑇_𝑥⁻¹𝑢𝜕.𝜕𝑢 (34)

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: ⟨𝑋_6𝑒⟩, ⟨𝑋_5𝑒⟩, ⟨𝑋_1𝑒+𝑋_4𝑒⟩, ⟨𝑋_4𝑒⟩, ⟨𝑋_3𝑒⟩, ⟨𝑋_2𝑒−𝑋_1𝑒⟩, ⟨𝑋_2𝑒⟩, and ⟨𝑋_1𝑒⟩.

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 ⟨𝑋_2𝑒−𝑋_1𝑒⟩ 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 Δ_𝑦𝑦ℎ−Δ_𝑦ℎ=0. This equation is then solved to get the invariant solution 𝑢=𝑐₁+𝑐₂2^{(𝑥(1)+𝑡(1))}. Similarly, corresponding to 𝑋_4𝑒, 𝑢=ℎ((𝑥⁽²⁾−𝑥⁽¹⁾𝜎_𝑥)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 Δ_𝑦𝑦ℎ+(1/2)𝑦Δ_𝑦ℎ=0. A further change of variable Δ_𝑦ℎ=𝑤(𝑦) reduces the equation to the following first-order difference equation with variable coefficient: 𝑤+1𝑦+12𝑦𝑤𝑦=0. (35)

To solve (35) we use Laplace method given in [3]. Let 𝑤(𝑦)=∫^𝑏_𝑎𝑡^𝑦−1𝑢(𝑡)𝑑𝑡, 𝑦𝑤(𝑦)=𝑡^𝑦𝑢(𝑡)|^𝑏_𝑎−∫^𝑏_𝑎𝑡^𝑦𝐷𝑢(𝑡)𝑑𝑡, and 𝑤(𝑦+1)=∫^𝑏_𝑎𝑡^𝑦𝑢(𝑡)𝑑𝑡. On substituting these values (35) becomes 12𝑡^𝑦𝑢(𝑡)|^𝑏_𝑎+^𝑏_𝑎𝑡^𝑦𝑢(𝑡)−12𝐷𝑢(𝑡)𝑑𝑡=0. (36)

Now there are two possibilities either 𝑢(𝑡)−(1/2)𝐷𝑢(𝑡)=0 and this gives 𝑢(𝑡)=𝑐𝑒^2𝑡 or 𝑐𝑡^𝑦𝑒^2𝑡=0; therefore, 𝑡→0 or 𝑡→−∞. So, 𝑤(𝑦)=𝑐∫⁰_−∞𝑡^𝑦−1𝑒^2𝑡𝑑𝑡, and one gets the following invariant solution: 𝑢=𝑐⁰_−∞𝑡^{((𝑥(2)−𝑥(1)𝜎𝑥)𝑡(−1)−1)}lnt2𝑥⁽¹⁾−𝜎_𝑥𝑡⁽⁻¹⁾𝑒^2𝑡𝑑𝑡. (37) 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.

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 Tables 2 and 3.

Table 2 

The optimal systems for discrete harmonic oscillator and respective invariant solutions.

Table 3 

The optimal systems for discrete centrifugal barrier and respective invariant solutions.

Remark 1. The solution corresponding to 𝑋_3𝑒 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 Section 2. 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.

Table 5 

Generating solutions for discrete harmonic oscillator.

Table 6 

Generating solutions for discrete centrifugal barrier.

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.

Acknowledgments

The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.