Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

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.


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][2][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][7][8][9][10][11][12][13][14][15][16][17][18][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.

Preliminaries
Let () be a continuous scalar function of  independent variables  = ( 1 ,  2 , . . .,   , . . .,   ) evaluated at finitely many points on a uniformly distributed lattice with spacing   > 0. Then partial finite differences are defined by Consequently, For example, for the factorial function gives Also the action of  −1   on  () is given by Consider a partial difference equation in the following form: where the operator   operates on a function () as follows: with   ≤   ≤   ,  = 1, 2, . . ., , and   ,   are some fixed integers.The shift operators    ,    are defined likewise.
A discrete vector field in its evolutionary form,   = (/) with characteristic  given by is an infinitesimal generalized symmetry generator of the difference equation ( 6) if it satisfies the infinitesimal symmetry criterion: Here,   () is the discrete analog of the ℎ prolongation of the vector field   .Prolonged vector field acts on the th extended lattice and therefore has the following form: Here, the coefficient functions    ,      , . . .are the discrete total variations of  given by where the total variation Δ   acts on functions of , , Δ  , . . .as follows:

Symmetry Algebra of Nonhomogeneous Discrete Heat Equation
In this section we summarize the results of [6].The difference equation is a discrete heat or diffusion equation.Here  is a function of space variable , time  and partial shifts   and   with respect to  and , respectively.Let the following be the generalized symmetry generator in the evolutionary form: If ( 14) is admitted by ( 13) then the infinitesimal symmetry criterion gives where is the second prolongation of   .Using (16) in (15) one gets the following set of determining equations: To obtain the symmetry generator ( 14) one needs to solve the set of equations ( 17) to (20) for unknown functions , , and .In [6] these functions have been found for three special source terms (, ,   ,   ).Here, we give the infinitesimal symmetry generators obtained in [6] and refer to the paper for details.

Free Heat Equation.
In this case (, ,   ,   ) = 0 and (13) takes the form The set of equations ( 17) to (20) are then solved to obtain the following symmetry algebra: 3.2.Nonzero Potential.In this case the potential  is taken to be a nonzero function  = (,  ).Plugging in this assumption on  in the set of equations ( 17) to (20) one gets where (,   ) is an arbitrary function to be known.And where (,   ) is an arbitrary function to be known.To find the unknown functions  and  one puts a condition on the potential function  such that the commutator [ (1) Δ  ()   −1  , ] vanishes and (24) becomes Using ( 25) in (20) and simplifying one gets The authors then consider the following two particular physically important models that obey the assumptions imposed on the function .

Discrete Harmonic Oscillator. The discrete harmonic oscillator follows the discrete heat equation with the following potential function:
Solving the set of determining equations ( 17) to (20) for above value of  one gets the following symmetry algebra: has the following symmetry algebra: We consider the following case where the potential function also depends on the variables ,   and yet satisfies the required condition.
Optimal systems Invariant solutions No invariant solution Table 3: The optimal systems for discrete centrifugal barrier and respective invariant solutions.

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.

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.

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.
Remark 1.The solution corresponding to  3 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.

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

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.

Table 1 :
Optimal systems for free heat equation and respective invariant solutions.

Table 4 :
Generating solutions for free heat equation.

Table 5 :
Generating solutions for discrete harmonic oscillator.