In 1984, Jerzy Popenda  introduced the difference operator defined on as . In 1989, Miller and Rose  introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the inverse fractional difference operator (   ). Several formula on higher order partial sums on arithmetic, geometric progressions and products of n-consecutive terms of arithmetic progression have been derived in  .
In 2011, M. Maria Susai Manuel, et al.   , extended the definition of to defined as for the real valued function v(k), . In 2014, the authors in  , have applied q-difference operator defined as and obtained finite series formula for logarithmic
function. The difference operator with variable coefficients defined as equation equation is established in  . Here, we extend the operator to a partial difference operator.
Partial difference and differential equations  play a vital role in heat equations. The generalized difference operator with n-shift values on a real valued function is defined as,
This operator becomes generalized partial difference operator if some . The equation involving with at least one is called generalized
partial difference equation. A linear generalized partial difference equation is of the form , then the inverse of generalized partial difference equation is
where is as given in (1), for some i and is given function.
A function satisfying (2) is called a solution of Equation (2). Equation (2) has a numerical solution of the form,
where , m is any positive integer. Relation
(3) is the basic inverse principle with respect to  . Here we form partial
difference equation for the heat flow transmission in rod, plate and system and obtain its solution.
2. Solution of Heat Equation of Rod
Consider temperature distribution of a very long rod. Assume that the rod is so long that it can be laid on top of the set of real numbers. Let be the temperature at the real position and real time of the rod. Assume that diffusion rate is constant throughout the rod shift value . By Fourier law of Cooling, the discrete heat equation of the rod is,
where . Here, we derive the temperature formula for at the general position .
Theorem 2.1. Assume that there exists a positive integer m, and a real number such that and are known then the heat Equation (4) has a solution of the form
Proof. Taking in (4) gives
The proof of (5) follows by applying the inverse principle (3) in (6). ,
Example 2.2. From (2) we get,
whose imaginary parts yield
Taking in (6), using (7) and (5),
The matlab coding for verification of (8) for , , , , as follows, .
Theorem 2.3. Consider (4) and denote and . Then, the following four types solutions of the Equation (4) are equivalent:
Proof. (a). From (4), we arrive the relation
By replacing by in (13) gives expressions for and . Now proof of (a) follows by applying all these values in (13).
(b). The heat Equation (4) directly derives the relation
Replacing by and substituting corresponding v-values in (14) yields (b).
(c). The proof of (c) follows by replacing by and by and
(d). The proof of (d) follows by replacing by and by and . ,
Example 2.4. The following example shows that the diffusion rate of rod can be identified if the solution of (4) is known and vice versa. Suppose that is a closed form solution of (4), then we have the relation
, which yields . Cancelling on both sides derives .
Theorem 2.5. Assume that the heat difference is proportional to i.e., . In this case the heat Equation (4) has a solution if and only if either or .
Proof. From the heat Equation (4), and the given condition, we derive
If, , then (15) becomes, which yields,
By rearranging the terms, we get
which yields either or and hence or . Retracing the steps gives converse. ,
3. Heat Equation for Thin Plate and Medium
In the case of thin plate, let be the temperature of the plate at position and time . The heat equation for the plate is
Theorem 3.1. Consider the heat Equation (16). Assume that there exists a positive integer m, and a real number such that and
the partial differences are known functions then the heat Equation (16) has a solution as,
Proof. Taking in (16), we arrive
The proof follows by applying inverse principle of in (18).
Consider the notations in the following theorem: also .
Theorem 3.2. Assume that is a solution of Equation (16), exist and denote , . Then the following are equivalent:
Proof. The proof of this theorem is easy and similar to the proof of the Theorem (2.3). From (16) and (1), we arrive
Now the proof of (a), (b), (c), (d) follows by replacing
by , by , and by , , and by , in (i), (ii), (iii), (iv) respectively. ,
The following diagrams (generated by MATLAB) are obtained by using 13, and taking , ,
(i) sine function; boundary values(BV) are , , ,
(ii) cosine function; BV are , , ,
(iii) sum of sine and cosine function; BV are , , respectively.
From the above diagrams, when the transmission of heat is known at the boundary points then the diffusion within the material under study can be easily determined.
The study of partial difference operator has wide applications in discrete fields and heat equation is one such. The core theorems (2.1), (2.3) and (3.2) provide the possibility of predicting the temperature either for the past or the future after getting the know the temperature at few finite points at present time. The above study helps us in making a wise choice of material(g) for better propagation of heat. In the converse, it also shows the nature of transmission of heat for the material under study. Thus in conclusion, we can say that the above research helps us in reducing any wastage of heat and also enables us in making a optimal choice of material (g).
 Ferreira, R.A.C. and Torres, D.F.M. (2011) Fractional h-Difference Equations Arising from the Calculus of Variations. Applicable Analysis and Discrete Mathematics, 5, 110-121.
 Susai Manuel, M., Britto Antony Xavier, G., Chandrasekar, V. and Pugalarasu, R. (2012) Theory and application of the Generalized Difference Operator of the nth kind (Part I). Demonstratio Mathematica, 45, 95-106. https://doi.org/10.1515/dema-2013-0347
 Britto Antony Xavier, G., Gerly, T.G. and Nasira Begum, H. (2014) Finite Series of Polynomials and Polynomial Factorials arising from Generalized q-Difference Operator. Far East Journal of Mathematical Sciences, 94, 47-63.
 Maria Susai Manuel, M., Chandrasekar, V. and Britto Antony Xavier, G. (2011) Solutions and Applications of Certain Class of α-Difference Equations. International Journal of Applied Mathematics, 24, 943-954.