1. Introduction
In 1984, Jerzy Popenda [1] introduced the difference operator
defined on
as
. In 1989, Miller and Rose [2] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the inverse fractional difference operator
( [3] [4] ). Several formula on higher order partial sums on arithmetic, geometric progressions and products of n-consecutive terms of arithmetic progression have been derived in [5] .
In 2011, M. Maria Susai Manuel, et al. [6] [7] , extended the definition of
to
defined as
for the real valued function v(k),
. In 2014, the authors in [6] , 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 [6] . Here, we extend the operator
to a partial difference operator.
Partial difference and differential equations [8] play a vital role in heat equations. The generalized difference operator with n-shift values
on a real valued function
is defined as,
(1)
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
(2)
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,
(3)
where
, m is any positive integer. Relation
(3) is the basic inverse principle with respect to
[6] . 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,
(4)
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
(5)
Proof. Taking
in (4) gives
(6)
The proof of (5) follows by applying the inverse principle (3) in (6). ,
Example 2.2. From (2) we get,
whose imaginary parts yield
(7)
Taking
in (6), using (7) and (5),
(8)
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:
(a)
(9)
(b)
(10)
(c)
(11)
(d)
(12)
Proof. (a). From (4), we arrive the relation
(13)
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
(14)
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
(15)
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
(16)
where
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,
(17)
Proof. Taking
in (16), we arrive
(18)
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:
(a)
(19)
(b)
(20)
(c)
(21)
(d)
(22)
Proof. The proof of this theorem is easy and similar to the proof of the Theorem (2.3). From (16) and (1), we arrive
(i)
(ii)
.
(iii)
.
(iv)
.
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.
4. Conclusion
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).