is as given in (1),
for some i and
is given function.
satisfying (2) is called a solution of Equation (2). Equation (2) has a numerical solution of the form,
, 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,
. 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
are known then the heat Equation (4) has a solution
of the form
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
in (6), using (7) and (5),
The matlab coding for verification of (8) for
Theorem 2.3. Consider (4) and denote
. Then, the following four types solutions of the Equation (4) are equivalent:
Proof. (a). From (4), we arrive the relation
in (13) gives expressions for
. Now proof of (a) follows by applying all these values in (13).
(b). The heat Equation (4) directly derives the relation
and substituting corresponding v-values in (14) yields (b).
(c). The proof of (c) follows by replacing
(d). The proof of (d) follows by replacing
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
on both sides derives
Theorem 2.5. Assume that the heat difference
is proportional to
. In this case the heat Equation (4) has a solution
if and only if either
Proof. From the heat Equation (4), and the given condition, we derive
, then (15) becomes,
By rearranging the terms, we get
which yields either
. 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
. 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
the partial differences
are known functions then the heat Equation (16) has a solution
in (16), we arrive
The proof follows by applying inverse principle of
Consider the notations in the following theorem:
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
in (i), (ii), (iii), (iv) respectively. ,
The following diagrams (generated by MATLAB) are obtained by using 13,
(i) sine function; boundary values(BV) are
(ii) cosine function; BV are
(iii) sum of sine and cosine function; BV are
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).
Cite this paper
Xavier, G. , Borg, S. and Meganathan, M. (2017) Discrete Heat Equation Model with Shift Values. Applied Mathematics, 8, 1343-1350. doi: 10.4236/am.2017.89099.
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