Discrete Heat Equation Model with Shift Values

We investigate the generalized partial difference operator and propose a model of it in discrete heat equation in this paper. The diffusion of heat is studied by the application of Newton’s law of cooling in dimensions up to three and several solutions are postulated for the same. Through numerical simulations using MATLAB, solutions are validated and applications are derived.


Introduction
In 1984, Jerzy Popenda [1] introduced the 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 [5].
In 2011, M. Maria Susai Manuel, et al. [6] [7], extended the definition of α ∆ to for the real valued function v(k), 0 >  .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 ( ) , , , , 0 This operator where for some i and ( ) Equation ( 2) has a numerical solution of the form, where ( ) 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.

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 ℜ of real numbers.Let ( ) , v k k be the temperature at the real position 1 k and real time 2 k of the rod.Assume that diffusion rate γ is constant throughout the rod shift value 0 >  . By Fourier law of Cooling, the discrete heat equation of the rod is, where . Here, we derive the temperature formula for ( ) Proof.Taking ( ) ( ) ( ) The proof of (5) follows by applying the inverse principle (3) in (6) Taking 6), using ( 7) and ( 5), The matlab coding for verification of ( 8) for
(b).The heat Equation (4) directly derives the relation (c).The proof of (c) follows by replacing The following example shows that the diffusion rate of rod can be identified if the solution ( ) , v k k of ( 4) is known and vice versa.Suppose that ( ) is a closed form solution of (4), then we have the relation , which yields Theorem 2.5.Assume that the heat difference . In this case the heat Equation (4) has a solution ( ) Proof.From the heat Equation ( 4), and the given condition, we derive ( ) which yields either ( ) ( ) and hence ( ) Retracing the steps gives converse.

Heat Equation for Thin Plate and Medium
In the case of thin plate, let ( ) , , v k k k be the temperature of the plate at position ( ) , v k k and time 3 k .The heat equation for the plate is where Consider the heat Equation ( 16).Assume that there exists a positive integer m, and a real number 3 0 >  such that ( ) , , v k k k ml − and the partial differences , , , , are known functions then the heat Equation ( 16) has a solution ( ) The proof follows by applying inverse principle of Consider the notations in the following theorem: ) ( ) Theorem 3.2.Assume that ( ) ( ) ( ) ( ) 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 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.

Conclusion
The study of partial difference operator has wide applications in discrete fields 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 and substituting corresponding v-values in (14) yields (b).