Numerical solution of heat equation through double interpolation

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 58-62
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Numerical solution of heat equation through double interpolation
P.Kalyani1
, P.S.Ramachandra Rao2
1
(Department of Mathematics, Kakatiya Institute of Technology and Sciences, Warangal, India)
2
(Department of Mathematics, Kakatiya Institute of Technology and Sciences, Warangal, India)
Abstract :In this article an attempt is made to find the solution of one-dimensional Heat equation with initial
and boundary conditions using the techniques of numerical methods, and the finite differences. Applying
Bender-Schmidt recurrence relation formula we found u(x ,t) values at lattice points. Further using the double
interpolation we found the solution of Heat equation as double interpolating polynomial.
Keywords - Boundary Value Problem , Finite Difference method, Double Interpolation.
I. INTRODUCTION
Boundary value problems occur very frequently in various fields of science and engineering such as
mechanics, quantum physics, electro hydro dynamics, theory of thermal expansions. For a detailed theory and
analytical discussion on boundary value problems, one may refer to Bender [1], Collatz [2], Na [3],Numerical
solution of boundary value problems by splines and numerical integration are discussed by P.S. Rama Chandra
Rao [4-5], Ravikanth. A.S.V. [6].
The finite difference method is one of several techniques for obtaining numerical solutions to the
boundary value problems. Especially to solve partial differential equations , in which the partial derivatives are
replaced by finite differences of two variables .Mortan and Mayer[7] and Cooper[8] provide a more
mathematical development of finite difference methods and modern introduction to the theory of partial
differential equation along with a brief coverage of numerical methods. Fletcher [9] described the method to
implement finite differences to solve boundary value problems.
In this paper we describe how to solve a one-dimensional heat equation using finite difference method
and double interpolation [10]. The heat equation is of fundamental importance in diverse scientific fields which
describes the distribution of heat (or variation in temperature) in a given region over time. In mathematics it is
the prototypical parabolic partial differential equation. In financial mathematics, the famous Black-Scholes
option pricing models differential equation can be transformed in to the heat equation allowing relatively easy
solution from a familiar body of mathematics. The diffusion equation, a more general version of the heat arises
in connection with the study of chemical diffusion and other related processes and a direct practical application
of the heat equation, in conjunction with the Fourier theory, in spherical co-ordinates, is the measurement of the
thermal diffusivity in polymers. The heat equation can be efficiently solved numerically using the Bender-
Schmidt method. Usually the solution of these equations is given in Fourier series form. Monte [11] applied a
natural analytical approach for solving the one dimensional transient heat conduction in a composite slab.
Lu [12] gave a novel analytical method applied to the transient heat conduction equation. In this paper
we found the solution of one-dimensional heat equation with certain initial and boundary conditions as a
polynomial.
II. DESCRIPTION OF THE METHOD
Consider one-dimensional heat conduction equation
(1)
Where t and x are the time and space co-ordinates respectively, in the region
With appropriate initial and boundary conditions.
If is any function of two independent variables x and t then we define
where . Here denotes the initial value of x, denotes
the initial value of t, and h, k are the intervals of differencing for the variables x and t respectively. Further r, s
are non negative integers.
Replacing the partial derivatives of (1) with finite differences we get Bender-Schmidt recurrence relation which
is given by

(2)
Where
By using boundary conditions and equation (2) we find values at lattice points (xr,ts) . Further using
double interpolation that is by finding the two way differences of various orders and substituting in the general
formula for double interpolation we find double interpolating polynomial of .
The following formulae are given in [10] .
2.1. Formulae
1. The operator is defined by the equation and the operator is
defined by .
2. Double or two way differences of various orders are defined by
Similarly
The general formula for the differences of different order is given by
(3)
Where m,n are positive integers.
General formula for double interpolation is
(4)
2.2. Formulation of the problem
We consider the following boundary value problem of one- dimensional heat equation.
(5)
Subject to the following boundary conditions
(6)
(7)
(8)
2.3. Solution of the problem
we take the interval of differencing of x as 1,
i.e., h=1.and the interval of differencing of t as
thus and
Drawing straight lines parallel to coordinate axis (t, x) we have 25 mesh points.

suppose
where . for (r, s = 0,1,2,3,4,5)
The boundary condition (2) gives
(9)
he boundary condition (3) gives
(10)
The initial condition (4) gives
(11)
The other values are obtained from the recurrence relation
for which are shown in the Table 1.
Since the first and last column values of Table 1 are zero
The differences of different orders of are zero.
i.e. . particularly
(12)
(13)
By considering the second column values of table 1 , the differences are obtained and shown in the
table 2.
From Table 2, we have
(14)
By considering the third column values of Table 1, the differences of are obtained and shown in the
table 3.
From Table 3, we have
(15)
Similarly by considering 4th
and 5th
columns we obtain
(16)
(17)
Repeating the above process for rows of table 1 , we obtain
(18)
(19)
(20)
(21)
(22)
(23)
The differences
are obtained from the formula (3) are given by
(24)
The formula for interpolating polynomial in two variables (4) up to 5th
difference (m=5) is

(25)
Substituting the values of from the equations (12), (18) and (24) in equation (25) and simplifying we
obtain
. (26)
The required interpolating polynomial of which is the approximate solution of heat equation (5) is
given by equation (26).
III. TABLES
Table 1. Function Table.
t x 0 1 2 3 4 5
0 0 24 84 144 144 0
½ 0 42 84 114 72 0
1 0 42 78 78 57 0
3/2 0 39 60 67.5 39 0
2 0 30 53.5 49.5 33.75 0
5/2 0 26.625 39.75 43.5 24.75 0
Table 2. Difference Table 1.
24
→ 18
42 → -18
→ 0 → 15
42 → -3 → -18
→ -3 → -3 → 32.625
39 → -6 → 14.625
→ -9 → 11.625
30 → 5.625
→ -3.375
26.625

Table 3. Difference Table2.
84
→ 0
84 → -6
→ -6 → -6
78 → -12 → 29.25
→ -18 → 23.25 → -70.25
60 → 11.25 → -41.25
→ -6.75 → -18
53.25 → -6.75
→ -13.5
39.75
.
IV. CONCLUSION
Problems considered in thermodynamics and engineering may yield parabolic equations. usually the
analytical solution of such equations are in Fourier series. In this we obtain the solution as a polynomial in two
variables.
REFERENCES
[1] Bender.C.M and Orszag.S.A., Advanced mathematics methods for scientist and engineers ( McGraw-Hill. New York,1978).
[2] Collatz, L.,The numerical treatment of differential equations ( Springer Verlag,Berlin,1996).
[3] Na,T.Y., Computational methods in engineering boundary value problems (Academic Press, New York,1979).
[4] Rama ChandraRao,P.S., Solution of a Class of Boundary Value Problems using Numerical Integration, Indian journal of
mathematics and mathematical sciences.vol.2.No2,2006,pp.137-146.
[5] Rama Chandra Rao,P.S., Solution of fourth order of boundary value problems using spline functions, Indian Journal of Mathematics
and Mathematical Sciences.vol.2.No1,2006, pp.47-56
[6] Ravikanth,A.S.V., Numerical treatment of singular boundary value problems, Ph.D. Thesis, National Institute of Technology,
Warangal, India,2002.
[7] K.W. Mortan and D.F. Mayers. Numerical solution of partial differential equations: An introduction ( Cambridge University press,
Cambridge,England.1994).
[8] Jeffery Cooper, Introduction to partial differential equations with matlab (Birkhauser, Bosten, 1998).
[9] Clive A.J,Fletcher, Computational techniques for fluid dynamics (Springer-Verlag,Berlin,1998).
[10] J.B. Scorborough, Numerical mathematical analysis ( Oxford and IBH Publishing Co. Pvt. Ltd.,1966)
[11] F.de Monte, Transient heat conduction in one-dimensional composite slab: A natural analytic approach, International Journal of
heat and mass transfer,Vol.43,2000,pp.3607-3619.
[12] X Lu,P Tervola and M Viljanen, An efficient analytical solution to the transient heat conduction in one dimensional hollow
composite cylinder, J.Phy.A.Math.Gen.Vol.38,2005,pp.10145-10155.

Numerical solution of heat equation through double interpolation

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