Comparative Study Of Numerical Method –Spline, Finite Difference And Finite Element For Solving One Dimensional Hyperbolic Partial Differential

Equation

A.K.PATHAK1 and H.D.DOCTOR2

1 Lecturer, Department of H & S.S., S.T.B.S. College of Diploma Engg., Surat.Email: anantkpathak@yahoo.com

2 Professor, Department of Mathematics, V.N.S.G.U., SuratEmail: harishdoctor@yahoo.com

Abstract:

The present work describes spline collocation method for solving flow of electricity cable in

transmission line. The mathematical problem gives rise to solve a partial differential equation with one

space variable which is of hyperbolic type. The method involves the solution of algebraic linear equation

which can be written in the matrix form is main advantage. The solution are obtained by spline explicit

and spline implicit methods and compared with Finite difference, Finite element and analytic solution to

demonstrate the justification and simplicity of the spline approximation.

Key words: Spline collocation, Finite difference, Finite Element Method, Partial differential equation

1 INTRODUCTION:

Two common questions are encountered while the numerical solution to the problem is obtained.

The first is about its acceptance whether it is sufficiently close to the true solution or not. If one has an

analytic solution then this can be answered very clearly but in either case it is not so easy. One has to be

careful while concluding that a particular numerical solution is acceptable when an analytic solution is

not available. Normally a method is selected which requires a minimum number of steps, consuming the

shortest computational time and yet one that does not produce an excessive errors.

2 SPLINE COLLOCATION METHOD:

For solving linear and nonlinear differential equation with the help of the numerical

methods required much computational work and time. Brickley [2] Suggested the method of spline

function containing truncated power polynomials to solve a linear boundary value problem Ahlberg et al

[1] used cardinal splines for solving differential equations. Doctor et al [4,5] have shown that the method

of spline collocation is quite useful for the solution of physical phenomena which give rise to linear

parabolic one dimensional partial differential equation. The method demonstrates the use of spline

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National Conference on Recent Trends in Engineering & Technology

function. Spline functions are piecewise polynomial and their successive derivatives are continuous.

They were used for data interpolation initially. In 1967, Blue [3] suggested the use of spline function for

the solution of B.V.P.

y” = f(x,y,y’) (1)

with boundary conditions

1G [Y(0),Y’(0)]

2G [Y(1),y’(1)] (1a)

The following recurrence relations were used.

S”( 1ix )+4s”( ix )+s”( )1ix = 6/ 2h (f( 1ix )-2f( ix )+f( )1ix (2)

3. SPLINE FORMULA TO SOLVE HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION

WITH ONE SPACE VARIABLES :

The general form of hyperbolic PDE with one space variable x and time variable t is given by

0 t ,L x 0 ; 2xu / 2 / 2c t 2u / 2 …(3)

with a Dirichilet boundary conditions, namely

u(0, t) = 0

u(L, t) = 0 …(4)

and two initial conditions at t = 0 (Cauchy conditions)

u(x, 0) = f(x)

)()0,(u t xgx …(5)

In equation (3), 2c is a constant term, it depends upon some physical quantities in case of

different problems.

Divide the region L x 0 into say n sub – intervals each of width h)(x such that

L x n . The subscript j denotes time and i for the positions. The points of subdivisions are

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National Conference on Recent Trends in Engineering & Technology

n(1)0 i ;xi . Let ji,u denote the solution of equation (3) at thj)(i, mesh point. Discretize the

left hand side of PDE (3) by the central difference formula like finite difference and right side by

second derivative of cubic spline S(x) i.e. )(xS i at the thj)(i, mesh point, one can get.

]u4uu[ -

u)6r(2u)12r-(8u)6r(2 )u4u(u

1-j1,i1-ji,1-j1,-i

j1,i2

ji,2

j1,-i2

1j1,i1ji,1j1,-i

…(7)

where ht / c r i = 1(1) n – 1

Above formula is known as cubic spline explicit formula at to solve hyperbolic PDE of the form

(3). It is clear that above formula is applied for all values of 1j . However, for j = 0, it becomes

]u4uu[ -

u)6r(2u)12r-(8u)6r(2 )u4uu

1-1,i1-i,1-1,-i

1,0i2

0i,2

01,-i2

11,i1i,11,-i

…(8)

where i = 1(1) n – 1

The system has a tri-diagonal matrix, which can be solved by any well-known method. After

calculating the values of u for j = 0, we apply equation (7) for 1j , we get (n – 1) simultaneous linear

equations in (n – 1) unknowns with tri-diagonal matrix, again 1r is the required condition for

convergence and stability of this cubic spline explicit method.

Implicit scheme is unconditionally stable i.e. stable for all the values for r. In implicit

method the discretization of the differential equation at any mesh point (i, j) is done by replacing time

derivative by the central difference formula as done in explicit scheme and the space derivative is

replaced by average of second derivatives of cubic spline S(x) at the th1)-(j and th1)(j level we get

)u4u 2(u

u)1-(3ru)4(6r -u)1-(3r

u)3r-(1u)6r(4u)3r-(1

j1,-iji,j1,i

1-j1,-i2

1-ji,2

j1,i2

1j1,-i2

1ji,2

1j1,i2

…(11)

where 1-n1(1)i ; h t / c r

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National Conference on Recent Trends in Engineering & Technology

The above equation (11) is known as cubic spline implicit formula to solve hyperbolic PDE of

the form (3). Like explicit scheme, described as above, the equation (11) gives (n – 1) simultaneous

linear equations in (n – 1) unknowns with the coefficient matrix of tri-diagonal form. For j = 0, here we

can also use the initial condition in similar manner described as above.

4. THE FLOW OF ELECTRICITY IN THE TRANSMISSION LINES :

This is the study of the derivation of partial differential equations from physical principals, we assume

the flow of electricity in a cable of transmission line. We assume the cable to be imperfectly insulated so

that there is both capacitance and current leakage to ground. Figure (4.1 a) shows such a cable with the

electromotive source x=0 and load x=1.

Figure(4.1a) Figure (4.1b)

i.e. 2222 tu / (LC) xu / …(12)

where u stands for either i(x, t). L is the inductance and C is the capacitance per unit length of the

transmission line i.e. cable. This equation is known as hyperbolic PDE with one space variable x and

time variable t.

Let l = length of the transmission line = 1 and using following initial and boundary conditions,

the solution of above equation (12) i.e. the current distribution is obtained as follows.

Dirichilet boundary conditions :

u(0, t) = 0

u(l, t) = 0 …(13)

Initial conditions (cauchy conditions at t = 0)

1 x 0 ; x sin 0)u(x,

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National Conference on Recent Trends in Engineering & Technology

U(x, 0) = 0 …(14)

We solve this problem with spline explicit -implicit method and compare the results with Finite

Difference Explicit - Implicit and Finite element method as well as exact solution with same boundary

and initial condition.

5. Result

0.000000

0.000002

0.000004

0.000006

0.000008

0.000010

0.000012

0.0 0.1 0.2 0.3 0.4 0.5

UFI-UEXT

USI-UEXT

Figure (6.1) Figure (6.2 )Comparison Of Error Analysis For Spline Comparison Of Error Analysis For Spline Explicit And Finite Difference Explicit Implicit And Finite Difference Implicit (at t = 0.04) (at t = 0.04)

0.0000000

0.0000002

0.0000004

0.0000006

0.0000008

0.0000010

0.0000012

0.0 0.1 0.2 0.3 0.4 0.5

X

ER

RO

R

USE-UEXT

USI-UEXT

0.000000

0.000001

0.000002

0.000003

0.000004

0.000005

0.000006

0.000007

0.000008

0.000009

0.000010

0.0 0.1 0.2 0.3 0.4 0.5

X

ER

RO

R UFEM-UEXT

USE-UEXT

USI-UEXT

Figure (6.4) Figure (5.8.2)Comparison Of Error Analysis For Spline Comparison Of Error Analysis For Spline

Explicit- Implicit(at t = 0.04) Explicit-Implicit And Finite Element Results

0.000000

0.000001

0.000002

0.000003

0.000004

0.000005

0.000006

0.000007

0.000008

0.000009

0.0 0.05 0.10 0.15 0.20 0.25

X

ER

RO

R

UFE-UEXT

USE-UEXT

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

0.000000

0.200000

0.400000

0.600000

0.800000

1.000000

1.200000

0.0 0.1 0.2 0.3 0.4 0.5

X

UUSE

USl

UEXT

Figure (6.5)Comparison Of Spline Explicit –Implicit Solution With Exact Solution

7 Conclusion:

It is noticed from graph and table that spline solution produces a reduced amount of error

than finite difference and finite element method. Analytic solution and spline solution represents a single

curve which indicates the accuracy and reliability of spline collocation technique and spline solution

are accurate up to five decimal places. Also spline collocation method required compact calculations and

besides these two methods, namely explicit and implicit method, implicit methods have more accurate

results than the explicit method. This justifies that spline collocation approach to finite difference, finite

element approximation as well as analytic solution.

REFERENCE: 1. AHLBERG, J.H., NILSON, E.N. AND WALS, J.N., J.N., The Theory of spline and their Application. Academic Press, N.Y.1967.2. BICKLEY, W.G.-Piecewise cubic Interpolation and Two point Boundary Value problem. Comp. J, 1968, 11,126.3. BLUE, J.L.-‘Spline function Methods for linear Boundary Value problem.” Communications of the ACM.1969, 12, No.6, 327. 4. BRUCH, J. C., AND G. ZYVOLOSKI: “Transient Two-Dimensional Heat Conduction Problems

Solved by the Finite Element Method,” International Journal for Numerical Methods in Engineering, 8,PP 3481-494 (1974).

5. DR. GREWAL B.S.:- Numerical Methods in Engineering and science, fifth ed, Khanna Publishers.

6. DOCTOR, H.D., and KALTHIA, N.L. - Spline Approximation of Boundary value Problems. Presented at Annual Conf.of Indian math.Soc, 1983, 497. DOCTOR, H.D.,, BULSARI,A.B.and KALTHIA, N.L.- Spline Collocation approach to Boundary Value Problems.Int.J.Number, Floids.1984, 4, 6,511.8. Fyfe, D.J.-The use of cubic splines in solutions of two point boundary value problems,

com.P.J.1969, 12,188-192.9. REDDY J.N., - An Introduction to the Finite Element Method, McGraw-Hill.

13-14 May 2011 B.V.M. Engineering College, V.V.Nagar,Gujarat,India

National Conference on Recent Trends in Engineering & Technology

10. SASTRY S.S., - “Introductory Methods of Numerical Analysis. Prentice-Hall of India Private Limited. New Delhi.

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National Conference on Recent Trends in Engineering & Technology