A Numerical Approach for Solving Nonlinear Boundary Value Problems in Finite Domain using Spline Col

IJIRST –International Journal for Innovative Research in Science & Technology| Volume 3 | Issue 01 | June 2016 ISSN (online): 2349-6010

A Numerical Approach for Solving Nonlinear Boundary Value Problems in Finite Domain using Spline Collocation Method Nilesh A. Patel Department of Mathematics Shankersinh Vaghela Bapu Institute of Technology, Gandhinagar, Gujarat

Jigisha U. Pandya Department of Mathematics Sarvajanik college of Engineering & Technology, Surat, Gujarat

Abstract A collocation method with quartic splines has been developed to solve third order boundary value problems. The proposed method tested on third order nonlinear boundary value problem. The solution of nonlinear boundary value has been obtained linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in a good agreement with the analytical solutions available in the literature. Based on the Spline Collocation Method, a general approximate approach for obtaining solution to nonlinear boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three point nonlinear problems.

Keywords: Third order differential equation, Quartic Spline, Hessen berg matrix, Quasilinearization Boundary value problem _______________________________________________________________________________________________________ I.

INTRODUCTION

In the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in engineering, applied mathematics and several branches.

However, it is usually difficult to obtain closed-form solutions for boundary value problems, especially for nonlinear boundary value problems. In most cases, only approximate solutions (either numerical solution or analytical solutions) can be expected.

Some numerical methods such as finite difference method [1], finite element method [2], spline approximation method [3], shooting method [4], and sinc-Galerkin method [5], have been developed for obtaining approximate solutions to boundary value problems.

In this paper, we will apply a spline collocation method approach, to obtain a solution to the wide class of nonlinear systems of boundary value problems. It is worth noting that the system we are studying is more general than the ones discussed in the below mentioned references as it includes the extra nonlinear terms. The spline method approach is widely utilized for the numerical solution of nonlinear problems arising in real world applications. The numerical analysis literature contains a few other methods developed to find an numerical solution of this problem. Al Said et al. [1] have solved a system of third order two point boundary value problems using cubic splines. Noor et al. [4] generated second order method based on quartic splines.

Other authors [2,3] generated finite difference using fourth degree quintic polynomial spline for this problem subject to other boundary conditions.

The governing equations here are highly nonlinear differential equations, which are solved by using the Quartic spline collocation method. In this way, the paper has been organized as follows. In section 2, we use the Quartic spline collocation method. Section 3, approximate solution for the governing equations and contains the results and discussion. The conclusions are summarized in section 4.

II. QUARTIC SPLINE COLLOCATION METHOD Consider equally spaced knots of partition π: a  x 0  x1  x 2  ........  x n  b on  a , b  . The quartic spline is defined by s ( x )  a 0  b0 ( x  x 0 ) 

1 2

c0 ( x  x0 )  2

1 6

d 0 ( x  x0 )  3

1

n 1

 ek ( x  x k )  24 4

(1)

k 0

Where the powers function ( x  x k )  is defined as  x  xk , ( x  xk )     0,

x  xk x  xk

(2)

All rights reserved by www.ijirst.org

318

A Numerical Approach for Solving Nonlinear Boundary Value Problems in Finite Domain using Spline Collocation Method (IJIRST/ Volume 3 / Issue 01/ 053)

And the boundary value problem is given by y "'( x )  p ( x ) y "( x )  q ( x ) y '( x )  r ( x ) y ( x )  m ( x )

(3)

Subject to boundary conditions

 0 y0   0 y n '   0 y n "   0

1 y0 '  1 y n   1 y n "   1  2 y0 "   2 y n   2 y n '   2

(4) To solve this boundary value problem substitute s(x), s'(x), s"(x), s"'(x) from quartic spline, then the boundary value problem becomes n 1



 ek   xi

k 0



 xk





1 2

p i  xi  x k

 2

1   d 0 1  p i ( x i  x 0 )  q i  x i  x 0 2 



1 6



2

q i  xi  x k



1 6

 3

ri  x i  x 0

1



24



3

ri  x i  x k



  4



  

1 2    c 0  p i  q i  x i  x 0   ri  x i  x 0   2    b0  p i  ri  x i  x 0   a 0  ri   m  x i  .W here i = 0,1,2,.....n.

(5) Thus for quartic spline and third order boundary value problem we get nine linear algebraic equations in nine unknowns a0, b0, c0, d0, e0, e1,.... e4. The matrix form of this system is given by AX = B Where X= [e4, e3, e2, e1, e0, d0, c0, b0, a0]T B    2 ,  1 ,  0 , m 5 , m 4 , m 3 , m 2 , m1 , m 0 

T

And the co-efficient matrix A is an upper Hessenberg matrix.

Quartic spline and third order boundary value problem, we take number of intervals n=5. Continuing this process we can say that "In general for higher degree spline and lower order boundary value problem i.e. for n th degree spline and (n-1)th order boundary value problem we can take (n+1) number of intervals at xi, i = 0(1) n and get a set of linear algebraic equations in (2n-1) unknowns."

III. SOLUTION BY USING COLLOCATION METHOD [9]Srivastava et al (1987), the basic equations governing the motion of two dimensional, steady incompressible viscous fluids past continuous surface in the presence of transverse magnetic field can be written in non-linear coupled equation as follow: f

'''

 ff

''



f  '

2

1 0

(6)

Subject to boundary conditions as, (7) f (0)  0, f (0)  0, f (1)  0.

We use quasilinearization technique to convert (4) into linear form with help of boundary conditions (5).

We get linear form as '

''' ' ' '' f i 1  f i 1 f i  2 f i f i 1  f i f i 1   1  f i f i  ( f i ) With boundary conditions as (5).The Quartic spline is given by '

s    a 0  b0 (   0 ) 

''

1 2

' 2

c 0 (   0 )  2

1 6

(8)

d 0 (   0 )  3

1

n 1

e 24

(   k )  4

k

(9)

k 0

After applying quasilinearization technique, we get linear differential equation (6).

All rights reserved by www.ijirst.org

319

A Numerical Approach for Solving Nonlinear Boundary Value Problems in Finite Domain using Spline Collocation Method (IJIRST/ Volume 3 / Issue 01/ 053) n 1



fi

e k [( i   k ) 

2

k 0

( i   k ) 

 d 0 [1  f i ( i   0 ) 

2

2 fi

'

'

( i   k )  3

6

'

( i   0 )  2

2

 c 0 [ f i  2 f i ( i   0 ) 

2 fi

fi

fi 6

fi

''

24

( i   k ) ] 4

''

( i   0 ) ] 3

''

2

( i   0 ) ] 2

 b0 [  2 f i  f i ( i   0 )] '

''

 a0 [ fi ]   1  fi fi  ''

''

f  '

2

i

(10)

An applying initial condition in (7), we get following equations a0  0 b0  0

(11)

a 0  b 0  0.5 c 0  0.1667 d 0  0.0417 e 0  0.0273 e1  0.0171e 2  0.0100 e 3  0.0054 e 4  0.0026 e 5  0.0011e 6  0.000337 e 7  0.000066 e8  0.0000 041e 9  0

Solve the above equations, substitute constants in (9) and we get solution for different value of  .

Graphical solution of given problem: Analytic solution

Spline solution

Fig. 1:

Fig. 2:

IV. CONCLUSION Solved the problem using quartic spline collocation method. This shows that spline method also gives nearest and accurate results. Also, the results are obtained three iterations, which show the reliability of the method. . Thus we can solve such type of problems using numerical method REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

E.A. Al-Said, M.A. Noor, "Cubic Splines Method for a System of Third Boundary Value Problems", Applied Mathematics and Computations 142 (2003) 195-204.

H.N.Calagar, S.H.Cagalar and E.H.Twizell, "The Numerical Solution of Third Order Boundary Value Problems with Fourth Degree B-Spline", International Journal of Computer Mathematics 71(1999) 373-381.

A. Khan and T. Aziz, "The Numerical Solution of Third Order Boundary Value Problems using quintic spline", Applied Mathematics and Computations137 (2003) 253-260.

M.A. Noor, E.A. Al-Said, "Quartic Spline Solutions of Third Order Obstacle Boundary Value Problems", Applied Mathematics and Computations, 153(2004) 307-316.

S. Valarmathi And N. Ramanujam , “A Computational Method For Solving Boundary Value Problems For Third-Order Singularly Perturbed Ordinary Differential Equations”, Appl. Math. And Comput. Vol. 129, Pp. 345-373, 2002.

R.E. Bellman and Kalaba, Quasilinearization And Nonlinear Boundary Value Problems, American Elsevier Publishing Company, Inc, New York, 1965.

Ahlberg, J. H., Nilson, E. N. And Walsh, J. H. Fundamental Properties Of Generalized Splines, Proc. Nat. Acad. Sci. U. S., 52, 1412-1419.

Berdyshev V.I.; Subbopim Ju .N: Numerical Methods of Approximation of Functions (Russian). Sverdlovsk. Ural.Publ.House, 1979, 118pp

All rights reserved by www.ijirst.org

320

A Numerical Approach for Solving Nonlinear Boundary Value Problems in Finite Domain using Spline Collocation Method (IJIRST/ Volume 3 / Issue 01/ 053) [9] [10] [11] [12] [13] [14]

Doctor, H. D. And Kalthia, N. L.,: Spline Approximations Of Boundary Value Problems, Presented At 49th Annual Conf. Of Indian Mathematical Society, 1983.

Doctor, H. D., Bulsari, A. B. And Kalthia, N. L.,: Spline Collocation Approach To Boundary Value 325 Dr. Grewal B.S.:- Numerical Methods In Engineering And Science, Fifth Ed, Khanna Publishers.

Gabusin V.N.; Bunykov M.A.: Mironov V.I.: Splines and Numerical Methods Of Approximation.

Anant K. Pathak,; Prof. Harish D. Doctor The Comparative Study Of Collocation Method – Spline, Finite Difference and Finite Element Method for Solving Partial Differential equations.

David J.Jeffrey, “An analytical approach for solving nonlinear boundary value problems in finite domains”.

All rights reserved by www.ijirst.org

321