International Journal of Scientific & Engineering Research, Volume 3, Issue 5, May-2012 1

ISSN 2229-5518

Some Linear Multi-step Methods for the

Initial Value Problems

y 

f (x, y, y) by

Perturbed Collocation

Kwanamu, J. A*. and Odekunle, M. R

*Department of Mathematical Sciences, Adamawa State University, Mubi, Nigeria

Email: jakwanamu@yahoo.com

Department of Mathematics & Computer Science, Modibbo Adama University of Technology Yola, Nigeria

Abstract-Two linear multi-step schemes for the numerical solutions of initial values problems of the type

y 

f (x, y, y) by

perturbed collocation using Legendre and Chebyshev polynomials as our approximating functions in tau methods of solution were developed. The schemes were found to perform very well when compared with existing known schemes and were also found to be stable.

Index Terms- Analytical solution, Canonical polynomials, Chebyshev-tau method, Legendre-tau method, Multi-step methods, Ordinary differential equations, Perturbed collocation

1 INTRODUCTION

Many of the real-life problems arising from the fields such as Engineering, Natural Sciences, Environmental and Social Sciences, Economics and other humanities can be modeled by either linear or non-linear (or quasi-linear) ordinary differential equations ‘’[1], [3], [4], [6]’’. Therefore, this calls for the use of efficient and effective numerical methods, particularly when such models are transformed to appropriate mathematical problems.
In this work, we shall derive some linear multi- step methods (LMM) for initial value problems (IVP) of second order o.d.e. by perturbed
These schemes were found to perform very well
but they were found to be incapable of handling problems of the type (1). Their problem was based on the fact that their method could not deal with the case when the RHS of 1 involves the first derivate. The fact that real-life problems are not limited to equations of the type (2) according to
‘’[2 ], [5 ]’’ we decided to find a way of overcoming this shortcoming so that we could have an algorithm that will handle these two forms of equations to a good degree of accuracy.

2 DERIVATION OF THE SCHEMES

In equation (1), we assume that
i. the equation has unique solution
collocation.
Consider the general form of this type of equation in the form
ii
iii

h xi 1 xi is constant

y(x0 )  y0 , y(xi j )  yi j , y(xi j )  yij , f (xi j , yi j )  f j j

y ( x)  f ( x, y, y)

axb

We then seek a solution of the form

N

y(0)  y0 , y(0)  y0

(1)

y ( x)  a Q ( x)

j 0

(5)

N j j

A special case of this problem is

y  f ( x, y)

axb

(2)
where N is the degree of the polynomial

yN ( x)

y(0)  y0 , y(0)  y0

[7] looked into equation (2) and came up with two schemes (3) and (4) for the solution of this problem

and Qj (x) , j = 0, 1, 2, … are canonical polynomials

[8] generated by the operator

d 2 d

using Legendre and Chebyshev polynomials respectively.

    1

dx 2 dx

(6)

yi  2  2 yi 1 yi

and

2

fi  2  4 fi 1 fi

6

(3)
Define

Q j (x)  x

yi  2  2 yi 1 yi

2

fi  2  2 fi 1 fi (4)

4

then, we can easily show that the following
recurrence relation holds

j

Qj (x)  x

j( j  1)Qj2 (x)  j(Qj1 (x);

j  0, 1, 2,...(7)

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Suppose N = 2 (for second order differential equations), then (5) can be written as

a1  2a2 xi 1

a  2a x f

fi 1 1T1 ( xi 1 )  2T2 ( xi 1 )

T ( x )  T ( x )

1 2 i

i 1 1 i

2 2 i

y2 (x)  a0 a1 (x  1)  a2 (x

 2x),

xi 1 x xi 1 (8)

a  2a x

f

T

( x

)  T

( x )

(17)
We define the residual function

1 2 i 1

i 1 1 1

i 1i

2 2 i 1

2

RN (x) as : RN (x)  N 1 PN 1 (x)  N PN (x)

N  1

(9)

y2 ( xi 1 )  a0 a1 ( xi 1  1)  a2 ( xi 1  2xi 1 )

which can be approximated by appropriate

y ( x )  a

a ( x

)  a ( x 2  2x )

2 i 0

1 i 1 2 i i

orthogonal polynomial. Substituting (8) in (1) we
have
The polynomials Ti defined in [-1, 1] are first transformed into the interval [xi-1, xi+1] then

a1  2a2 (x  1)  2a2 f (x, y, y)  R2 (x)

2.1 Legendre-tau Approximant

(10)
substituted into (15). The resulting equations are then solved for the unknowns so that (5) can be written as
We substitute (9) in (10) for N =2 to obtain

y ( x

)  4

2  h h 2

y

f  2 f f

(18)

a1  2a2 x f (x, y)  1 P1 (x)  2 P2 (x) (11)

2 i 1

(2  h) i

 2  h

i 1

2(2  h)

i 1 i i

where PN
is Legendre polynomial of degree N.
On replacing i by i + 1 we have
Collocating (11) at

 4   2 

2

2

(19)

y y

h h

y   

x xi 1, x xi

and

x xi 1

and

i  2

(2  h) i 1

 2  h i

f

2(2  h) i  2

fi 1

fi

interpolating (80) at

x xi 1

and

x xi we

Equation (19) is again an implicit discrete formula
obtain the following perturbed collocation
used in solving (1). The error constant of the

4

equations
method is

h

62  h

and it is also of order 2. The

a1  2a2 xi 1 f i 11 P1 ( xi 1 )  2 P2 ( xi 1 )

a1  2a2 xi f i1 1 P1 ( xi i )  2 P2 ( xi )

a1  2a2 xi 1 f i 1 1 P1 ( xi 1 )  2 P2 ( xi 1 )

y ( x )  a a ( x  1)  a ( x 2  2x )

(12)
stability region of the method is [-, 0].

3 NUMERICAL EXPERIMENT

Two problems are used in our experiment.The first problem is of the type (2) while the second is of the type (1).
Problem 1

y ( x )  a

a ( x )  a

( x 2  2x )

y  y;

y(0)  1, y(0)  1, x [0,1], h = 0.1

2 i 0 1 1 2 1 i

The polynomials P1 and P2 defined in [-1, 1] are then transformed into the interval [xi-1, xi+1].
Analytical solution is y = ex
Problem 2
Equations (12) are then solved for ai , i = 0, 1, 2 and

y  y  2 y  sin x,

y(0)  4, y(0)  4, x [0,1], h  0.1

j , j = 1, 2 so that (8) becomes


Analytical solution is y = 25 ex  1 e 2 x  1 cos x

y ( x

)  4

y  (2  h) y

h [ f

 4 f f ]

(13)

6 12 4

2 i 1

(2  h) i

(2  h)

i 1

3(2  h)

i 1

i i 1

h ≠ - 2. Replacing i by i + 1 in (13) we have

y  4 y

 (2  h) y h [

 4  ]

(14)

i 2

(2  h)

i 1


(2  h) 1

f i 2

3(2  h)

f i 1 f i

Equation (14) is an implicit discrete formula used in solving (1). The error constant of the method is

h 4 and it is of order 2. Its region of stability

62  h

covers [-12, 0].

2.2 Chebyshev-tau Approximant

We define RN(x) in Chebyshev term as

R N (x)  

N 1 x)  x

 

(15)
If N = 2, we can write (8) as
a  2a x f (x, y, y)  τ1T1(x) + τ 2T2(x)

1 2

(16)
Again, collocating (16) at

x xi 1 , x xi

and

x xi 1 we obtain

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TABLE 1

SOLUTION OF PROBLEM (1) USING GENERAL CASES (14) AND (19) AND THE SPECIAL CASES (3) AND (4)

General cases Special cases

X Legendre

(equation 2.6)

Chebyshev

(equation 3.6)

Legendre

(equation 1.3)

Chebyshev

(equation 1.4)

0.0 0.000000000×100 0.000000000×100 0.000000000×100 0.000000000×100

0.1 1.304000000x10-6 1.304000000x10-6 1.304000000x10-6 1.304000000x10-6

0.2 1.054078312x10-2 1.053654149x10-2 2.316000000x10-6 6.769332500x10-6

0.3 1.163983975x10-2 1.163063555x10-2 1.249756167x10-5 2.216197250x10-5

0.4 2.506330703x10-2 2.476434219x10-2 5.887433587x10-4 8.922730200x10-4

0.5 3.596509010x10-3 4.514135785x10-3 1.771619300x10-3 2.677146135x10-3

0.6 0.090041670×100 8.825350582x10-2 3.596509010x10-3 4.514135785x10-3

0.7 1.418157777x10-1 1.383179677x10-1 7.711124430x10-3 8.822160573x10-3

0.8 2.045757502x10-1 1.971799740x10-1 1.515289005x10-2 2.002504479x10-2

0.9 2.76941288x10-1 2.6221680972x10-1 3.087834911x10-2 4.265570100x10-2

1.0 3.578826883x10-1 3.310201873x10-1 5.741798914x10-2 8.143455555x10-2

TABLE 2


ERRORS IN PROBLEM (1) USING GENERAL CASES (14) AND (19) AND THE SPECIAL CASES (3) AND( 4)
General cases Special cases
X Legendre

(equation 2.6)
Chebyshev
(equation 3.6)
Legendre
(equation 1.3)
Chebyshev
(equation 1.4)
0.0 0.000000000×100 0.000000000×100 0.000000000×100 0.000000000×100
0.1 1.304000000x10-6 1.304000000x10-6 1.304000000x10-6 1.304000000x10-6
0.2 1.054078312x10-2 1.053654149x10-2 2.316000000x10-6 6.769332500x10-6
0.3 1.163983975x10-2 1.163063555x10-2 1.249756167x10-5 2.216197250x10-5
0.4 2.506330703x10-2 2.476434219x10-2 5.887433587x10-4 8.922730200x10-4
0.5 3.596509010x10-3 4.514135785x10-3 1.771619300x10-3 2.677146135x10-3
0.6 0.090041670×100 8.825350582x10-2 3.596509010x10-3 4.514135785x10-3
0.7 1.418157777x10-1 1.383179677x10-1 7.711124430x10-3 8.822160573x10-3
0.8 2.045757502x10-1 1.971799740x10-1 1.515289005x10-2 2.002504479x10-2
0.9 2.76941288x10-1 2.6221680972x10-1 3.087834911x10-2 4.265570100x10-2

1.0 3.578826883x10-1 3.310201873x10-1 5.741798914x10-2 8.143455555x10-2

TABLE 3

SOLUTION OF PROBLEM (2) USING GENERAL CASES (14) AND (19) AND THE SPECIAL CASES (3) AND (4)


X

General cases

Special cases

Legendre

(equation 2.6)

Chebyshev

(equation 3.6)

Legendre

(equation 1.3)

Chebyshev

(equation 1.4)

0.0

1.904656430x10-1

1.904656430x10-1

0.000000000×100

0.00000000×100

0.1

3.894125485x10-2

3.344156277x10-2

7.315453222x10-3

7.315453222x10-3

0.2

1.033313860x10-1

8.218303910x10-2

1.025823425x10-2

1.033286126x10-2

0.3

1.875686188x10-6

1.363872455x10-1

9.919233170x10-2

9.989583045x10-2

0.4

2.856462596x10-1

1.859934026x10-1

1.600338640x10-1

1.784071329x10-1

0.5

3.931299927x10-1

2.229072769x10-1

3.109254963x10-1

3.110114269x10-1

0.6

5.067881753x10-1

2.404480508x10-1

4.883858890x10-1

5.234922800x10-1

0.7

6.243019529x10-1

2.329074939x10-1

7.179635114x10-1

7.180717350x10-1

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0.8

7.440382163x10-1

1.951967200x10-1

1.008817549×100

1.008366028×100

0.9

8.648718558x10-1

1.225596309x10-1

1.369320215×100

1.369457881×100

1.0

1.904656430x10-1

1.90465643x10-1

1.8125685339×100

1.812514091×100

TABLE 4

ERRORS IN PROBLEM (2) USING GENERAL CASES (14) AND (19) AND THE SPECIAL CASES (3) AND (4)


X

General cases

Special cases

Legendre

(equation 2.6)

Chebyshev

(equation 3.6)

Legendre

(equation 1.3)

Chebyshev

(equation 1.4)

0.0

1.904656430x10-1

1.904656430x10-1

0.000000000×100

0.00000000×100

0.1

3.894125485x10-2

3.344156277x10-2

7.315453222x10-3

7.315453222x10-3

0.2

1.033313860x10-1

8.218303910x10-2

1.025823425x10-2

1.033286126x10-2

0.3

1.875686188x10-6

1.363872455x10-1

9.919233170x10-2

9.989583045x10-2

0.4

2.856462596x10-1

1.859934026x10-1

1.600338640x10-1

1.784071329x10-1

0.5

3.931299927x10-1

2.229072769x10-1

3.109254963x10-1

3.110114269x10-1

0.6

5.067881753x10-1

2.404480508x10-1

4.883858890x10-1

5.234922800x10-1

0.7

6.243019529x10-1

2.329074939x10-1

7.179635114x10-1

7.180717350x10-1

0.8

7.440382163x10-1

1.951967200x10-1

1.008817549×100

1.008366028×100

0.9

8.648718558x10-1

1.225596309x10-1

1.369320215×100

1.369457881×100

1.0

1.904656430x10-1

1.90465643x10-1

1.8125685339×100

1.812514091×100

4 DISCUSSION OF RESULTS

The errors defined as

en


y( x)  yn

are
illustrated in tables 2 and 4 for problems 1 and 2 respectively. The special schemes (3) and (4) perform better than the general schemes (13) and (14) when applied to solve special second order ordinary differential equation (2). On the contrary, they proved weak to handle the general cases of ordinary differential equation (1) as illustrated in table 1. However, the general schemes (13) and (14) attempted to solve special case (2) of ordinary differential equation averagely and proved much better in handling general case (1) compared with the special schemes (3) and (4). Nevertheless, the special and the general schemes cover same region of stability and same order but differs in error constant.

5 Conclusion

The results of this work as illustrated by problems (1) and (2) show that an algorithm for the solution of initial value problem using the combination of (3) and (14) or (4) and (19) is possible. If the problem is of the type (1), either (3) or (4) is called on to solve the problem but if the problem is of the type (2), either (14) or (19) is called on. The combination of these schemes will act as a veritable tool for the numerical solution of initial value problems.

References

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[2] F. C. Gerald, Applied numerical analysis, (Second

Edition) USA: Adision Wesley Publishing Company.

1980.

[3] B. D. Gupta, Mathematical physics. Jangpura: New

Delhi Publishing House. 1978.

[4] M. K. Jain, Numerical solution of differential equation. (second edition) New Delhi: Willy Eastern. 1987.

[5] E. Kreyszig, Engineering mathematics. London: John

Willy and Sons. 2001.

[6] J. D. Lambert, Numerical method for ordinary differential system. The initial value problem. London: John Willey and Sons.

1991.

[7] J. O. Oladele, P. Onumanyi, & R. O. Ayeni Derivation of linear multi-step methods for the initial value problems for o.d.es by perturbed collocation. Abacus, the Journal of Mathematical Association of Nigeria, 25 (2), 370. 1997.

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