A New Block Integrator for the Solution of Initial Value Problems of ...

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), pp. 92-100

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper

A New Block Integrator for the Solution of Initial Value Problems of First-Order Ordinary Differential Equations M.R. Odekunle1, A.O. Adesanya2 and J. Sunday3,* 1, 2 3

Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria

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

* Corresponding author, e-mail: ([email protected], [email protected]) (Received: 1-6-12; Accepted: 14-7-12)

Abstract: We develop a continuous linear multistep method using interpolation and collocation of the approximate solution for the solution of first order ordinary differential equation with a constant step-size. The approximate solution is combination of power series and exponential function. The independent solution was then derived by adopting block integrator. The properties of the method was investigated and found to be zerostable, consistent and convergent. The integrator was tested on numerical examples and found to perform better than the existing methods. Keywords: Integrator, Collocation, Approximate Solution, Zero-Stability, Consistence, Convergence, Power Series, Exponential Function.

1. Introduction: In this article, we propose a new numerical integrator for the solution of first-order ordinary differential equation of the form,

y ' = f ( x, y ), y (a) = η ∀ a ≤ x ≤ b

(1)

where f is continuous within the interval of integration. We assume that f satisfies Lipchitz condition which guarantees the existence and uniqueness of solution of (1). This class of problems arises mainly in the study of dynamical systems and electrical networks. According to Olver (2008) and Kandasamy et al. (2005), equation (1) is used in simulating the growth of populations, trajectory of a particle, simple harmonic motion, deflection of a beam etc. With the advent of the modern high speed electronic digital computers, the numerical integrators have been successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as physics, biophysics, atmospheric sciences and geosciences, see Jain et al. (2007).

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), 92-100.

93

Many numerical integration schemes to generate the numerical solution to problems of the form (1) have been proposed by several authors. The single-step methods include those developed by Fatunla (1986), Aashikpelokhai (1991), Ibijola (1997, 2000), Kama and Ibijola (2000), Ibijola and Ogunrinde (2010), Ibijola and Sunday (2010, 2011), Ibijola, Bamisile and Sunday (2011), to mention a few. These methods were constructed by representing the theoretical solution y ( x) to (1) in the interval,

[ xn , xn+1 ] , n ≥ 0 by linear and

non-linear polynomial interpolating functions.

On the other hand, authors like Butcher (2003), Zarina et al. (2005), Awoyemi et al. (2007), Areo et al. (2011), Ibijola, Skwame and Kumleng (2011) have all proposed linear multistep methods (LMMs) to generate numerical solution to (1). These authors proposed methods in which the approximate solution ranges from power series, Chebychev’s, Lagrange’s and Laguerre’s polynomials. The advantages of LMMs over single step methods have been extensively discussed by Awoyemi (2001). In this paper, we propose a continuous LMM, in which the approximate solution is the combination of power series and exponential functions.

2. Methodology: Construction of the New Block Integrator Interpolation and collocation procedures are used by choosing interpolation point s at a grid point and collocation points r at all points giving rise to ξ = s + r − 1 system of equations whose coefficients are determined by using appropriate procedures. The approximate solution to (1) is taken to be a combination of power series and exponential function given by, 3

4

y ( x) = ∑ a j x + a4 ∑ j

j =0

α jx j

(3)

j!

j =0

with the first derivative given by, 3

4

j =0

j =1

y '( x) = ∑ ja j x j −1 + a4 ∑

α j x j −1 ( j − 1)!

(4)

where a j , α j ∈ ℝ for j = 0(1)4 and y ( x) is continuously differentiable. Let the solution of (1) be sought on the partition π N : a = x0 < x1 < x2 < . . . < xn < xn +1 < . . .< xN = b , of the integration

interval [ a, b ] with a constant step-size h , given by, h = xn +1 − xn , n = 0,1,..., N . Then, substituting (4) in (1) gives, 3

4

j =0

j =1

f ( x, y ) = ∑ ja j x j −1 + a4 ∑

α j x j −1 ( j − 1)!

(5)

Now, interpolating (3) at point xn + s , s = 0 and collocating (5) at points xn + r , r = 0(1)3 , leads to the following system of equations,

 α 2 xn2 α 3 xn3 α 4 xn4  yn = a0 + a1 xn + a x + a x + a4  1 + α xn + + +  2 6 24    α 3 xn2 α 4 xn3  f n = a1 + 2a2 xn + 3a3 xn2 + a4  α + α 2 xn + +  2 6   2 2 n

3 3 n

 α 3 xn2+1 α 4 xn3+1  f n +1 = a1 + 2a2 xn +1 + 3a3 xn2+1 + a4  α + α 2 xn +1 + +  2 6  

(6) (7) (8)

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 α 3 xn2+ 2 α 4 xn3+ 2  + f n + 2 = a1 + 2a2 xn + 2 + 3a3 xn2+ 2 + a4  α + α 2 xn + 2 +  2 6    α 3 xn2+3 α 4 xn3+3  f n +3 = a1 + 2a2 xn + 3 + 3a3 xn2+3 + a4  α + α 2 xn +3 + +  2 6  

94

(9) (10)

Solving (6)-(10), for a j ' s, j = 0(1)4 , using Gaussian elimination method and substituting back into (3) gives a continuous linear multistep method of the form, 3

y ( x ) = α 0 ( x ) yn + h ∑ β j ( x ) f n + j

(11)

j =0

where f n + j = f (( xn + jh), y ( xn + jh)), yn + j = y ( xn + jh) , α 0 , β j are polynomials. The coefficients of yn and f n + j are given as,

   1 4 3 4 3 2 4 2 4  (t − 4t α − 4t + 18t α + 4t − 24tα ) β0 = −  24α 4  1 4 3 4 3 2 4 2 − − + + (3 t 8 t 12 t 24 t 12 t ) β1 = α α  24α 4  1  4 3 4 3 2 4 2 (3t − 4t α − 12t + 6t α + 12t ) β2 = − 4  24α  1 4 3 2  (t − 4t + 4t ) β3 =  24α 4

α0 = 1

(12)

where t = ( x − xn ) h . Evaluating (11) at t = 1, 2,3 gives,

 1    1   1   1  yn+1 = yn + h  f − (2α 4 + 3) fn+2 +  (16α 4 + 3) fn+1 +  (10α 4 −1) fn  4  n +3 4  4  4   24α   24α   24α   24α   (13)

h (14) [ fn+2 + 4 f n+1 + fn ] 3  3    1   9   1  = yn + h  4  f n + 3 −  (54α 4 − 27) f n + 2 +  4  f n +1 +  (18α 4 − 9) f n  (15) 4  4   24α   8α   24α   8α  

yn + 2 = yn + yn + 3

Writing equations (13)-(15) in block form gives,

A(0) Ym = ey n + hdf ( y n ) + hbF ( Ym )

(16)

where Ym = [ yn +1 , yn + 2 , yn + 3 ] , y n = [ yn − 2 , yn −1 , yn ] T

T

F ( Ym ) = [ f n +1 , f n + 2 , f n + 3 ] , f ( y n ) = [ f n − 2 , f n −1 , f n ] , T

T

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), 92-100.

(0)

A

1 0 0 =  0 1 0 ,  0 0 1

 0 0 0 1    e = 0 0 1 , d = 0  0 0 1  0 

 1 4  24α 4 (16α + 3)  4 b= 3   9  8α 4

0 0 0

1 (10α 4 − 1) 4 24α 1 3 1 (18α 4 − 9) 4 24α

95

       

1 1  (2α 4 + 3) 4 24α 24α 4   1 0   3  1 3  4 α (54 27) − 24α 4 8α 4  −

Determination of α α is determined such that equations (13)-(15) will be consistent.

Definition 1 (Consistency) A CLMM (11) is said to be consistent, if it satisfies the following conditions, (i)

the order p ≥ 1 k

(ii)

∑α j =0

(iii)

j

= 0 , and

ρ '(1) = σ (1) , where ρ ( z ) and σ ( z ) are respectively the first and second characteristic polynomials.

3. Remark: Condition (i) is a sufficient condition for the associated block integrator (16) to be consistent (Jator, 2007). From (13),

ρ ( z) = z − 1

(17)

and

σ ( z) =

1 1 1 1 (10α 4 − 1) + (16α 4 + 3) z − (2α 4 + 3) z 2 + z3 4 4 4 4 24α 24α 24α 24α

(18)

Hence,

ρ (1) = 0, ρ '(1) = 1 1 1 1 1 24α 4 4 4 4 (10 − 1) + (16 + 3) − (2 + 3) + = =1 σ (1) = α α α 24α 4 24α 4 24α 4 24α 4 24α 4 Therefore, for any value of α , the integrator will be consistent, i.e. α ∈ ( −∞, ∞) .

(19)

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), 92-100.

96

Analysis of Basic Properties of the New Block Integrator Order of the New Block Integrator Let the linear operator L { y ( x); h} associated with the block (16) be defined as,

L { y ( x); h} = A(0)Ym − eyn − hdf ( yn ) − hbF (Ym )

(20)

expanding using Taylor series and comparing the coefficients of h gives,

L { y ( x); h} = c0 y ( x) + c1hy '( x) + c2 h2 y ''( x) + ... + c p h p y p ( x) + c p +1h p +1 y p +1 ( x) + ...

(21)

Definition 2 The linear operator L and the associated continuous linear multistep method (11) are said to be of order p if c0 = c1 = c2 = ... = c p and c p +1 ≠ 0. c p +1 is called the error constant and the local truncation error is given by,

tn+ k = c p +1h( p +1) y ( p +1) ( xn ) + O(h p + 2 )

(22)

For our method, taking α = 1 gives,

h ( f n +3 − 5 f n + 2 + 19 f n+1 + 9 f n ) 24 h yn+ 2 = yn + ( f n + 2 + 4 f n +1 + f n ) 3 h yn+3 = yn + (9 f n +3 + 27 f n+ 2 + 27 f n+1 + 9 f n ) 24

yn +1 = yn +

Thus,

9  24 1 0 0   yn +1  1  1 L { y ( x); h} = 0 1 0   yn + 2  − 1 [ yn ] − h     3 0 0 1  yn +3  1  9  24

19 5 1 − 24 24 24   f n   f n +1  4 1   0 =0   f n+ 2  3 3  27 27 9   f n +3    24 24 24 

(23)

Expanding (23) in Taylor series gives,

 ∞ ( h) j j 9h ' ∞ h j +1 j +1  19 j 5 1  yn − yn − yn − ∑ yn  (1) − (2) j + (3) j  ∑ 24 24 24  24  j =0 j !  j =0 j ! 0  ∞ (2h) j j    h ' ∞ h j +1 j +1  4 j 1 j  yn − yn − yn − ∑ yn  (1) + (2)  ∑  = 0 3 3 3  j =0 j !  j =0 j !    ∞  0 j j +1 ∞ (3 h ) 3 h h 9 9 3   ∑ ynj − yn − yn' − ∑ ynj +1  (1) j + (2) j + (3) j   8 8 8 8   j =0 j !  j =0 j !

(24)

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97

Hence, c0 = c1 = c2 = c3 = c4 = 0, c5 = [ −2.64( −02) − 1.11( −02) 3.75( −02) ] . Therefore, the new T

block integrator is of order four.

Zero Stability Definition 2 The block integrator (16) is said to be zero-stable, if the roots zs , s = 1, 2,..., k of the first characteristic polynomial ρ ( z ) defined by ρ ( z ) = det( zA (0) − e) satisfies z s ≤ 1 and every root satisfying z s ≤ 1 have multiplicity not exceeding the order of the differential equation. Moreover, as

h → 0, ρ ( z ) = z r − µ ( z − 1) µ where µ is the order of the differential equation, r is the order of the matrices A (0) and e (see Awoyemi et al. (2007) for details). For our new integrator,

1 0 0 0 0 1 ρ ( z ) = det z 0 1 0 − 0 0 1 = 0 0 0 1 0 0 1

(25)

ρ ( z ) = z 2 ( z − 1) = 0, ⇒ z1 = z2 = 0, z3 = 1 . Hence, the new block integrator is zero-stable.

Convergence The new block integrator is convergent by consequence of Dahlquist theorem below. Theorem (Dahlquist, 1956) The necessary and sufficient conditions that a continuous LMM be convergent are that it be consistent and zero-stable.

Numerical Implementations Problem 1 Consider a linear first order IVP,

y ' = − y, y (0) = 1, 0 ≤ x ≤ 2, h = 0.1

(26)

with the exact solution,

y ( x) = e − x

(27)

This problem was solved by Umar and Yahaya (2010). They adopted a block method of order four to solve the problem. We compare the result of our new block integrator (16) (which is of order four) with their result as shown in table 1.

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), 92-100.

98

Table 1: Performance of the New Block Integrator on y ' = − y, y (0) = 1, 0 ≤ x ≤ 1, h = 0.1

x

Exact Solution

0.1

0.9048374180359595 0.9048371857175925

2.3231(-07)

2.5292(-06)

0.2

0.8187307530779818 0.8187306524074074

1.0067(-07)

2.0937(-06)

0.3

0.7408182206817178 0.7408187895625000

3.2505(-07)

2.0079(-06)

0.4

0.6703200460356393 0.6703195798065572

4.6622(-07)

1.6198(-06)

0.5

0.6065306597126334 0.6065303190001389

3.4071(-07)

3.1608(-06)

0.6

0.5488116360940264 0.5488111544782534

4.8161(-07)

2.7294(-06)

0.7

0.4965853037914095 0.4965847405085258

5.6328(-07)

2.5457(-06)

0.8

0.4493289641172216 0.4493285145544428

4.4956(-07)

2.1713(-06)

0.9

0.4065696597405991 0.4065691245561064

5.3518(-07)

3.1008(-06)

1.0

0.3678794411714422 0.3678788624630127

5.7870(-07)

2.7182(-06)

Computed Result of the Error in the New Error in New Block Integrator (16) Block Integrator Mohammed & Yahaya (16), Order 4 (2010), Order 4

Problem 2 Consider a linear first order IVP,

y ' = xy, y (0) = 1, 0 ≤ x ≤ 2, h = 0.1

(28)

with the exact solution,

y ( x) = e

x2 2

(29)

This problem was solved by Badmus and Mishelia (2011). They adopted a self-starting block method of order six to solve the problem. We compare the result of our new block integrator (16) (which is of order four) with their result as shown in table 2. Table 2: Performance of the New Block Integrator on y ' = xy, y (0) = 1, 0 ≤ x ≤ 1, h = 0.1

x

Exact Solution

0.1

1.0050125208594010 1.0050130448437500

Computed Result of the Error in the New Error in New Block Integrator Block Integrator Badmus & Mishelia (16) (16), Order 4 (2010), Order 6 5.2398(-07)

5.29(-07)

Int. J. Pure Appl. Sci. Technol., 11(1) (2012), 92-100.

99

0.2

1.0202013400267558 1.0202015091666667

1.6913(-07)

1.77(-07)

0.3

1.0460278599087169 1.0460287323437500

8.7243(-07)

8.99(-07)

0.4

1.0832870676749586 1.0832900774746079

3.0098(-06)

3.09(-06)

0.5

1.1331484530668263 1.1331581997020368

1.7466(-06)

1.91(-06)

0.6

1.1972173631218102 1.1972215342118766

4.1710(-06)

4.48(-06)

0.7

1.2776213132048868 1.2776309597974733

9.6465(-06)

1.02(-05)

0.8

1.3771277643359572 1.3771345633320713

6.7989(-06)

7.74(-05)

0.9

1.4993025000567670 1.4993154138013334

1.2913(-05)

1.44(-05)

1.0

1.6487212707001286 1.6487478460826666

2.6575(-05)

2.93(-05)

4. Conclusion In this paper, we have proposed a new block numerical integrator for the solution of first-order ordinary differential equations. The block integrator proposed was found to be zero-stable, consistent and convergent. The new integrator was also found to perform better than some existing methods.

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