a generalized scheme for the numerical solution of initial value

International Journal of Pure and Applied Mathematics Volume 88 No. 1 2013, 1-13 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v88i1.1

AP ijpam.eu

A GENERALIZED SCHEME FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS BY THE RECURSIVE FORMULATION OF TAU METHOD K. Issa1 § , R.B. Adeniyi2 1 Department

of Mathematics Kwara State University Malete Ilorin, NIGERIA 2 Department of Mathematics University of Ilorin Ilorin, NIGERIA

Abstract: The generalization of the recursive form of the tau method for both overdetermined and non-overdetermined ordinary differential equations of the initial value type is the main thrust of the work reported here. This will facilitate an automation of this variant of the method and consequently an efficient utilization of the technique. Results from the numerical experiment confirm the validity and effectiveness of the derived scheme. AMS Subject Classification: 34K28 Key Words: Lanczos Tau method, Chebyshev polynomials, initial value problems, Lanczos-Ortiz canonical polynomial, ordinary differential equations 1. Introduction The tau method of Lanczos, first proposed 1938 to solves the linear ordinary Received:

May 3, 2013

§ Correspondence

author

c 2013 Academic Publications, Ltd.

url: www.acadpubl.eu

2

K. Issa, R.B. Adeniyi

differential equations Ly(x) ≡

Nr m X X r=0

Prk x

k

k=0

L∗ y(xrk ) ≡

m−1 X

!

y r (x) =

σ X

f r xr ,

a ≤ x ≤ b,

(1.1)

r=0

ark y (r) (xrk ) = αk ,

k = 1(1)m,

r=0

by seeking an approximant

yn (x) =

n X

ar x r ,

r < +∞,

r=0

of y(x) which is the exact solution of the corresponding perturbed system Lyn (x) =

σ X

fr xr + Hn (x)

(1.2)

r=0

L∗ yn (xrk ) = αk ,

k = 1(1)m

(see [1]-[6], [8]). In (1.1) L is the differential operator, y r (x) is the derivative of order r of y(x),Prk (x), σ, αk , xrk , a and b are real constants and Hn (x) =

m+s−1 X i=0

τi+1 Tn−m+i+1 (x) =

m+s−1 X

τi+1

i=0

n−m+i+1 X

Cr(n−m+i+1) xr

(1.3)

r=0

is the perturbation term. The parameters τi , i = 1(1)(m + s) in (1.3) are to be determined. n X (n) r −1 2x − a − b Cr x = Cos nCos Tn (x) ≡ (1.4) b−a r=0 is the chebyshev polynomial valid in the interval [a, b]. The number s, called the overdetermination number of (1.1), is given by s = max {Nr − r > 0 | 0 ≤ r ≤ m}

(1.5)

In 1956 (see [11]), Lanczos introduced the use of canonical polynomials {Qr (x)}, r = 0, 1, . . . into the tau method. The difficulty involved in their construction was removed by Ortiz in 1969 (see [13]), when he reported a recursive generation of the polynomials, Qr (x) defined by LQr (x) = xr

(1.6)

A GENERALIZED SCHEME FOR THE NUMERICAL...

3

2. Generalization of Lanczos-Ortiz Canonical Polynomials For the purpose of generalizing the tau approximant of the solution of y(x) to equation(1.1), we shall briefly review in this section the work of Yisa (2012), see [14]. We consider canonical polynomials for initial value problem (1.1), for the cases m = 1, 2, . . . , s = 1, 2, . . . and then generalize Qn (x) ∀m and s. Case m = 1, s = 1: From(1.1), we have Ly(x) ≡ (P0,0 + P0,1 x) y(x) + P1,0 + P1,1 x + P1,2 x

2

′

y (x) =

σ X

f r xr

r=0

⇒ Lxr = (P0,0 + rP1,2 ) xr+1 + (P0,0 + rP1,1 ) xr + rP1,0 xr−1 Since LQr (x) = xr , we have Qr+1 (x) =

xr − rP1,0 Qr−1 (x) − (P0,0 + rP1,1 ) Qr (x) , P0,1 + rP1,2

r ≥ 0.

(2.1)

Case m = 1, s = 2: P0,0 + P0,1 x + P0,2 x

⇒ Qr+2 (x) =

2

2

y(x) + P1,0 + P1,1 x + P1,2 x + P1,3 x

3

′

y (x) =

σ X

f r xr

r=0

xr − rP1,0 Qr−1 (x) − (P0,0 + rP1,1 ) Qr (x) − (P0,1 + rP1,2 ) Qr+1 (x) . P0,2 + rP1,3

Case m = 1, s = 3: We have Qr+3 (x) = xr − rP1,0 Qr−1 (x) − (P0,0 + rP1,1 ) Qr (x) − (P0,1 + rP1,2 ) Qr+1 (x) − (P0,2 + rP1,3 ) Qr+2 (x) . P0,2 + rP1,4

Generally, for case m = 1, s = s we have Qn (x) =

P xn−s − [ sj=0 (P0,j−1 + (n − s)P1,j ) Qn−s+j−1(x)] P0,s + (n − s)P1,s+1

,

n = r + s, r ≥ 0, (2.2) where P0,−1 = 0 and P0,s + (n − s)P1,s+1 6= 0. Case m = 2, s = 1: From (1.1), we have

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K. Issa, R.B. Adeniyi P2,0 + P2,1 x + P2,2 x2 + P2,3 x3 y ′′ (x) + P1,0 + P1,1 x + P1,2 x2 y ′ (x) σ X f r xr , + (P0,0 + P0,1 x) y(x) = r=0

Qr+1 (x) = xr − [(P0,0 + rP1,1 + r(r − 1)P2,2 ) Qr (x) + (rP1,0 + r(r − 1)P2,1 ) Qr−1 (x) + r(r − 1)P2,0 Qr−2 (x)] . P0,1 + rP1,2 + r(r − 1)P2,3

Case m = 2, s = 2: For this case we obtained Qr+2 (x) =

1 (xr − [(r(r − 1)P2,3 + rP1,2 + P0,1 )Qr+1 (x) P0,2 + rP1,3 + r(r − 1)P2,4 + (P0,0 + rP1,1 + r(r − 1)P2,2 )Qr (x) + (rP1,0 + r(r − 1)P2,1 )Qr−1 (x) + r(r − 1)P2,0 )Qr−2 (x)]).

Generally, for case m = 2, s = s We have P xn−s − [ sj=0 (P0,j−1 + rP1,j + r(r − 1)P2,j+1 ) Qr+j−1 (x)] , Qn (x) = P0,s + rP1,s+1 + r(r − 1)P2,s+2 n = r + s, r ≥ 0 (2.3) where P0,−1 = 0 and P0,s + rP1,s+1 + r(r − 1)P2,s+2 6= 0. Case m = m, s = s: Continuing the process above, then derived the general canonical polynomial for both overdetermined and non-overdetermined ordinary differential equations (see Yisa [14]). Qn (x) = 1 Pm

k=0 k!

n−s k

Pk,k+s

 



 m X n − s  j! Pj,j−k  Qn−s−k (x) xn−s −   j k=1 j=k    m s−1 X  X n − s  j! Pj,j+k  Qn−s+k (x) . (2.4) +  j 

k=0

m X

j=0

Theorem 1. (see Yisa [14]) If Ly(x) ≡

m Nr X X r=0

k=0

Prk xk

!

y r (x) =

σ X r=0

fr xr = f (x)


5

where Pk (x) is a k-th degree polynomials, y r (x) is the r-th derivative of y(x) and f (x) is a polynomial of degree σ,is an m-th order ordinary differential equation, then the canonical polynomial associated with its differential operator ! Nr m X X dr L= Prk xk dxr r=0 k=0

is Qn (x) = 1 Pm

k=0 k!

n−s k

Pk,k+s

 





 m X n − s  xn−s −  j! Pj,j−k  Qn−s−k (x)  j k=1 j=k    m s−1 X  X n−s  j! Pj,j+k  Qn−s+k (x) . +  j m X

k=0

j=0

3. Derivation of Recursive Formulation of the Tau Approximant In this section, the tau approximant for the recursive form using the generalized canonical polynomial is considered for varying orders and degrees. From the perturbed equation(1.2) Lyn (x) =

σ X

fr xr + Hn (x),

r=0

L∗ yn (xrk ) = αk , k = 1(1)m, where Hn (x) =

m+s−1 X

τi+1 Tn−m+i+1 (x) =

m+s−1 X

τi+1

r=0

i=0

i=0

n−m+i+1 X

is the perturbation term. Case m = 1, s = 1: We have Lyn (x) =

σ X r=0

f r xr +

1 X i=0

τi+1

n+i X r=0

Cr(n+i) xr

Cr(n−m+i+1) xr

6


=

σ X

fr LQr (x) +

r=0

=L

σ X

τi+1

fr Qr (x) +

n+i X

1 X

τi+1

n X

n+i X

)

Cr(n+i) Qr (x)

r=0

i=0

fr Qr (x) + τ1

Cr(n+i) LQr (x)

r=0

i=0

( σ X r=0

⇒ yn (x) =

1 X

Cr(n) Qr (x) + τ2

Cr(n+1) Qr (x).

r=0

r=0

r=0

n+1 X

Now, inserting (2.4); expand the resulting equation and equate the coefficient of undetermined canonical polynomial (i.e Q0 (x)) to zero, to obtain τ1

n X

Cr(n) Pr + τ2

n+1 X

Cr(n+1) Pr +

fr Pr = 0.

(3.1)

r=0

r=0

r=0

σ X

Take P0 = 1. Equation (3.1) is the coefficient of undetermined canonical polynomial (i.e Q0 (x)) and for ease generalization we assume Q0 (x) = 1, to obtain yn (x) =

σ X

fr qr (x) +

r=1

1 X

τi+1

n+i X

Cr(n+i) qr (x),

(3.2)

r=1

i=0

where qn = Qn (x) − Pn and Pn =

−P0,0 Pn−1 − (n − 1)P1,0 Pn−2 − (n − 1)P1,1 Pn−1 . P0,1 + (n − 1)P1,2

(3.3)

Case m = 1, s = 2: From (1.2), we have yn (x) =

σ X

fr Qr (x) + τ1

r=0

n X

Cr(n) Qr (x)

+ τ2

r=0

n+1 X

Cr(n+1) Qr (x)

r=0

+ τ3

n+2 X

Cr(n+2) Qr (x).

r=0

Again, insert (2.4); expand the resulting equation and equate the coefficient of undetermined canonical polynomials (i.e Q0 (x), Q1 (x)) to zero, to obtain τ1

n X r=0

Cr(n) Pr + τ2

n+1 X r=0

Cr(n+1) Pr + τ3

n+2 X r=0

Cr(n+2) Pr +

σ X r=0

fr Pr = 0.

(3.4)


7

Take P0 = 1, P1 = 0 for coefficient of Q0 (x), for coefficient of Q1 (x) take P0 = 0, P1 = 1 and for yn (x) we assume Q0 (x) = Q1 (x) = 1,and take P0 = P1 = 1, to obtain n+i 2 σ X X X Cr(n+i) qr (x), (3.5) τi+1 fr qr (x) + yn (x) = r=2

r=2

i=0

where

Pn = −(n − 2)P1,0 Pn−3 − (P0,0 + (n − 2)P1,1 ) Pn−2 − (P0,1 + (n − 2)P1,2 ) Pn−1 , P0,2 + (n − 2)P1,3 and qn = Qn (x) − Pn . Case m = 2, s = 1: Insert (2.4) in (1.2); expand the resulting equation and equate the coefficient of undetermined canonical polynomial (i.e Q0 (x)) to zero, to obtain τ1

n−1 X

Cr(n−1) Pr + τ2

r=0

n X

n+1 X

Cr(n) Pr + τ3

r=0

Cr(n+1) Pr +

σ X

fr Pr = 0.

(3.6)

r=0

r=0

Take P0 = 1 for the coefficient of Q0 (x), and for yn (x) we assume Q0 (x) = P0 = 1, then we obtain the solution yn (x) =

σ X

fr qr (x) +

r=1

and yn′ (x)

=

σ X r=1

2 X

τi+1

+

2 X

Cr(n−1+i) qr (x)

(3.7)

Cr(n−1+i) Q′r (x).

(3.8)

r=1

i=0

fr Q′r (x)

n−1+i X

τi+1

i=0

n+i X r=1

Case m = 2, s = 2: Insert (2.4) in (1.2); expand the resulting equation and equate the coefficient of undetermined canonical polynomials (i.e Q0 (x) and Q1 (x)) to zero to obtain τ1

n−1 X r=0

Cr(n−1) Pr + τ2

n X r=0

Cr(n) Pr + τ3

n+1 X r=0

Cr(n+1) Pr + τ4

n X

Cr(n+2) Pr

r=0

+

σ X r=0

fr Pr = 0

8


for the coefficient of Q0 (x), take P0 = 1 and P1 = 0, and for the coefficient of Q1 (x), take P1 = 1, P0 = 0 and for yn (x) we assume Q0 (x) = Q1 (x) = P0 = P1 = 1, then we obtain the solution yn (x) =

σ X

fr qr (x) +

r=2

and yn′ (x) =

σ X

3 X

τi+1

r=2

3 X

Cr(n−1+i) qr (x)

(3.7)

r=1

i=0

fr Q′r (x) +

n−1+i X

τi+1

n+i X

Cr(n−1+i) Q′r (x).

r=1

i=0

Observing the trend in the tau approximant of the first and second order ordinary differential equations presented above, we derived the general formula yn (x) =

σ X

fr qr (x) +

r=s

m+s−1 X

τi+1

n−m+i+1 X

Cr(n−m+i+1) qr (x).

(3.9)

r=s

i=0

Assume Qr (x) = Pr = 1, r = 0(1)(s − 1) ynλ (x) =

σ X

fr Qλr (x) +

r=s

m+s−1 X

τi+1

i=0

τi+1

Cr(n−m+i+1) Qλr (x) = αλ ,

r=s

i=0

m+s−1 X

n−m+i+1 X

n−m+i+1 X

Cr(n−m+i+1) Pr

λ = 0(1)(m − 1), (3.10) +

σ X

fr Pr = 0,

(3.11)

r=0

r=0

where qn (x) = Qn (x)−Pn and equation (3.11) is the coefficient of undetermined canonical polynomials.    m m  X X −1 n−s  Pn = Pm j! Pj,j−k  Pn−s−k n−s  j k! P k,k+s k=0 k k=1 j=k    s−1 X m  X n − s  j! + Pj,j+k  Pn−s+k . (3.12)  j k=0

j=0

Take Pr = 1, when equating the coefficient of Qr (x) to zero, otherwise Pr = 0, r = 0(1)(s − 1). The formula also valid for non-overdetermined differential equation, by replacing s = 0 in (2.4),Pr = 0, ∀r in (3.9) and neglect (3.11) and (3.12).


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4. Numerical Experiments In this section we present some numerical experiments to illustrate the numerimax

cal results presented in this paper. The exact errors are defined as εn =a ≤ x ≤ b {|y(xk − yn (xk |}, where {xk } = (0.01k), k = 0(1)100. Problem 1. Ly(x) = y ′ (x) + 2xy(x) = 4x, y(0) = 3, with analytical solution y(x) = 2 + exp(−x2 ). Comparing with (1.2), P0,0 = 0, P0,1 = 2, P1,0 = 1, P1,1 = P1,2 = 0, f0 = 0, f1 = 4, σ = 1, m = 1, s = 1. For degree 5 (i.e n = 5), applying equations (3.9) and (3.11), we have y5 (x) = 3 +

3356272x2 29952x3 2100992x4 779264x5 164x − − + − . 3360543 3360543 1120181 3360543 3360543

For degree 6 (i.e n = 6), we have y6 (x) = 3 −

35308x 159018016x2 10087168x3 112491776x4 − − + 160301129 160301129 160301129 160301129 48468992x5 3788800x6 − + . 160301129 160301129

For degree 7 (i.e n = 7), we have y7 (x) = 3 −

56902112x2 210560x3 29105920x4 560x − − + 56926267 56926267 56926267 56926267 403456x5 11161600x6 3588096x7 − − + . 56926267 56926267 56926267

Problem 2. Ly(x) = y ′ (x) − x2 y(x) = 0,

y(0) = 1,

with analytical solution y(x) = exp( 13 x3 ). For degree 5, 6 and 7 approximate solution, we have y5 (x) = 1 +

1012427712x2 13794322304x3 9264926976x4 42887122x − + − 26442910625 26442910625 26442910625 26442910625 6901011968x5 , + 26442910625

10


y6 (x) = 1 −

5102898720x 172910611296x2 795917236736x3 + + 20423863454407 20423863454407 2917694779201 4385551601664x5 3231878008832x6 3494579851008x4 − + , + 20423863454407 20423863454407 20423863454407

54229310046x 2553117942456x2 81717617906288x3 − + 1148232742907190 1148232742907190 229646548581439 44581151086848x5 165543248190464x6 108270560916864x4 + − − 1148232742907190 229646548581439 1148232742907190 99073874491392x7 + . 1148232742907190

y7 (x) = 1 +

respectively. Problem 3. Ly(x) = y ′′ (x) − 2(1 + 2x2 )y(x) = 0,

y(0) = 1,

y ′ (0) = 0,

with analytical solution y(x) = exp(x2 ). For degree 5, 6 and 7 approximate solution, we have y5 (x) = 1 +

y6 (x) = 1 +

y7 (x) = 1 +

15314739712x3 22091646208x4 19855060080x2 + − 21441580663 21441580663 21441580663 23992840192x5 + . 21441580663 14339349817120x2 3116501231104x3 54597872159488x4 − + 14137226504329 14137226504329 42411679512987 31339103866880x6 46782632538112x5 + . − 42411679512987 42411679512987 28873617687738500x2 1809682067903230x3 + 28953089655559100 28953089655559100 22820567800864700x5 4638442954437370x4 + + 28953089655559100 28953089655559100 20305651810889700x6 11916952869830600x7 − + , 28953089655559100 28953089655559100

respectively. For experiment 4 and 5, we consider non-overdetermined type.


11

Problem 4. Ly(x) = 2(1 + x)y ′ (x) + y(x) = 0,

y(0) = 1,

0 ≤ x ≤ 1,

−1

with analytical solution y(x) = (1 + x) 2 , P0,0 = 1, P1,0 = 2, P1,1 = 2, m = 1, f0 = 0. For degree 5, 6 and 7 approximate solution, we have y5 (x) = 1 −

y6 (x) = 1−

88160x3 44800x4 10752x5 169546x 124272x2 + − + − , 339323 339323 339323 339323 339323

59392456x 44274072x2 34792960x3 23717120x4 645120x5 + − + − 118799927 118799927 118799927 118799927 6988231 2365440x6 + , 118799927

9239102x 6918816x2 5653568x3 4442368x4 14224896x5 + − + − 18478633 18478633 18478633 18478633 92393165 6 5992448x 1171456x7 + − . 92393165 92393165 Problem 5.

y7 (x) = 1 −

y ′′ (x) − (1 − x)y ′ (x) + y(x) = 0,

y(0) = 1,

y ′ (0) = 1,

0 ≤ x ≤ 1,

2

with analytical solution exp(x − x2 ). For degree 5, 6 and 7 approximate solution, we have y5 (x) = 1 + x +

y6 (x) = 1 + x −

2168x2 3315848x3 815488x4 675584x5 − − + , 3272391 9817173 9817173 9817173

230528x2 504319192x3 173802752x4 149575936x5 − − + 515794089 1547382267 1547382267 1547382267 51200x6 , − 5546173

57032x2 42921645520x3 10673371904x4 − − 42919912639 128759737917 128759737917 5 1202865152x6 455455744x7 6122414848x + − . + 128759737917 42919912639 42919912639 Below are the maximum errors obtain for each experiment and degree.

y7 (x) = 1 + x −

12

K. Issa, R.B. Adeniyi Experiment Problem 1 Problem 2 Problem 3 Problem 4 Problem 5

ε5 3.57 × 10−5 2.07 × 10−4 1.06 × 10−2 3.08 × 10−5 9.42 × 10−5

ε6 1.97 × 10−5 2.96 × 10−5 1.23 × 10−3 4.24 × 10−6 2.62 × 10−5

ε7 1.10 × 10−6 4.50 × 10−6 1.40 × 10−4 6.64 × 10−7 1.81 × 10−7

Table 1: Errors of Experiments 1-5 5. Conclusion The generalized form of the recursive formulation of the tau method for initial value problems in ordinary differential equations for both overdetermined and non-overdetermined types has been presented. Evidently, the overdetermined type is more cumbersome to derive than for the case of non-overdetermined type. This study will facilitate an automation of this variant of the tau method towards a software development of the technique. The derived scheme has been implemented on some selected test problems. Numerical evidences therefrom confirm its accuracy and effectiveness.

References [1] R.B. Adeniyi, A.I. Ma’ali, An error estimation of the tau method for some ordinary differential equations, IOSR J. Math., 2, No. 1 (2012), 32-40. [2] R.B. Adeniyi, A.I.M. Aliyu, On the tau method for a class of nonoverdetermined second order differential equations, Research J. of Appl. Sci., 3, No. 6 (2008), 438-446. [3] R.B. Adeniyi, K. Issa, An analogue of the tau method for ordinary differential equations, Global J. Maths. and Stat., 2 (2010), 161-170. [4] R.B. Adeniyi, E.O. Edungbola, On the tau method for certain overdetermined first order differential equations, J. Nig. Ass. Mathematical Physics Soc., 12 (2008), 399-408. [5] R.B. Adeniyi, E.O. Edungbola, On the recursive formulation of the tau method for class of overdetermined first order equations, Abacus J. Math. Assoc. Nig., 34, No. 2B (2007), 249-261.


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[6] R.B. Adeniyi, B.M. Yisa, On the variants of the tau methods for solutions of IVPs in first ordinary differential systems, Kenya J. of Sci., Series A, 15, No. 1 (2012), 12-26. [7] L. Fox, Numerical Solution of Ordinary and Partial Differential Equations, Pergamen Press Oxford (1968). [8] M.K. El-Daou, A tau method with the perturbed boundary conditions for certain ODEs, Numerical Algorithms, 38 (2005), 31-45. [9] M.K. El-Daou, E.L. Ortiz, Weight subspaces of the tau method and orthogonal collocation, J. Math. Anal. Appl., 326 (2007), 622-631. [10] L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, University Press Oxford (1968). [11] C. Lanczos, Applied Analysis, New Jersey (1956). [12] C. Lanczos, Trigonometric interpolation of empirical and analytic functions, J. Math. and Physics, 17 (1938), 123-199. [13] E.L. Ortiz, The tau method, SIMAJ. Numer. Anal., 6 (1969), 480-492. [14] B.M. Yisa, R.B. Adeniyi, Generalization of canonical polynomials for overdetermined m − th order ordinary differential equations (ODEs), IJERT Journal, 1, No. 6 (2012), 1-15.

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