Journal of Applied Sciences & Environmental Management, Vol. 9, No. 3, 2005, pp. 121-124

Superiority of Legendre Polynomials to Chebyshev Polynomial in Solving Ordinary Differential Equation

*¹AKINPELU, FO; ²ADETUNDE, L A;  ³OMIDIORA, E O.

¹ Ladoke Akintola University Of TechnologyDepartment Of Pure And Applied Mathematics Ogbomoso, Oyo State Nigeria.
² University Of AgricultureDepartment Of Mathematical SciencesAbeokuta, Ogun State, Nigeria.
³ Ladoke Akintola University Of Technology,Department of Computer Science & Engineering,Ogbomoso, Oyo State.

Code Number: ja05071 

ABSTRACT:

In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential    equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev polynomial T*_n(X), defined with the help of hypergeometric series T*_n(x) =F ( -n, n, ½ ;X) and later choosing Legendre polynomial P*_n (x) define by the series P*_n (x) = F ( -n, n+1, 1;X); with the help of an auxiliary set of Canonical polynomials Q_k in order to find the superiority between the two polynomials. Numerical examples are given which show the superiority of Legendre polynomials to Chebyshev polynomials. @JASEM

The so-called Canonical polynomials introduced by Lanczos(A) have hitherto been used in application to the Tau method for the solution of ordinary differential equation via Legendre polynomials and Chebyshev polynomials.

In this paper, we described how canonical polynomials can easily be constructed as basis to the solution of first order differential equations. From a computational point of view, the canonical polynomials are attractive, easily generated, uing a simple recursive relation and its associated conditional of the given problem via Legendre and Chebyshev polynomials is of great importance.

The paper of Oritz(B) gives an account of the theory of the Tau method which it subsequently uses in the problems considered to illustrate the effectiveness and superiority of Legendre polynomials to Chebyshev polynomials.

THE METHOD USED

IN THIS SECTION, WE GENERATE CANONICAL POLYNOMIALS FOLLOWING LANCZOS(A), WE DEFINE CANONICAL POLYNOMIALS Q_K(X); K = 0 WHICH ARE UNIQUELY ASSOCIATED WITH THE OPERATOR. CONSIDER A LINEAR DIFFERENTIAL EQUATION

Y'-Y=0, Y(0)=1 .................. 1

GENERATING THE CANONICAL POLYNOMIALS,

L=D/DX - 1

LX^K= KX^K-1 - X^K

THUS,

LX^K - KLQ_K-1(X) - LQ_K(X) - LQ_K(X)=0

FROM THE LINEARITY OF L, AND THE EXISTENCE OFL- ', WE HAVE

Q_K(X) + X^K = KQ _K-1(X)

SINCE DQ_K(X) = X^K

FROM THE BOUNDARY CONDITION

DY(X) = 0 => X^K = 0

Q_K(X) = KQ_K-1(X)

IT FOLLOWS THAT

Q_K(X) = K!S_K(X)

FOR THE DIFFERENTIAL EQUATION CONSIDER

CHEBYSHEV POLYNOMIALS 

WE RECALL SOME WELL-KNOWN PROPERTIES OF THE CHEBYSHEVPOLYNOMIALS:

T_N*(X) = F(-N,N,1/2,X)  T_N*(X) = COS N(COS-1(X),        -1 = X = 1  WHERE X = COS 0.   TO EVALUATE THE FIRST FEW POLYNOMIALS, WE FOLLOW

T₀(X) = T₀(COS 0 ) = 1

T₁(X) = T₁(COS 0) = X

WE NOW MAKE USE OF THE RECURSIVE RELATION

T_N+1(X) = 2XT_N(X) - T_N-1(X)

TO GENERATE OTHERS FOR N=1,2,3,...

LEGENDRE POLYNOMIALS

LEGENDRE POLYNOMIALS P_N*(X), DEFINED BY THE HYPERGEOMETRIC SERIES

P_N*(X) = F(-N, N+1, 1:X) = F(α,β,δ: X)

=> P_N*(X) = 1 +  αβX   +    α(α +1) β(β +1)X²   +  +

δ                    δ (δ+1)

α (α+1)( a+2) ….+( α+N) β (β+1) …( β+N)X

δ (δ+1)( d+2)            +…..+          ( δ+N)

WHERE N=0

P₀^*(X) = 1

WHEN N=1 => P₁^*(X) = 1-2X

WHEN N=2 => 1 - 6X + 12X² ETC.

THE TAU METHOD

ORITZ (B) GIVES AN ACCOUNT OF THE THEORY OF TAU METHOD; SUCH IS APPLIED TO THE FOLLOWING BASIC PROBLEM.

LY(X)=P_M(X)Y^(M)(X)+.......+P₀(X)Y(X) = F(X); A=X=B; Y^M(X) STANDS FOR THE DERIVATIVE OF ORDER M OF Y(X) AND Y(X)=Y_N(X)= S A X = S A Q (X)

WHERE Q_K(X) IS THE CANONICAL POLYNOMIAL. HERE, WE NEED A SMALL PERTURB TERM WHICH LEADS TO THE CHOICE OF CHEBYSHEV POLYNOMIALS WHICH OSCILLATES WITH EQUAL AMPLITUDE IN THE RANGE CONSIDERED.

P_N(X) = tT_N^*(X) WHERE T_N^*(X) IS THE SHIFTED CHEBYSHEV POLYNOMIAL WHICH ARE OFTEN USED WITH THE TAU METHOD AND

_N

T_N^*(X) =∑C_K^NX^K WHERE C_K^N ARE COEFFICIENTS OF X^K. WE

^K=0

ASSUME HERE THAT A TRANSFORMATION HAS BEEN MADE SUCH THAT A=0 AND B=1 TO SIMPLIFY MATTER FURTHER IN ORDER TO GET THE SHIFTED CHEBYSHEV POLYNOMIAL I.E

T₀*(X)=1, T₁^* (X)=X=(1-2q), T₂*(X)=1- 8q + 8q²=1-8X+8X²

RESULT AND DISCUSSION

CONSIDER THE DIFFERENTIAL EQUATION

Y'-Y = 0, Y(0) = 1 ...... 1

WHICH DEFINES THE EXPONENTIAL FUNCTION.

Y(X) = E^X = 1+X+X²/2+X³/3+..2

WHICH CONVERGES IN THE ENTIRE COMPLEX PLAIN. IF WE TRUNCATE THE TAYLOR SERIES

Y_N(X) = 1+X+X²/2!+.........+X^N/N!+.......3

THIS FUNCTION SATISFIES THE DIFFERENTIAL EQUATION

Y'_N-Y_N = X^N/N!    4

SUPPOSE WE ARE SOLVING 1 IN THE RANGE OF (0,1).

NOW BY CHOSING CHEBYSHEV POLYNOMIALS T_N*(X) DEFINED WITH THE HELP OF THE HYPERGEOMETRIC SERIES T_N*(X) = F(-N, N, 1/2; X) AS THE ERROR TERM ON THE RIGHT HAND SIDE OF (1) WE THEREFORE SOLVE THE DIFFERNTIAL EQUATION

Y'_N-Y_N = δT_N*(X) ...............5

BY INTRODUCING CANONICAL POLYNOMIAL QK.QK(X) IS DEFINED BY

Q'_K - Q_K(X) = X^K

=> Q_K(X) = -K!S_K(X) ............. 6

IF WE DENOTE ITS PARTIAL SUM OF THE FIRST K+1 TERMS OF THE TAYLOR SERIES BY SK(X) SUCH THAT

S_K(X) = 1+X+X²/2!+.......+X^N/N!… ………...7   WRITING OUT POLYNOMIALS T*N(X) EXPLICITLY AS

_N

T_N*(X) = C_N°+C_N¹X+C_N²X²+.........+C_N^NX^N =∑C_K^NX^K …….8

^K=0

BY SUPERPOSITION OF LINEAR OPERATION WE HAVE

_N

Y_N(X) = -τΣC_K^NK!S_K(X) ..........9

^K=0

SATISFY THE BOUNDARY CONDITION

Y_N(0) = 1, WILL YIELDS

_N

-          τΣC_K^NK!S_K(0) = 1

^K=0

1

- τ=                           _N

ΣC_K^NK!

^K=0

THE FINAL SOLUTION BECOMES

_N

∑C_K^NK!S_K(X)

Y_N(X) =      ^K=0                                           .....10

_N

∑C_K^NK!

^K=0

WHEN N = 4

T₄^*(X) = 1-32X+160X²-256X³+128X⁴

_N

∑C_K⁴K!S_K(X)

Y₄(X) =                ^K=0

_N

∑ C_K⁴K!

^K=0

WHERE

₄

^∑C_K⁴K!SK(X) =

^K=0

C₀⁴0!S₀(X)+C₁⁴1!S₁(X)+C₂⁴2!S₂(X)+C₃⁴3!S₃(X)+C₄⁴4!S₄(X)

₄

^∑C_K⁴K!SK(X) = C₀⁴0!+C₁⁴1!+C₂⁴2!+C₃⁴3!+C₄⁴4!

^K=0

_N

S_K(X) = 1+X+X²/2!+..........+X^N/N! =   ^∑X^K/K!

^K=0

=>   S₀(X) = 1, S₁(X) = 1+X, S₂(X) = 1+X+X²/2!

S₃(X) = 1+X+X²/2!+X³/3!, S₄(X) = 1+X+X²/2!+X³/3!+X⁴/4!

HENCE

Y₄(X) = 1325+1824X+928X²+256X³+128X⁴                   .11

1825

THE ABOVE SOLUTION LOOKS LIKE WEIGHTED AVERAGE OF THE PARTIAL SUMS S_K(X). THIS WEIGHTING IS VERY EFFICIENT IF X = 1 WE OBTAIN

Y₄(1) = 4961/1825 = 2.718356..12

THE EXACT VALUE

Y₄(1) = E¹ = 2.7182818284          .13

HENCE ERROR = EXACT VALUE - APPROXIMATE VALUE.

ERROR = -7.4*10^-5

WHEREAS THE UNWEIGHTED PARTIAL SUM S₄(1) GIVES

65/24 = 2.70832

WITH ERROR = 1.0 * 10^-2

HERE, WE SEE THE GREAT INCREASED CONVERGENCE THUS OBTAINED.

HOWEVER, THE RANGE (0, 1) IS ACCIDENTAL NOW TESTING WITH ANALYTIC FUNCTIONS WHICH ARE DEFINED AT ALL POINTS OF THE COMPLEX PLANE EXCEPT FOR SINGULAR POINTS. HENCE,OUR AIM WILL BE TO OBTAIN Y(Z) WHERE Z MAY BE CHOSEN AS ANY NON-SINGULAR COMPLEX POINT.

IN VIEW OF THIS, WE CHOOSE OUR ERROR POLYNOMIAL IN THE FORM T_N*(X/Z) AND SOLVE THE GIVEN DIFFERENTIAL EQUATION ALONG THE COMPLEX RAY WHICH CONNECTS THE POINT X=0 WITH THE POINT X=Z. THEN SOLVING THE DIFFERENTIAL EQUATION

DY_N(X) = jT_N*(X/Z) .............. 14

BY CONSIDERING Z MERELY AS A GIVEN CONSTANT, WE FINALLY SUBSTITUTE FOR X THE END-POINT X=Z OF THE RANGE IN WHICH T_N*(X/Z) IS USABLE.

HENCE,

_N

T_N*(X/Z) = ∑C_K^NX^K

^K=0    Z^K                         .............. .15

WE OBTAIN

_N

∑   C_K^NS_K(Z)K!

Y_N(Z) =        ^K=0     Z^K

T_N*(-1/Z)                                                                         .............................. 16

THE PREVIOUS APPROXIMATIONS HAVE NOW TURNED INTO RATIONAL APPROXIMATIONS GIVING THE SUCCESSIVE APPROXIMATES AS THE RATIO OF TWO POLYNOMIALS OF ORDER N.

WHEN N=4, WE HAVE

_N

∑  C_K⁴S_K(Z)K!

Y₄(X) =       ^K=0      Z^K

_N                                                                                 .........17

∑C_K⁴K!

^K=0      Z^K

=  3072+1536Z+320Z²+32Z³+Z⁴

3072-1536Z+320Z²-32Z³+Z⁴          ..17A

NOW REPLACING THE COEFFICIENT C_K^N OF THE CHEBYSHEV POLYNOMIAL BY THE CORRESPONDING COEFFICIENT OF THE LEGENDRE POLYNMIAL P_N* (X) DEFINED THE HYPERGEOMETRIC SERIES P_N* (X) = F(-N, N+1, 1; X)

HENCE

_N

∑   P_K^NS_K(Z)

Y^NP(Z) =       ^K=0       Z^K

_N                                                                  ....18

∑   P_K^N

^K=0    Z^K

WHEN N = 4

_N

∑    P_K⁴S_K(Z)

Y₄^P(Z) =         ^K=0     Z^K

_N                                                                      .18A

∑     P_K⁴

^K=0   Z^K

P₄*(X) = 1-20X+180X²-840X³+1680X⁴

WE NOW HAVE Y₄^P(Z) = 1680+840Z+180Z²+20Z³+Z⁴

1680-840Z+180Z²-20Z³+Z⁴

PUTTING Z = 1, WE OBTAIN

Y₄^P(1) = 2721/1001 = 2.71828172...........19

WHEREAS THE EXACT VALUE = 2.7182818284

HENCE THE ERROR =V = 1.1*10^-7

COMPARING THE RESULT OF CHEBYSHEV WITH LEGENDRE WE DISCOVER THAT LEGENDRE SOLUTION GIVE MUCH CLOSER E-VALUE THEN THE VALUES OBTAINED BY THE CHEBYSHEV WEIGHTING.

IF WE PROCEED BY PUTTING Z = I, WE OBTAIN SUCCESSIVE APPROXIMATIONS OF

E^I = COS1+ISIN1 = 0.54030231+0.84147098I

IN THE CASE N = 4 CONSIDERED

Y₄^P(I) = 1501+820I

1501-820I

= 1580601+2461640I

2925401

Y₄^P(I) = 0.540302338+0.841470964I

ERROR h = -3*10^-8+2*10^-8I

WHEREAS THE WEIGHTING BY CHEBYSHEV COEFFICIENT YIELDS

Y₄^C(I) = 2753+1504I

2753-1504I

=   5316993+8281024I

9841025

Y₄^C(I) = 0.5402885+0.8414798I

ERROR η= 1.4*10^-5-0.9*10^-5I

SEE TABLE 1 FOR SOME NUMERICAL RESULTS FOR THE ERROR ESTIMATES BASED ON THE EXAMPLE 1, WHEN X = 1

EXAMPLE 2

Y'(1+X) = 1, Y(0) = 0.

THE EXACT VALUE (SOLUTION) => Y(X) = LOG(1+X)

=> Y(X) = X-X²/2+X³/3

FOLLOWING THE ILLUSTRATION OF EXAMPLE 1 WE HAVE CANONICAL POLYNOMIAL BECOMES

Q_K(X) = (-1)_K-1S_K(X)

THE PERTURBED TERM BECOMES

Y'(1+X) = 1+jT*N(X)

_N

Y^N(X) = j∑C_K^N(-1)_KS_K(X)

^K=1

WHERE

_N

S_K(X)  =  ∑   (-1)^K+1X^K

^K=0       K

_N

T_N*(X) = ∑C_K^NX^K

^K=0

HENCE WE HAVE THE TABLE FOR THE RESULT OF EXAMPLE

CONCLUSIONS:

The polynomials of legendre and chebyshev has been described. The two method is shown to be accurate efficient and general in application for sufficiently solution y(x) and for tau polynomial approximation y_n(x).

the result obtained in the present work demonstrate the effectiveness and superiority of legendre polynomials to chebyshev polynomials for the solution of order linear differential equation. The variants of the error estimated described the case of reciprocal radii in which the point x = 0 becomes a singular point of our domain legendre polynomial fail to give better value than the chebyshev polynomials even of the end point x = 1. By excluding, however the point x = 0 by defining our range as (e,1) which by a simple linear transformation can then be changed back to the standard range (0,1). The condition that our domain shall contain no singular points is now satisfied.

in the vicinity of singularity p_n*(x) (i.e. the legendre polynomials) gives larger errors than the t_n*(x) (i.e. chebyshev) for small values of n. As n increases, the polynomials p_n*(x) compete with t_n*(x) with increasing accuracy to the t_n*(x) for the purpose of end point approximation. 

Acknowledgement:

The authors wish to thank ayeni e.o and tunde okewale for performing the numerical experiments. The subject matter of this paper has been discussed with numerous individuals and the authors are grateful to them all. The following deserve particular mention: r.o ayeni (lautech), tejumola, dosu ojengbede, omosoju, eegunyomi, and opoola t.o (of university of ilorin)

REFERENCES