JOURNAL OF
COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 67 (1996) 271-276
Additive parameters methods for the numerical integration of
y" =f(t,y,y')
A. Sesappa Rai a, U. A n a n t h a k r i s h n a i a h b'*
aDepartment of Mathematics, Manipal Institute of Technology, Manipal 576119, Karnataka State, India b Department of Mathematics, Karnataka Regional Engineering College, Surathkal 574157, Karnataka State, India
Received 8 April 1994; revised 10 November 1994
Abstract
In this paper numerical methods for the initial value problems of general second order differential equations are derived. The methods depend upon the parameters p and q which are the new additional values of the coefficients of y' and y in the given differential equation. Here, we report a new two step fourth order method. As p tends to zero and q ~> (2~/h) 2 the method is absolutely stable. Numerical results are presented for Bessel's, Legendre's and general second order differential equations.
Keywords: General second order initial value problems; Additive parameters; Absolutely stable AMS classification: 65L05
1. Introduction
We consider the general second order differential equation
y" = f ( t , y , y ' ) (1.1)
subject to the initial conditions
y(to) = Yo, y'(to) = Y'o (1.2)
For finding the numerical solution of (1.1) by finite differences, we can either reduce (1.1) to a system of two first order differential equations and then apply the standard methods for first order equations or alternatively, have direct methods without reducing (1.1) into a system. Various single step direct methods have been proposed in the literature, the most notable among them being the classical Runge-Kutta-Nystr6m method, the Hubolt method, the Wilson 0-method and the
* Corresponding author. E-mail: [email protected].
0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 0 4 2 7 ( 9 4 ) 0 0 1 2 7 - 8
2 7 2 A. Sesappa Rai, U. Ananthakrishnaiah /Journal of Computational and Applied Mathematics 67 (1996) 271-276 N e w m a r k average acceleration m e t h o d which are a few linear multistep m e t h o d s used by struc- tural engineers in the analysis of structure (see [3]).
The additive parameters m e t h o d s are the numerical m e t h o d s which contain arbitrary para- meters p a n d q. T h e m e t h o d s d e p e n d u p o n the parameters p a n d q which are the new additional values of the coefficients of y' a n d y. As p tends to zero a n d q/> (2re/h) 2, the m e t h o d is absolutely stable when applied to the test e q u a t i o n
y" + py' + qy = 0. (1.3)
The m e t h o d has been applied to solve Bessels, Legendre's and general second order differential equations. The coefficients of ~ , + 1, 4 . and 4 , _ 1 are in terms of the parameters p a n d q.
2. D e r i v a t i o n o f the m e t h o d s
We write (1.1) in the form
y" + py' + qy = 4 ~ ( t , y , y ' ) , (2.1)
where
4 ( t , y , y ' ) = f ( t , y , y ' ) + py' + qy (2.2)
with p > 0, q > 0 a n d p2 _ 4q < 0.
Eq. (2.1) can be written as
y" + py' + qy = g(t), (2.3)
where g(t) is an a p p r o x i m a t i o n to ~(t, y, y').
T h e general solution of (2.3) is
y(t) = A e ~'t + B e ~'~t + - - (e ~'~t-z) - e~'"-zJ)O(z)dz, (2.4)
0"2 - - 0-1 .
where 0-1 = u + iv a n d 0-2 = U - iv are the complex roots of the characteristic e q u a t i o n
m 2 -4- prn + q = 0; where u = - p / 2 and v = x/(4q - p:)/2; A and B are the arbitrary constants.
Differentiating (2.4) with respect to t, we have
y'(t) = a x A e ~'' + a 2 B e ~ t -~ (0-ze ~ t - z ) - a l e ° ' ~ t - z ) ) 9 ( z ) d z . (2.5) 172 -- 0-1 .
Eliminating A a n d B by substituting t = t , + l , t. a n d t . - 1 in (2.4) and (2.5) we obtain y(t,+ 1) - (e `~h + e " h ) Y ( t . ) + e - V n y ( t . - 1)
_ 1 ft.+, (e~"+l-z) _ e~'~"+'-'))(O(z) + e-PhO(2t. _ z))dz, (2.6)
0 " 2 - - 0-1 !,~t.
y'(t.+ O - (e `~h + e " h ) y ' ( t . ) + e - p h y ' ( t . - 1 ) 1 f t t"+~
_ (0-ze,,~,.+,-z) _ 0-1e,,,.+l-~))(9(z) + e-Pho(2t. _ Z ) ) d z . (2.7)
0 - 2 - - o"1 .
A. Sesappa Rai, U. Ananthakrishnaiah / Journal of Computational and Applied Mathematics 67 (1996) 271-276 273 I m p l i c i t M u i t i s t e p M e t h o d
W e replace ~(t, y, y') with the N e w t o n B a c k w a r d difference i n t e r p o l a t i o n p o l y n o m i a l of degree k at the p o i n t s t,+~, t,, t~_a, ..., t,-k+~
(t - t n + 1)
O(t) = 4'.+ t + h 174'.+ 1 + ( t - - t . + O ( t - t.) V24'.+ 1 + ...
2!h 2 + (t - t . + O ( t -- t . ) . . . ( t - - t . - k + 2 ) 17k4'.+1
k ! h k
(t -- tn+ 1)(t -- t n ) " ' " ( t - - t n - k + 1) (2.8) + (k + 1)! 4'~k+,~((); t. < ( < t.+ l-
N e g l e c t i n g the error t e r m in (2.8) a n d s u b s t i t u t i n g 9(t) in (2.6) we o b t a i n Y.+ I - ( e°'h + e~2h)yn + e-PhYn- t
f;
[e~21t-s)h e~,lt-s~h] y. ( _ 1)m[(:) (:)
S1
- - - - - - "}- e - p h Vm4'.+ ads, (2.9)
17"1 0"2 m=O
w h e r e (t - t . ) / h = s.
W h e n m = 2 this c a n be written in the f o r m
Y.+ I - - ( ea'h + ea2h)Yn + e - V h Y . - t = ao4'.+ t + at4'. + a 2 4 ' n - 1 , (2.10) w h e r e 4'. = py'. + qy. + f ( t . , y . , y ' . ) .
T h e coefficients ao, at a n d a2 can be d e t e r m i n e d using the u n d e t e r m i n e d coefficients m e t h o d . P u t t i n g y . = 1, y. = t., y. = t. z a n d t. = 0, t 2 = 0, we get
ao + a l + a2 = - [ I - (e *'n + e ~ h ) + e - P h ] , 1 q
(p + qh)ao + pal + (p - qh)a2 = h(1 - - e - P h ) ,
(2hp + 2 + qh2)ao + 2at + ( - 2 h p + 2 + qh2)a2 = h2(1 + e-Ph).
Solving ao, al a n d a2 we o b t a i n
ao = ~ 1 [2(1 - - ( e alh + e °2h) + e - P h ) ( p 2 - - q)
- - pqh(3 - - ( e "~h + e a 2 h ) - - e - p h ) d - 2 q 2 h 2 ] , at = 2qah 2 [4(1 - (e *'h + e ~h) + e-Ph)(p 2 - q) 1
- 4pqh(1 - e - ' h ) + 2q2hZ(e ~'h + e'~h)], a2 -- 2qah 2 [2(1 - (e `'h + e *~h) + 1 e-Ph)(p 2 -- q)
(2.11a) (2.11b) (2.11c)
-- pqh(1 + (e ~'h + e ~2h) - 3e -"h) + 2q2h2e-ph].
274 A. Sesappa RaL U. Ananthakrishnaiah/Journal of Computational and Applied Mathematics 67 (1996) 271-276 Similarly, replace g(t) using N e w t o n B a c k w a r d difference p o l y n o m i a l (2.8) in (2.7) a n d neglecting the error term we get
Y'.+I -- ( ealh + e~2h)Y'. + e-phy'.-1
,,
fl E(:s) (,:) ]
-- __ [0-2ea2(1-s) h _ f f l e ~ l ( 1 - s ) hI ~ ( - - 1 ) m q- e - p h I7mdpn+ldS.
0"1 0"2 m=O
(2.12) W h e n m = 2 using multistep implicit f o r m this can be written as
Y'.+ I - (e ~'h + e~'h)Y'. + e-phy'.-1 = boq~.+ 1 -k- b 1 ¢ . + b 2 ¢ . - a . (2.13) The coefficients bo, bl a n d b2 can be f o u n d o u t using the u n d e t e r m i n e d coefficients m e t h o d
bo + bl + bz = 0, (2.14a)
(p + qh)bo + p b l + (p - qh)b2 = (1 - (e ~'h + e ~:h) + e-Ph), (2.14b) (2hp + qh 2 + 2)bo + 2b~ + ( - 2 h p + @2 + 2)bz = 2h(1 - e-Ph). (2.14C) Solving bo, ba a n d b2 w e get
bo - 2qEh 2 [(1 - (e aln + e ~h) + 1 e-ph)(qh -- 2p) + 2hq(1 - e - P h ) ] ,
bl - 2q2h 2 [(1 1 - (e ~h d- e a2h) -t- e-Ph)4p -- 4hq(1 - e - P h ) ] ,
b2 - 2q2h 2 [(1 - (e ~lh + e ~h) + 1 e-ph)(qh + 2p) -- 2hq(1 - e - P h ) ] .
3. Truncation error and order of the methods
The linear difference o p e r a t o r L is defined b y
L [ y ( t . ) , h ] = Coy(t.) + C l h y ' ( t . ) + C2h~y"(t.) + ...
W e get Co = C1 = C2 = Ca = C4 = C5 = 0 a n d C6 = (49/23232) w h e n p = 0 a n d q = (2re/h)2;
49 h6y6(t. ) + O(hS). (3.1)
t r u n c a t i o n error = 23232
4. Stability
A p p l y i n g the m e t h o d to the test e q u a t i o n (1.3), we o b t a i n
Y.+I - - ( ealh + ea~h)Y. + e t ~ l + ~ ) h Y n - 1 = 0. (4.1)
A. Sesappa Rai, U. Ananthakrishnaiah /Journal of Computational and Applied Mathematics 67 (1996) 271-276 275
Definition. The linear multistep m e t h o d is said to be absolutely stable if the roots of the character- istic equation is of m o d u l u s less t h a n one for all values of the step length h.
The characteristic root ~ of the test equation (1.3) is given by
32 - - (e ~rlh + e~2h)¢ + e (al+ad)h = 0 .
Since at and o'2 are complex roots (i.e., aa, 2 = u _ iv) we obtain
¢2 __ 2 e u h ( c o s v h ) ~ + e2uh : O.
F r o m this, we find the roots e uh : e -phI2 <~ 1.
(4.2)
(4.3)
~1,2 = e u h ( cOSvh +--isinvh) are of modulus less than one since
5. Numerical results
Numerical results are presented for the following initial value problems to illustrate the order, accuracy and implementational aspects of the m e t h o d s (2.10) and (2.13).
Problem 1. Consider Legendre's equation
(1 - t 2 ) y " - 2ty' + n(n + 1)y = 0, (5.1)
when n = 4, y(t) = p4(t) = (35t 4 - 30t 2 + 3)/8.
The problem is solved using the m e t h o d s (2.10) and (2.13). The absolute errors in the numerical solution for the problem (5.1) are tabulated in Table 1.
Problem 2. Consider Bessel's differential equation
t2y '' + ty' + (t 2 - 0.25)y = 0, (5.2)
with y(t) = jl/z(t) = x/(2/rtt)sin t as the exact solution.
The problem is solved using the m e t h o d s (2.10) and (2.13). The a b s o l u t e errors in y(t) are presented in Table 1.
Table 1
h E r r o r at t = 8.0 when p = 0.1 and q = (100~/h) 2
P r o b l e m 1 P r o b l e m 2 P r o b l e m 3
O(h 4) O(h 4) O(h 4)
2 - 3 0.66865323( - 06) 0.69488559( - 09) 0.63807596( - 06)
2 - 4 0.42011379(-07) 0.43222648(-10) 0.42313559(-07)
2 - 5 0.26266207(-08) 0.26905145( - 11) 0.27246756(-08)
2 -6 0.16370905(-09) 0.16792123(-12) 0.17303137(-09)
2 -7 0.18189894(-10) 0.10491608(-13) 0.10857093(-10)
276 A. Sesappa RaL U. Ananthakrishnaiah/Journal of Computational and Applied Mathematics 67 (1996) 271-276 Problem 3. We consider the general second order differential equation
(1 + t)y" + 2y' - (1 + t)y = 0 (5.3)
with exact solution y(t) = et/(1 + t).
The problem is solved using the methods (2.10) and (2.13). The absolute errors in the solution for the problem (5.3) are given in Table 1, which shows that they are of order four.
6. Conclusion
The numerical results presented for linear problems show that the methods are of order four and absolutely stable when the parameters p and q are chosen as the coefficients of
y'
and y in the given differential equation.Acknowledgements
We are very thankful to the referee for his valuable comments and suggestions.
References
[1] U. Ananthakrishnaiah, Adaptive methods for periodic initial value problems of second order differential equations, J. Comput. Appl. Math. 8 (1982) 101-104.
[2] U. Ananthakrishnaiah, A class of two-step p-stable methods for the accurate integration of second order periodic initial value problems, J. Comput. Appl. Math. 14 (1986) 455-459.
[3] K.J. Bathe and E.L. Wilson, Stability and accuracy of direct integration methods, Earthquake Eng. Struct. Dyn.
1 (1973) 283-291.
[4] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).
[5] M.K. Jain, R.K. Jain and U. Ananthakrishnaiah, P-stable methods for periodic initial value problems for second order differential equations, BIT 19 (1979) 347-355.
[6] J.D. Lambert, Computational Methods in Ordinary Differential Equations (Wiley, London, 1973).
[7] J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, Inst. Maths. Appl.
18 (1976) 189-202.
