based on a local representation of the theoretical solution to the initial value problem by a nonlinear ...... Jain, M. K., Iyengar, S. R. K. and Jain, R. K. 2007.
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3 (2): 211-216 © Scholarlink Research Institute Journals, 2012 (ISSN: 2141-7016) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016)
On the Derivation, Convergence, Consistence and Stability of a New Numerical Integrator 1
M.R. Odekunle and 2J. Sunday
1
Department of Mathematics and Computer Science, Modibbo Adama University of Technology, Yola- Nigeria 2 Department of Mathematical Sciences, Adamawa State University, Mubi- Nigeria Corresponding Author: J. Sunday ___________________________________________________________________________ Abstract For any numerical integrator to be efficient, ingenious and computationally reliable, it is expected that it be convergent, consistent and stable. In this paper, we develop a new numerical integrator which is particularly well suited for solving initial value problems in ordinary differential equations. The algorithm developed is based on a local representation of the theoretical solution to the initial value problem by a nonlinear interpolating function (comprising of the combination of polynomial, exponential and cyclometric functions). We further test whether or not the integrator satisfies the conditions for convergence, consistence and stability. From the analysis presented, it is obvious that the new numerical integrator can provide accurate solution to the original differential equation. __________________________________________________________________________________________ Keywords: numerical integrator, convergence, consistence, stability and initial value problem (IVP) __________________________________________________________________________________________ INTRODUCTION problems of the form (1) having oscillatory or It is a known and documented fact that a given linear exponential solutions. This method was based on the or non-linear equation does not have a complete local representation of the theoretical solution solution that can be expressed in terms of a finite y ( x) to the IVP (1) in the interval xn , xn1 by a number of elementary functions (Ross, 1964; Humi nonlinear polynomial interpolating function and Miller, 1988). It is also a known fact that one of ( x ) the ways to solve such problem is to seek an F ( x) a0 a1 x breale , where approximate solution by means of various a0 , a1 and b are real undetermined coefficients, perturbation methods. It must be stated here that the above procedure will only hold for limited ranges of while and are complex parameters. Other the system parameters and/or the independent schemes include those developed by Ademiluyi variables (Mickens, 1994). As reported in Mickens (1987), Ibijola (1997), Kama and Ibijola (2000), (1994), for arbitrary values of the system parameters Wazwaz (2000), Ibijola and Ogunrinde (2010), at the present time, only numerical integration Ibijola and Sunday (2010), and Ibijola, Bamisile and technique can provide accurate solutions to the Sunday (2011) to mention a few. original differential equations.
Studies have shown that the Interpolants used by the authors above basically consist of the combination of a polynomial and exponential function. Having seen the performance of these schemes, we are motivated and challenged to investigate what happens if a nonlinear Interpolant that consists of the combination of polynomial, exponential and cyclometric (trigonometric) functions is used to derive a new numerical integrator. We shall state without proof, the theorem that guarantees the existence and uniqueness of solution of the IVP (1).
In this paper, we develop a new numerical integrator capable of solving equations of the form,
y ' f ( x, y), y ( x0 ) y0 , x > 0, x a, b (1) and its equivalent system. We further test for its convergence, consistence and stability. Many numerical integration schemes to generate the numerical solutions to problems of the form (1) have been proposed and developed by several authors. Most of these integrators were developed by representing the theoretical solution y( x ) to equation (1) by an interpolating function (linear or nonlinear) f ( x) . This type of construction was first reported in Fatunla (1976). He proposed a numerical integrator which is particularly well suited to solve
Theorem 1 (Lambert, 1973; Fatunla, 1988) Let f ( x , y ) be defined and continuous for all points
( x, y ) 211
in
the
region
D
defined
by
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016)
( x, y) : a x b, y
Definition 4 (Consistence) A numerical scheme with an increment function ( x n , y n ; h) is said to be consistent with the initial
where
a and b finite, and let there exist a constant L
x, y, y such ( x, y ) and ( x , y ) are both in D ;
such
that
for
every
that
value problem (1), if ( x n , y n ;0) f ( x n , y n ). The concept of consistency of one-step method is very crucial in the sense that it controls the magnitude of the local truncation error.
f ( x, y ) f ( x, y ) L y y (2) Then if is any given number, there exist a unique
Definition 5 (Jain et al, 2007) A method is stable, if the cumulative effect of all errors, including round-off errors is bounded, independent of the mesh points, i.e. if for any initial error E0 , there exist constant K and h0 0 such that, when the method is applied to the initial value problem (1) with mesh size h (0, h0 0 , the
solution y (x ) of the initial value problem (1), where
y( x ) is continuous and differentiable for all ( x, y ) in D . The inequality (2) is known as a Lipschitz condition and the constant L as a Lipschitz constant.
Definition 1 (Lambert 1991) A Numerical Scheme or Numerical Method or Numerical Integrator (sometimes shortened to ‘scheme’ or ‘method’ or ‘integrator’) is a difference equation involving a number of consecutive approximations y n j , j 0,1,2, .. . , k from which
ultimate error E n satisfies the following inequality;
E n KE 0 , K
Derivation of the New Numerical Integrator Let us assume that the theoretical solution y ( x) to the initial value problem (1) can be locally
it will be possible to compute sequentially the sequence
y
n
(4)
n 0,1,2, . . . , N . The integer k is
represented in the interval, xn , xn 1 , n 0 by the
called the step number of the scheme. If, k 1 , the method is called one-step, while if k 1 , the method is called a multi-step or k step .
non-linear interpolating function, (5) F ( x) a0 a1 x a2 x 2 a3e x b sin x where a0 , a1 , a2 , a3 , b, and are undetermined
Definition 2 (Henrici 1962) Any method for solving differential equations in which the approximation y n 1 to the solution at the
coefficients. Let yn 1 be the numerical estimate to
point x n 1 can be calculated if only x n , y n , and h are known, is called a ONE-STEP METHOD. It is a common fact to write the functional dependence y n 1 on the quantities x n , y n , and h in the form;
x xn 1 and that f n f ( xn , yn ) . Let also, xn 1 x0 ( n 1) h, n 0,1, 2,... (6) At the points x xn and x xn 1 we require that, ,
the
theoretical
solution
y ( xn 1 ) at the point
F ( xn ) a0 a1 xn a2 xn2 a3e xn b sin xn (7)
(3) y n 1 y n h ( x n , y n ; h ) where ( x n , y n ; h) is called the increment function.
and
F(xn1) a0 a1xn1 a2 xn21 a3exn1 bsin xn1 (8)
Definition 3 (Jain et al, 2007) A method is convergent if, as more grid points are taken or step-size is decreased, the numerical solution converges to the exact solution, in the absence of round-off errors. Alternatively, a numerical scheme is said to be convergent if for all initial value problem (1) satisfying the hypothesis of Lipschitz condition,
If F ( xn ) and F ( xn 1 ) coincide with yn and yn 1 (i)
( x ) denotes the ith total derivative of f ( x, y ) with respect to x , and also respectively and that F
adopt the Maclaurin series expansion for the exponential function, we have, r x nr 2 x n2 3 x n3 e xn 1 xn ... (9) r! 2! 3! r 0 We now substitute the first few terms of equation (9) into (7), this gives, 2x2 3x3 F(xn) yn a0 a1xn a2xn2 a3(1xn n n ) bsin xn (10) 2! 3! Differentiating equation (10) with respect to x at xn gives,
Max y ( x n ) y n 0 as h 0
0 n N
One of the conditions a numerical method must satisfy if it is to be convergent is that, it has to be a sufficiently accurate representation of the differential system (Lambert, 1991).
212
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016) Substituting equations (15), (16) and (17) into equation (11), we have,
3xn2 ) bcos xn fn (11) 2 y '' F(2) (xn ) 2a2 a3 (2 3xn ) bsin xn fn' (12)
y' F'(xn ) a1 2a2xn a3( 2xn
y ''' F (3) ( xn ) a3 3 b cos xn f n2
(13)
y iv F (4) ( xn ) b sin xn f n3
(14)
1 a1 f n f n' ( f n2 f n3 cot ( x n )) x n f n3 x n (18) 2 1 x x ( f n2 f n3 cot ( x n )) 2 n n f n3 cot ( x n ) 2
Using the assumption that
From equation (14),
F ( xn ) yn and F ( xn 1 ) yn 1 , we subtract
f3 b n sin xn
equation (7) from (8), this gives us,
(15)
yn1 yn a1(xn1 xn)a2(xn21 xn2)a3(exn1 exn ) b(sinxn1 sinxn) (19)
Using the fact that,
Substitute equation (15) in (13), we have,
xn1 xn h
1 a3 3 ( f n2 f n3 cot ( xn ))
(20) x x 2xnh h sin(xn1) sin(xn h) sin xn cos(h) cos xn sin(h) 2 n1
(16)
Putting equations (15) and (16) into equation (12), we have, (17) 1 ' 1 a f ( f 2 f 3 co t ( x )) x f3 2
2
n
n
n
n
n
n
2 n
2
So that equation (19) becomes,
yn 1 y n a1h a2 (2 xn h h 2 ) a3 e xn ( e h 1) b (sin xn cos( h) cos xn sin( h)) sin xn Substituting the values of a1 , a2 , a3 , and b in equation (21), we have, ' 1 3 2 3 fn fn ( fn fn cot(xn )) xn fn xn yn1 yn h 2 ( f 2 f 3 cot(x )) 1 xn xn f 3 cot(x ) n n n 2 n n 2
Convergence of the New Numerical Integrator The following theorem, whose proof may be found in Henrici (1962) page 71, states necessary and sufficient conditions for the method (3) to be convergent.
(22)
1 1 fn' ( fn2 fn3 cot(xn )) xn fn3 (2xnh h2 ) 2
Theorem 2 (Lambert 1973, Fatunla 1988) (i) Let the increment function ( xn. y n ; h) be continuous jointly as a function of its three arguments, in the region D defined by
f 3 (sin xn cos(h) 1 3 ( fn2 fn3 cot(xn ))exn (eh 1) n sin xn cos xn sin(h)) sin xn
Note that, (23) fn3 3 (sin xn cos(h) cos xn sin(h)) sin xn fn cos(h) cot(xn )sin(h) 1 sin xn
x a, b ,
We shall now substitute the first few terms of the
y ( , ), h 0, h0 , where h0 > 0 and let there exists a constant M , (ii) Let ( xn. y n ; h) satisfy a Lipschitz
h
series expansion for cos( h),sin( h), and e in equation (24) into (22), (h )2r ( 1) r r0 (2 r )! 2 r 1 ( h ) s in ( h ) ( 1) r ( 2 r 1 ) ! r0 ( h ) r h e (r)! r0
co s(h )
condition of the form, (24)
n
( xn , y ; h) ( xn , yn ; h) M yn yn (26)
for all points ( xn , yn ; h ) and ( xn , yn ; h) in the region D . Then the method (3) is convergent if and only if it is consistent. This is in turn the necessary and sufficient condition for the convergence of the method (25).
Finally, we have our new numerical integrator as, ' 1 3 2 3 fn fn ( fn fn cot(xn )) xn fn xn 1 xn xn2 3 2 3 ( fn fn cot(xn )) 2 fn cot(xn ) 2 ' h 1 3 2 3 fn ( fn fn cot(xn )) xn fn (xn ) yn1 yn h 2 2 1 h h 2 3 xn ( f f cot( x )) e n n n 2 (2!)( ) 3! 2 4 6 h h3 h5 h h h 3 fn cot(xn) 1 3! 5! 7! 2! 4! 6!
(21)
We shall now check whether or not our new numerical integrator satisfies the condition for convergence in equation (26).
(25)
Proof If we expand equation (25), we have the following set of equations,
213
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016) 2 x f 3 cot(xn ) 2 3 ' xn fn xn2 fn2 n n xn fn cot(xn) fn xn fn 2 2 2 2 3 3 fn xn fn xn fn fn cot(xn) xn fn cot(xn) 3 xn fn 2 2 2 (27) 2 3 ' 2 2 xn fn cot(xn) 3 hfn 2 2 ' hfn xn fn f cot( x ) x f x f n n n n n n 2 2 2 2 3 3 x hf x f cot(xn ) hfn cot(xn) 2 3 yn1 yn h n n n n xn fn cot(xn) 2 2 x x x xnhfn3 cot(xn) xn fn3 hfn3 e n fn2 e n hfn2 e n h2 fn2 2 2 2 2 2 6 xn 3 xn 3 xn 2 3 e fn cot(xn ) e hfn cot(xn ) e h fn cot(xn) 2 2 6 3 5 2 4 6 3 h h h 3 h h h fn fn cot(xn )1 6 120 5040 2 24 720
(xn , yn; h) f (xn , yn ) Bf ' (xn , yn ) (C D) f 2 ( xn , yn ) (E F) f 3( xn , yn ) (36)
(xn, yn;h) f (xn, yn)Bf '(xn, yn)(CD) f 2(xn, yn)(EF) f 3(xn, yn) (37)
Thus, we have f (xn,yn)f (xn,yn)Bf '(xn, yn) f ' (xn,yn) (38) (xn,yn;h)(xn, yn;h) 2 2 3 3 (CD)f (xn, yn) f (xn,yn)(EF)f (xn, yn) f (xn, yn)
Let C D G and E F H , then equation (38) reduces to, f (xn, yn) f (xn, yn) B f ' (xn, yn) f '(xn, yn) (39)
(xn, yn;h) (xn, yn;h) 2 2 3 3 G f (xn, yn) f (xn, yn)H f (xn, yn) f (xn, yn)
Let y be defined as point in the interior of the
interval whose end points are y and y , if we apply the Mean Value Theorem, we have
Collecting like terms and simplifying, we have
yn1 yn h fn Bfn' Cfn2 Dfn2 Efn3 Ffn3 (28)
' f ( x , y ) n f ' ( x n , y n ) f ' ( x n , y n ) ( y n y n ) (40) y n 2 f ( x , y ) n f 2 ( x n , y n ) f 2 ( x n , y n ) ( y n y n ) y n 3 f ( xn , y ) 3 3 f ( xn , yn ) f ( x n , y n ) ( y n yn ) y n f ( x n , y n ) f ( x n , y n )
where,
B C
h 2
(29)
x2 x h 1 xn h n n 2 2 2 2
D
(30)
e xn e xn h e xn h 2 2 2 6
(31)
xn h3 h5 2 24 720
(32)
E
f ( xn , y ) ( yn yn ) y n
If we define,
f ( xn , y n ) yn ( xn , yn )D f ' ( xn , yn ) L1 sup yn ( xn , yn )D 2 f ( xn , y n ) L2 sup yn ( xn , yn )D f 3 ( xn , yn ) L3 sup ( xn , yn )D y n L sup
x2 cot(xn) cot(xn ) x cot(xn ) hcot(xn ) F n 2 cot(xn ) n 2 2 xnhcot(xn ) exn cot(xn ) exn hcot(xn ) (33) 2 xn cot(xn ) 2 2 2 h2 h4 h6 exn h2 cot(xn) cot(xn )1 6 6 120 5040 Equation (28) is therefore written as,
yn1 yn h fn Bfn' (C D) fn2 (E F) fn3 (34)
(41)
which is the form (3). The increment function is given by,
(xn , yn ; h) fn Bfn' (C D) fn2 (E F) fn3 (35) From equation (35), If we put these relations in equation (39), we obtain f ( xn , y ) f ' ( xn , y ) f 2 ( xn , y ) ( y n y n ) B ( y n y n ) G ( y n y n ) y y y n n n ( x n , y n ; h ) ( x n , y n ; h ) 3 H f ( xn , y ) ( y y ) n n yn
f ( xn , y n ) f ' ( xn , yn ) sup ( y y ) B sup ( yn yn ) n n y n y n ( x n , y n ) D ( x n , y n ) D ( x n , y n ; h ) ( x n , y n ; h ) 2 3 G sup f ( x n , y n ) ( y y ) H sup f ( x n , y n ) ( y y ) n n n n ( x n , y n ) D y n ( x n , y n ) D y n Therefore, we have, ( x n , y n ; h ) ( x n , y n ; h ) L ( y n y n ) B L1 ( y n y n ) G L 2 ( y n y n ) H L3 ( y n y n ) 214
(42)
(43)
(44)
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016)
(xn, yn;h)(xn, yn;h) LBL1 GL2 HL3 (yn yn) (45)
< , >0. If the two numerical estimates
Taking the absolute value of both sides of (45), we have (46) (xn, yn;h)(xn, yn;h) (LBL1 GL2 HL3) (yn yn)
are generated by the interpolation scheme (3), we have (52) yn 1 yn h ( xn , yn ; h)
Let,
M ( L BL1 GL2 HL3 )
ln 1 ln h ( xn , ln ; h) (47)
yn 1 ln 1 K
then we have, n
n
( xn , y ; h) ( xn , yn ; h) M y yn
(48)
(54)
is the necessary and sufficient conditions that our numerical scheme (25) be stable and convergent.
Therefore, we say that our new numerical integrator (25) is convergent and hence is Lipchitzian.
Proof Let,
Consistency of the New Numerical Integrator According to Fatunla (1988), a numerical scheme with an increment function ( xn , yn ; h ) is said to be consistent with respect to an initial value problem (1) if and only if, (49) ( xn , yn ;0) f ( xn , yn ) From equation (25), ' 2 3 1 3 fn fn ( fn fn cot(xn)) xn fn xn 2 1 x x ( fn2 fn3 cot(xn)) 2 n n fn3 cot(xn) (50) 2 h 1 fn' ( fn2 fn3 cot(xn)) xn fn3(xn ) (xn, yn;h) 2 2 2 h h 2 3 xn 1 ( f f cot( x )) e 2 n n n (2!)() 3! h h3 h5 h2 h4 h6 fn3 cot(xn)1 3! 5! 7! 2! 4! 6! ' 2 3 1 3 fn fn ( fn fn cot(xn)) xn fn xn 1 x x2 ( fn2 fn3 cot(xn)) 2 n n fn3 cot(xn) 2 (51) 1 3 ' 2 3 (xn, yn;0) fn ( fn fn cot(xn)) xn fn (xn) f (xn, yn) 2 3 exn 3 ( fn fn cot(xn)) 2 fn cot(xn)
(53)
The condition that
yn1 yn h f (xn , yn ) Bf ' (xn , yn ) Gf 2 (xn, yn ) Hf 3( xn , yn ) (55)
ln1 ln h f (xn , ln ) Bf ' (xn , ln ) Gf 2 (xn , ln ) Hf 3 (xn , ln ) (56)
Then, f (xn, yn) f (xn,ln) Bf '(xn, yn) f '(xn,ln) (57) yn1 ln1 yn ln h 2 2 3 3 Gf (xn, yn) f (xn,ln)Hf (xn, yn) f (xn,ln)
Applying the Mean Value Theorem as before, with the assumption that y is a point in the interior of the interval whose end points are y and l , we have f (xn, y) f '(xn, y) (yn ln) B (yn ln) yn y (58) yn1 ln1 yn ln h n 2 3 Gf (xn, y) (y l ) H f (xn, y) ( y l ) n n n n yn yn f (xn, yn) f ' (xn, yn) (yn ln) B sup (yn ln) sup (59) (xn,ln)D yn (xn,ln)D yn yn1 ln1 yn ln h 2 3 f ( x , y ) f ( x , y ) G sup n n n n (yn ln) H sup (yn ln) ( xn,ln )D yn (xn,ln)D yn
yn1 ln1 yn ln hL(yn ln)BL1(yn ln)GL2(yn ln)HL3(yn ln) (60) yn1 ln1 yn ln h L BL1 GL2 HL3 ( yn ln ) (61)
Taking the absolute value of both sides of (61) gives, yn1ln1 (yn ln)hL ( BL1 GL2 HL3)(yn ln) 1hL ( BL1 GL2 HL3) yn ln (62)
0, K 1h(LBL1 GL2 HL3), y(x0) andl(x0) , Given
then
yn 1 ln1 K
(63)
We therefore conclude that our new numerical integrator (25) is stable and hence convergent.
Therefore, the new numerical integrator (25) is consistent since it satisfies equation (49).
CONCLUSION Since the new numerical integrator (25) satisfies the conditions for convergence, consistence and stability, it is obvious that the result of this numerical integrator will approximate the solutions of the problems it is designed for.
Stability of the New Numerical Integrator Theorem 3 (Lambert 1973, Fatunla 1988) Let yn y ( xn ) and ln l ( xn ) denote two different numerical solutions of differential equation (1) with the initial conditions specified as y( x0 ) and l ( x0 ) respectively, such that 215
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(2):211-216(ISSN: 2141-7016) REFERENCES Ademiluyi, R. A. 1987. New Hybrid Methods for Systems of Ordinary Differential Equations, Ph. D Thesis, University of Benin, Nigeria.
Lambert, J. D. 1991.Numerical Methods for Ordinary Differential Systems: The Initial Value Proble, John Willey and Sons, New York. Mickens, R. E. 1994.Non-Standard Finite Difference Models of Differential Equations, World Scientific, Singapore.
Fatunla, S. O. 1976. A New Algorithm for Numerical Solution of Ordinary Differential Equations, Computer and Mathematics with Applications, 2:247253.
Ross, S. L. 1964. Difference Equations, Blaisdell; Waltham, MA.
Fatunla, S. O. 1988. Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic Press Inc, New York.
Wazwaz, A. M. 2000. A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operation, Appl. Math. Comp. 3:53-69.
Henrici, P. 1962. Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York. Humi, M. and Miller, W. 1988. Second Course in Ordinary Differential Equations for Scientists and Engineers, Springer-Verlag; New York. Ibijola, E. A. 1997. New Scheme for Numerical Integration of Special Initial Value Problems in Ordinary Differential Equations, Ph. D Thesis Submitted to the Department of Mathematics, University of Benin, Nigeria. Ibijola, E. A. and Sunday, J. 2010. A Comparative Study of Standard and Exact Finite-Difference Schemes for Numerical Solution of Ordinary Differential Equations Emanating from the Radioactive Decay of Substances, Australian Journal of Basic and Applied Sciences, 4(4): 624-632. Ibijola, E. A. and Ogunrinde, R. B. 2010. On a New Numerical Scheme for the Solution of IVPs in ODEs, Australian Journal of Basic and Applied Sciences, 4(10): 5277-5282. Ibijola, E. A., Bamisile, O.O. and Sunday, J. 2011. On the Derivation and Applications of a New OneStep Method Based on the Combination of Two Interpolating Functions, American Journal of Scientific and Industrial Research, 2(3): 422-427. Jain, M. K., Iyengar, S. R. K. and Jain, R. K. 2007. Numerical Methods (for Scientific and Engineering Computation), 5th edition, New age international (p) limited publishers, New Delhi. Kama, P. and Ibijola, E. A. 2000. On a New OneStep Method for Numerical Solution of Ordinary Differential Equations, International Journal of Computer Mathematics, Vol. 78, issue 4 (UK). Lambert, J. D.1973. Computational Methods in Ordinary Differential Equations, John Willey and Sons, New York.
216
