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Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

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Extension of generalized recursive Tau method to non-linear ordinary differential equations

Q1 K. Issaa'b'*, R.B. Adeniyia

a Department of Mathematics, University ofIlorin, Nigeria b Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Ilorin, Nigeria

Received 14 April 2014; received in revised form 1 December 2014; accepted 14 January 2015

Abstract 3

In a recent paper, we reported a generalized approximation technique for the recursive formulation of the Tau method. This 4

paper is concerned with an extension of that discourse to non-linear ordinary differential equations. The numerical results show 5

that the method is effective and accurate. 6

© 2015 Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access article under 7 the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Lanczos Tau method; Chebyshev polynomials; Initial value problems; Canonical polynomial; Ordinary differential equations 8

1. Introduction 10

In 1981, Ortiz and Samara [1] proposed an operational technique for the numerical solution of non-linear ordinary 11

differential equations with some supplementary conditions based on the Tau method [2]. Recently, considerable work 12

has been done both in the development of the technique, its theoretical analysis and numerical applications. The 13

technique has been described in a series of papers [3-6,1], for the case of linear ordinary differential eigenvalue 14

problems. Yisa and Adeniyi [7] reported the construction of generalized canonical polynomials while Issa and 15

Adeniyi's [8] reported generalized approximation for the recursive formulation of the Tau method for the solution of 16

ordinary differential equations, their earlier works are further extended to non-linear ordinary differential equations. 17

2. Recursive formulation of Tau approximant 18

In this section, we review the Tau approximant for the recursive form (see [8]) using the generalized Canonical 19

polynomials Qn (x) (see [7]) to solve the mth order ordinary differential equation of the form: 20

Peer review under responsibility of Nigerian Mathematical Society. * Corresponding author at: Department of Mathematics, University of Ilorin, Nigeria. E-mail addresses: issakazeem@yahoo.com (K. Issa), raphade@unilorin.edu.ng (R.B. Adeniyi).

http://dx.doi.org/10.1016/j.jnnms.2015.02.002

0189-8965/©© 2015 Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access article under the

CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/)._

Please cite this article in press as: Issa K, Adeniyi RB. Extension of generalized recursive Tau method to non-linear ordinary differential equations. Journal of the Nigerian Mathematical Society (2015), http://dx.doi.Org/10.1016/j.jnnms.2015.02.002

ARTICLE IN PRESS

K. Issa, R.B. Adeniyi / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

m / Nr

Ly (x ) = £ £ Pr,kxk\ y(r )(x ) = Y, frxr, a < x < b

r=0 \k=0 m —1

L * y (xrk ) = ^2 arky(r )(xrk ) = ak, k = 1(1)

by seeking an approximant

(2.1a) (2.1b)

yn (x) = arxr, r <

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

Lyn (x) frxr + Hn (x)

L * yn (xrk ) = ak, k = 1(1)m

(2.2a)

(2.2b)

where ak, fk, Pr,k, Nr; r = 0(1)m, k = 0(1)Nr are real integers, y(r) denote the derivatives of order r of y(x), the perturbation term Hn (x) in (2.2a) is defined by:

m +s— 1 m+s—1 n—m+i + 1

Hn (x ) = ^2 Ti + 1 Tn—m+i + 1 (x ) = ^2 Ti + 1 ^2 C

i =0 i =0 r=0

(n—m+i + 1) xr

(2.2c)

and C( ) is the coefficient of xr in the nth degree Chebyshev polynomial Tn (x); that is,

Tn(x) = cos (n cos-= ]Tc(n)xr

The t's are fixed parameters to be determined and s, the number of overdetermination of (2.1a), is defined by:

s = max {Nr — r > 0 | 0 < r < m}. For different orders m and s (that is m = 1, 2,... and s = 1, 2,...) we have

m+s —1 n—m+i + 1

yn(x) = ^2 frqr(x) + ^2 Ti+1 X c

(n—m+i+1)

qr (x ).

(2.3a)

Assume Qr (x) = Pr = 1, r = 0(1)(s - 1) and

m+s—1 n-m+i + 1 a

£ Ti + 1 £ C(n—m+i + 1) Pr + £ frPr = 0 i =0 r=0 r =0

where Eq. (2.3b) is the coefficient of undetermined Canonical polynomials, qn (x) = Qn (x) — Pn,

(2.3b)

EE J '

n — s

£k l(n ks)Pk ,k+s =k V J

PJ, J—k I Pn—s—k

s— 1 / m

+ E Ej

k=0 j=0

PJ,J+k\ Pn—s+k

(2.3c)

K. Issa, R.B. Adeniyi / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

we assume Pr = 1, r = 0, 1 ... s — 1, when equating the coefficient of Qr (x ) to zero, otherwise Pr = 0, Pr = 1, r = 0, 1 ... s — 1, see [8] and

Qn (x ) =

E k ! (" k s ) Pk,k+s

s—1/ m

+ El Ej !

k=0 \ j=o

n — s j

EE j !

,k=i \j=k

Pj, j+k Qn—s+k (x)

Pj,j—k) Qn—s—k(x)

(2.3d)

is the generalized Canonical polynomial, see [7]. 3. Methodology

In this section, we shall consider an extension of the generalized recursive formulation of the Tau method to non-linear problem. For this purpose, we shall employ the Newton-Kantorovich linearization process to non-linear differential equation of the form

G(x, y (x), y'(x), ..., y(m)(x)) = £ frxr

and the process of Newton-Kantorovich linearization, derived from the Taylor series expansion in several variables of G, is given by:

^ d G ,3 G „3 G (m ) d G r

G + A-v 97 + △ y'â? + △ y â? + ■ ■ ■ + △ y(m) dymi = E frx

where Aylk = yk+1 — ylk, i = 0, 1, ..., m.

We seek kth iterative approximant of the form:

yn,k (x ) = ar,kxr.

The form (2.2) corresponding to Eq. (3.2) is

d Gn,k

d yn,k

E( yJn,k+1(x ) — yJn,k (x »-j = E frxr — Gn,k + Hn,k (x )

m+s —1

Hn,k (x ) = Ti + 1,kTn—m+i + 1(x )

Gn^k = G (x, yn,k (x ), y'n k (x ), ..., y^m^x )), k = 0, 1, ....

The number of overdetermination s, for the non-linear problems, unlike in the case of linear problems, depends on n and can be very large depending on the degree of non-linearity of the problem under consideration. The iterative process is repeated until

!£n,k — Hn,k+1l < Tolerance Value

Hn,k = max{| yk (x) — yn,k (x)| : a < x < b}.

4 K. Issa, R.B. Adeniyi / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

1 The iterative scheme needs a suitable choice of an initial approximation yn,o (x) from linearization process, for a rapid

2 convergence. In most problems, the initial approximation yn,o (x) is taken to be the simplest polynomial satisfying

3 the associated condition in (2.1b). In some cases, good choice of initial approximation is obtained from the given

4 differential equation itself (see Example 4.1).

5 4. Numerical examples

6 We shall consider here four problems of interest for the illustration of the method of the preceding section. It is to

7 be noted that the results presented in the Tables below were obtained using mathematica 7.0 package.

8 We illustrate the accuracy of the presented method by computing (maximum error) for various numerical

9 examples.

10 Example 4.1 (Variable Coefficients First Order Homogeneous Problem, see [9]). Consider the first order non-linear

11 problem:

12 y' (x) + (2x - 1)y2 (x) = 0 (4.1)

13 y (0) = 1, 0 < x < 1 (4.2)

14 with exact solution y (x) = (x2 — x + 1)-1.

15 For this problem, we have

16 Gk = G(x, yk(x), yk(x)) = y'k(x) + (2x — 1)y|(x)

17 so that

d Gk , d Gk

18 Gk + △ yk ^ + △ y'k -^T = 0 (4.3)

dyk dyk

19 where

. . . d Gk d Gk

20 △ yk = yk +1 — yk, △ yk = yk +1 — yk, — = 2 yk(2x — ^ -ITT = 1

dyk dyk

21 substitute in (4.3), leads to the linearized problem

22 yk +1 + (4x — 2) ykyk+1 = (2x — 1) yf (4.4)

23 from (3.4), we have

24 y'n,k +1 + (4x — 2) yn,kyn,k+1 = (2x — 1) y^k + Hn,k (x)

25 where

26 Hn,k(x) = £ Ti + 1,kTn+i (x).

27 Thus, the sequence of linearized Tau problem to be solved is:

28 y'nk+1 + (4x — 2)yn,kyn,k+1 = (2x — 1)ylk + £ Ti+1,kTn+i (x) (4.5a)

29 yn,k+1 (0) = 1, k = 0, 1,2,.... (4.5b)

30 For the choice of initial approximation, we have from (4.1), y' (0) = 1 (since y (0) = 0). Hence, if we assume an initial

31 approximation of the form y = a + bx, then we get y(x) = x + 1 by using y (0) = 0 and y' (0) = 1 to determine a

32 and b. So, we choose yn,0(x) = x + 1 and then compute the approximant solution.

33 First Iteration (k=0): we have from (4.4)

34 y 1 (x) + (4x2 + 2x — 2) y 1 (x) = 2x3 + 3x2 — 1 (4.5)

35 yj(0) = 1.

K. Issa, R.B. Adeniyi / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx Table 4.1

Maximum error for Example 4.1.

Iteration (k) Degree 5 (§5,k) Degree 6 (§6,k) Degree 7 (£7^)

k = 0 9.37 x 10-2 9.39 x 10-2 9.41 x 10-2

k = 1 2.23 x 10-3 4.64 x 10-4 4.44 x 10-4

k = 2 2.27 x 10-3 1.67 x 10-4 1.70 x 10-4

Comparing (4.5) with (2.1a), we have m = 1 s = 2, and so the perturbed problem (4.5) becomes

/9 3 2

yn>1(x) + (4x + 2x - 2)yn,1(x) = 2x3 + 3x - 1 + n Tn (x) + T2Tn+1 (x) + T3 Tn+2(x). Applying (2.3), we obtained the solution for degrees 5, 6 and 7 as:

13734241x 2477864x2 15478136x3 2400320x4 1142176x5

y5,1 (x) = 1 + ....... + ................. + ______ +

y 6,1 (x ) = 1 +

14036168 5263563 5263563 1754521 5263563

758745347x 1599024368x2 923988736x3 15803064192x4

754661830 54946095104x5

7923949215 1584789843 5926750208x6

2641316405

y7,1 (x ) = 1 +

7923949215 2641316405 109326648889x 643141379x2

1645570879x3 15100542224x4

109205132560 13650641570 2730128314

7700535256x5 14043641024x6 1679799488x7 + ........ +

6825320785

6825320785 6825320785 1365064157 respectively. Second Iteration (k=1): We similarly obtained

y5,2(x) = 1 + 0.982633448x + 0.398369665x2 - 2.746545092x3 + 1.345127453x4 + 0.020877660x5 y62(x) = 1 + 1.001540930x - 0.073718784x2 - 0.27552491x3 - 3.801892850x4 + 4.714527741x5 - 1.564467919x6

y7,2(x) = 1 + 1.001763174x - 0.081717596x2 - 0.214195149x3 - 3.996928214x4 + 5.014854129x5 - 1.786985423x6 + 0.0636669636x7

and for Third Iteration (k=2) we have

y5,3(x) = 1 + 0.982318x + 0.405095x2 - 2.77481x3 + 1.3874x4 + 5.28099 x 10-6x5 y6,3(x) = 1 + 1.001699849x - 0.078836193x2 - 0.241731719x3 - 3.889122757x4 + 4.811985927x5 - 1.603995100x6

y7,3(x) = 1 + 1.001699864x - 0.0788367194x2 - 0.241727690x3 - 3.889135527x4 + 4.812005511x5 respectively.

1.60401x6 + 4.111601230 x 10-6x7

Table 4.1 is the maximum error for each iteration, and this is in good agreement with the result obtained in the literature [9].

Example 4.2 (A Constant Coefficients Second Order Homogeneous Problem, see [10]). y'' (x) - y (x) y' (x) = 0 y (0) = 0, y (1) = tanh

with analytic solution tanh( ), 0 < x < 1 after linearized, we have

y'k+1 - yky'k +1 - ykyk +1 = -yky'k

6 K. Issa, R.B. Adeniyi / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Table 4.2

Maximum error for Example 4.2.

Iteration (k) §5,k §6,k §7,k

k = 0 1.14 x 10- 5 9.22 x 10- 6 9.42 x 10- 6

k = 1 5.91 x 10- -6 2.64 x 10- -7 3.95 x 10- -8

k=2 5.91 x 10- -6 2.64 x 10- -7 3.99 x 10- -8

Table 4.3

Maximum error for Example 4.3.

Iteration (k) §5,k §6,k §7,k

k=0 5.18 x 10- -3 5.17 x 10- 3 5.16 x 10- 3

k=1 1.31 x 10- -4 2.86 x 10- 5 3.77 x 10- 6

k=2 1.31 x 10- -4 2.39 x 10- 5 3.78 x 10- 6

1 yk +i(0) = 0, yk+! = tanh^-pj , 0 < x < 1,

2 see Table 4.2 for the maximum error.

3 Example 4.3 (A Constant Coefficients First Order Non-homogeneous Problem, see [9]).

4 y'(x) - (y (x))2 = 1

5 y (0) = 1

6 with analytic solution y (x) = tan x, 0 < x < 4.

7 This leads to the linearized problem

8 y'+i - 2ykyk+1 = 1 - yk2

9 yk +1(0) = 0

10 see Table 4.3 for the maximum error.

11 Example 4.4. Consider the linear initial value problem y''(x) + 2y'(x) + y (x) = 0, y (0) = 1, y' (0) = 0.

12 The closed form of the solution is y (x) = e-x (1 + x) and the approximate solution using the Adomian decompo-

13 sition method (see [11]) is:

y(x) = 1 - 2x2 + 1 x3 - 1 x4 + ±x5 + O(6).

The presented method gives the following results:

, 1510240 2 2978560 3 3032960 4 499712 5

y5(x ) = 1--x 2 +--x 3--x 4 +--x 5

' 3023161 9069483 27208449 27208449

19830276760 2 13202664320 3 4880385280 4 1181708288 5 154245120 6

y6(x) = 1--x 2 +--x 3--x4 +--x 5--x 6

^ 39662689081 + 39662689081 39662689081 + 39662689081 39662689081

40269705965496 2 26843943252224 3 10051654400256 4 2642224435200 5

y7(x ) = 1--x 2 +--x 3--x +--x 5

y 80539624942061 + 80539624942061 80539624942061 + 80539624942061

501023258624 6 54334488576 7

-x +--x

80539624942061 80539624942061

see Table 4.4 for the maximum error.

Table 4.2 is the maximum error for each iteration, and this gives better result compared to the result obtained by the cubic spline using collocation method (see [10]).

Table 4.3 is the maximum error for each iteration, and this is in good agreement with the result obtained in the literature [9].

Table 4.4

Maximum error for Example 4.4.

7 3.07 x 10-8

Table 4.4 is the maximum error, and this gives better result compared to the result obtained by the Adomian 1

decomposition method (see [11]). 2

5. Conclusion 3

The generalized form of the recursive formulation of the Tau method for initial value problems and boundary value 4

problems has been applied to non-linear problems and to linear differential equation, we obtained the maximum error 5

for degrees 5, 6 and 7. Our results were compared with some existing results and we found that they are in good 6

agreement. 7

Finally, because of the accuracy and simplicity of the method presented in this study, we recommend its application 8

in finding approximate solution to non-linear differential equations. 9

Uncited references 10

[12], [13], [14], [15], [16] and [17]. 11

References 12

[1] Ortiz EL, Samara H. An operational approach to the Tau method for the numerical solution of non-linear differential equations. Computing 13 1981;27:15-25.

[2] Ortiz EL. The Tau method. SIAM J Numer Anal 1969;6:480-92. 14

[3] Liu KM, Ortiz EL. Numerical solution of Ordinary and Partial function-differential eigenvalue problems with the Tau method. Computing, 15 (wien) 1989;41:205-17.

[4] Liu KM, Ortiz EL. Tau method approximation of differential eigenvalue problems where the spectral parameter enters non-linearly. J Comput 16 Phys 1987;72:299-310.

[5] Liu KM, Ortiz EL. Approximation of eigenvalues defined by ordinary differential equations with the Tau method. In: Ka gestrm B, Ruhe A, 17 editors. Matrix pencils. Berlin: Springer-Verlag; 1983. p. 90-102.

[6] Liu KM, Ortiz EL. Eigenvalue problems for singularly perturbed differential equations. In: Miller JJH, editor. Proceedings of the BAIL II 18 conference. Dublen: Boole Press; 1982. p. 324-9.

[7] Yisa BM, Adeniyi RB. Generalization of canonical polynomials for overdetermined mth order ordinary differential equations(ODEs). IJERT 19 2012;1(6):1-15.

[8] Issa K, Adeniyi RB. A generalized scheme for the numerical solution of initial value problems in ordinary differential equations by the 20 recursive formulation of Tau method. Int J Pure Appl Math 2013;88(1):1-13.

[9] Adeniyi RB, Onumanyi P. Error estimation in the numerical solution of ODE with the Tau method. Comput Math Appl 1991;21(9):19-27. 21

[10] Taiwo OA. Comparison of collocation methods for the solution of second order non-linear boundary value problems. Int J Comput Math Q4 22 2005;82(11):1389-401.

[11] Onur Kymaz, Seref Mirasyedioglu. A new symbolic computational approach to singular initial value problems in the second order ordinary. 23 Appl Math Comput 2005;171:1218-25. differential equations.

[12] Adeniyi RB, Edungbola EO. On the tau method for certain overdetermined first order differential equations. J Niger Assoc Math Phys Soc 24 2008;12:399-408.

[13] Adeniyi RB, Edungbola EO. On the recursive formulation of the tau method for class of overdetermined first order equations. Abacus J Math 25 Assoc Nig 2007;34(2B):249-61.

[14] Fox L, Parker IB. Chebyshev Polynomials in numerical analysis. University Press Oxford; 1968. 26

[15] Lanczos C. Applied analysis. New Jersey: Prentice-Hall; 1956. 27

[16] Lanczos C. Trigonometric interpolation of empirical and analytic functions. J Math Phys 1938;17:123-99. 28

[17] Ortiz EL, Samara H. Numerical solution of differential eigenvalue problems with an operational approach to the Tau method. Computing 29 1983;31:95-103.