A two step block hybrid Adam Moulton method of uniform order five is presented for the solution of stiff initial value problems. The individual schemes that made up the block method are obtained from the same continuous scheme which is applied to provide the solutions of stiff initial value problems on non overlapping intervals. The constructed block method is consistent, zero – stable and A – stable. Numerical results obtained using the new block method show that it is superior for stiff systems and competes well with existing ones. Keywords: stiff ODEs, Block Method, Adam Moulton method, Stabilit

G. M, Kumleng,

O, Adee S.

Y, Skwame

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.3, No.9, 2013  77 Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations Kumleng, G. M1* Adee S. O 2 Skwame Y 3  1. Department of Mathematics, University of Jos, P.M.B. 2084, Plateau, State, Nigeria 2. Department of Mathematics, Modibbo Adama  University of Technology, Yola, Adamawa state, Nigeria 3. Department of Mathematical Science, Adamawa State University, Mubi, Nigeria * E-mail of the corresponding author: kumleng_g@yahoo.com  Abstract A two step block hybrid Adam Moulton method of uniform order five is presented for the solution of stiff initial value problems. The individual schemes that made up the block method are obtained from the same continuous scheme which is applied to provide the solutions of stiff initial value problems on non overlapping intervals. The constructed block method is consistent, zero – stable and A – stable. Numerical results obtained using the new block method show that it is superior for stiff systems and competes well with existing ones.  Keywords: stiff ODEs, Block Method, Adam Moulton method, Stability  1. Introduction         Let us consider the stiff initial value problem          0 0'( ) ( , ( )),    ( )y x f x y x y x y= =               (1) on the finite interval 0[ , ]NI x x=  where 0 0: [ , ]   and  : [ , ]m m mN Ny x x f x x→ ℜ × ℜ → ℜ  is continuous and differentiable. It is a well known fact that (1) always occur in engineering and control systems. Fatunla (1998) noted that mathematical formulation of new models of physical situations in engineering and sciences often lead to systems of the form (1) and as such there was need to generate techniques to conveniently cope with these type of problems. The search for better methods for solving these stiff systems leads to the discovering of the Backward differentiation formulae (BDF) by Curtiss and Hirschfelder (1952). Since then, most of the improvements in the class of linear multistep methods have been based on the BDF, because of its special properties. Cash (1980) introduced the extended BDF. Ebadi et al. (2010) worked on the hybrid BDF with one additional off-grid point introduced in the first derivative of the solution to improve the absolute stability region of the method.  Akinfenwa et al.(2011) has proposed an A-stable block hybrid Adam Moulton method incorporating four off –step points to improve on the accuracy of the two step Adam Moulton method for the solution of stiff ordinary differential equations. However, this method has improved the accuracy of the two step Adam Moulton method, unfortunately it entails greater computational efforts due to the many off –step points involved. In this paper, an attempt is made to reducing the computational efforts involved in Akinfenwa et al (2011) in solving stiff ordinary differential equations while retaining the accuracy of the method. We achieved this by incorporating just two off–step collocation points at 5 74 4 and n nx x+ + in the last subinterval 1 2[ ,  ]n nx x+ + of the two step Adam Moulton method as compared to the four ( 1 2 4 53 3 3 3 , ,  and  n n n nx x x x+ + + +) in Akinfenwa et. al (2011). The individual schemes that constitute the new two step block hybrid Adam Moulton method are obtained from the same continuous scheme which is applied to provide the solution for (1). Milne (1953) was the first to introduced block methods for use only as a means of obtaining starting values for predictor – corrector algorithms. Many researchers have also worked on the block methods (see Rosser, (1967), Shampine, et al. (1969) ) for general use. The block method is self starting and so does not require starting values or special predictors to predict the off-grid values.  The paper is presented as follows: In section 2, we discuss the derivation of the block method and obtain the continuous scheme of the method which is used in generating the members of the block method. In section 3, the stability analysis of the block method is presented. In section 4, we briefly discuss the implementation of the method. In section 5, we give some concluding remarks.  2. Derivation of the Block Method This section is concerned with the derivation of the two step hybrid block Adam Moulton method using the idea of interpolation and collocation proposed by Lambert (1973, 1991).The idea was later referred to as the multistep Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.3, No.9, 2013  78 collocation method by Lie and Norsett (1989). Onumanyi et al. (1994, 1999) has also worked on the multistep collocation method.       The general k – step Linear multistep method is given as         0 0k kj n j j n jj jy h fα β+ += ==∑ ∑                        (2) From (2), we obtain the general form of the k- step Adam Moulton as        10kn k n k j n jjy y h fβ+ + − +=− = ∑                            (3) We obtain the continuous scheme for the two step Adam Moulton method incorporating the two off-step points 5 74 4 and n nx x+ + as  ( )5 5 7 74 4 4 41 1 0 1 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ( )n n n nn ny x x y h x f x f x f x f x fα β β β β β+ + ++ += + + + + +          (4) From  Onumanyi  et al. (1994), we obtain the continuous coefficients 5 74 41 0 1 2( ),  ( ),  ( ),  ( ),  ( ) and ( )x x x x x xα β β β β β and ( 4 ) becomes  •      2 3 4 52 3 42 3 4 52 3 42 3 4 52 3 4571 201 211 12 81 2100 140 210 35 17535 131 20 16122145 3 9 3 15616 224 928 152 128225 15 45 15 75( ) ( )                   +( )                   +( )n nh h h hnh h h hh nh h hy x y h fh fh fλ λ λ λλ λ λ λλ λ λ λλ++= + − + − + − +− + − + −− + − + 542 3 4 572 3 442 3 4 52 3 4376 160 736 136 128315 21 63 21 10579 35 836 8 82180 12 18 3 15                   +( )                   +( )                   h nh h hnh h h hh fh fλ λ λ λλ λ λ λ+++− + − + − − + − +          (5) Evaluating (5) at the following points 5 74 40,  ,   ,2h h hλ =  yields the following discrete schemes which constitute the two step hybrid block Adam Moulton method.   • ( )( )( )5 74 45 5 74 4 47 5 74 4 4 100800  11 1 263001 1 21200  2631 12 11713 17080 17248 752011095 +1521 2765  -33 8 -1570 4909 315 5586 9 4200 12hhhn n n n nn nn n n nn n nn n n nn nhnn ny f f f f ff f f f ff f fyy yy yy yf f+ + ++ ++ + ++ + ++ + ++ + ++ +++− = + − + −− = +− = + + −− = ( )5 74 40 20 1280 2912 2720 3853 n n nn nf f f f f+ ++ ++ + + +                                 (6) The new block method is of order ( )5,5,5,5 T  and error constant of ( )253 131 159 1115200 9830400 3276800 38400, , , T− −.  3. Stability Analysis From (6), we arrange the block method as a matrix finite difference equation of the form                      (1) (0) (1)1u u uA y A y hB F+ − =           (7) where     ( )( )( )5 74 41 34 45 74 41 1 211 2, , ,, , ,, , ,Tu n nn nTu n nn nTu n nn nY y y y yY y y y yF f f f f+ + ++ +− + ++ ++ +=== Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.3, No.9, 2013  79 for 0,... and 0,...u n= =  and the matrices  (1) (0) (1), ,A A B  are  4 by 4 matrices whose entries are given by the coefficients of  (6).  3.1 Zero stability block method According to Fatunla (1994), a block method is said to be zero stable if 1, 2,...,ij kλ =  specified as                                ( )0( ) 0ki k iiAρ λ λ −== =∑                     (8)  satisfies ( ) 1jρ λ λ= ≤ , the multiplicity must not exceed two.  The block method (6) is expressed in matrix equation form to obtain         01 0 0 01 1 0 0,1 0 1 01 0 0 1A  − = − −     10 0 0 10 0 0 00 0 0 00 0 0 0A   =    The first characteristic polynomial of the new block method ( 6) is given by      )det()( )1()0( AA −= λλρ                      (9)         substituting  (0) (1)A and A  into the characteristic polynomial (9)  gives                1 0 0 0 0 0 0 11 1 0 0 0 0 0 0( )1 0 1 0 0 0 0 01 0 0 1 0 0 0 0ρ λ λ      −   = −   −   −    0 0 10 00 00 0λλ λλ λλ λ−−=−−                       4 3λ λ= −  3( 1)λ λ= − = 0 Therefore, 1 2 3 41, 0λ λ λ λ= = = = . Following Henrici (1962), the new block method (6) is zero stable and consistent since its order 5 1p = > . The method is therefore convergent. 3.2 Region of Absolute Stability of the Block method The absolute stability region of the block method is obtained using Chollom et al. (2007) and is as shown below,  4. Numerical Experiment In this section, some numerical experiments are presented in order to ascertain the performance of our new block method (6). The problems considered are the ones solved by Akinfenwa et. al. (2011) and the results obtained shall be compared with this method .  Example 1 Consider the stiffly linear problem '1 1 2 1'2 1 2 229998 59994                y (0) 19999 19997                    y (0) 00 10y y yy y yt= − − == + =≤ ≤ The eigenvalues  of  the system are 1 210000  and  1 with exact solutionλ λ= − = −  Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.3, No.9, 2013  80 1000011 9999100002( ) (29997 19998 )( )x xx xy x e ey x e e− −− −= −= −   Example 2 Consider the Stiffly nonlinear problem             ( )1 1 21 1 2 122 1 2 2 2' 2          (0) 1'                       (0) 10 10y y y yy y y y ytε ε− −= − + + == − − =≤ ≤     where 610ε −= , the smaller ε is, the more serious the stiffness of the system. The exact solution is   21 2 2( ) ( ) ,  ( ) xy x y x y x e−= =  5. Conclusion The numerical results obtained in Table1 and 2 for the problems solved suggest that the new proposed block hybrid method (6) is suitable for solving stiff problems and perform competitively well with less computational effort compared with the method of Akinfenwa et al (2011).  References Arai, T., Aiyama, Y., Sugi, M. & Ota, J. (2001), “Holonic Assembly System with Plug and Produce”, Computers in Industry 46, Elsevier, 289-299. Akinfenwa, O., Jator, S. & Yao, N. (2011), “Implicit two step continuous hybrid block methods with four off-steps points for solving stiff ordinary differential equations”, Proceedings of the international conference on computational and Applied mathematics, Bangkok, Thailand 2011. Cash, J. R. (1980), “On the Integration of Stiff systems of ODEs using extended backward  differentiation formulae”  Numer. Math. 34, 235 – 246. Chollom,, J. P.,  Ndam, J. N. & Kumleng, G. M. (2007), “On Some Properties of the Block Linear Multistep Methods”, Science World Journal. 2, 11 – 17 Curtiss, C.F. & Hirschefelder, J, O. (1952), “Integration of Stiff equations”, Proceedings of the National Academy of Sciences of the United States of America, 38. no.3,235. Ebadi, M. & Gokhale, M. Y. (2010), “Hybrid BDF methods for the numerical solutions of ordinary differential equations”,  Numer. Algor, 55, 1 -17 Fatunla, S. O. (1988), “Numerical methods for IVPs” ,Academic Press ,USA.      Fatunla, S. O. (1991),”Block Method for Second order Differential Equations”, International Journal of Computer Mathematics. 41, 55 – 63. Henrici, P. (1962), “Discrete Variable Method in Ordinary Differential Equations”. New York John Wiley  and Sons Ltd. 285p Lambert, J. D. (1973), “Computational Methods in Ordinary Differential Equations”. New York. John Willey and Sons. Lambert, J. D. (1991), “Numerical Methods for Ordinary Differential Systems”. New York. John Willey and Sons. Lie, I. & Norsett, S. P. (1989), “Super Convergence for Multistep Collocation”, Mathematics of Computation. 52(185), 65 – 79. Milne, W. E.(1953), “Numerical solution of differential equations”, John Wiley and sons.  Onumanyi, P., Awoyemi, D. O., Jator, S. N. & Sirisena, U.W. (1994), “New Linear       Multistep Methods with continuous Coefficients for first order Initial Value  Problems”, Journal of the Nigerian Mathematical Society.13, 37 – 51. Onumanyi, P., Sirisena,U.W. & Jator, S.N. (1999), “Continuous finite Difference Approximations for solving Differential Equations”, International Journal of Computer Mathematics .72, 15 – 27. Rosser, J.B. (1967), “A Runge-Kutta for all seasons”,SIAM,REV. 9, 417– 452. Shampine, L. F. & Watts, H.A. (1969), “Block Implicit One Step Methods”. Math.Comp. 23, 731-740.   Journal of Natural Sciences Research                                                                                                                                                www.iiste.org ISSN 2224-3186 (Paper)   ISSN 2225-0921 (Online) Vol.3, No.9, 2013  81  Fig.1 Absolute Stability Region of the New Block Method The absolute stability region consists of the set of points in the complex plane outside the enclosed figure. Therefore, the block method is A - stable since the left –half complex plane is contained in S. where { }: ( ) 1S z R z= ∈ ≤ . Table 1.  Example 1 Absolute  Error For  New  Block  Hybrid Method With Two Off-Steps  Compared  With  Akinfenwa Et Al.(2011) With Four Off-Steps  At The  End Point   T = 10  Table 2:  Example 2 Absolute  Error For  New  Block  Hybrid Method With Two Off-Steps Compared  With  Akinfewa Et Al.(2011 ) With Four Off-Steps  At The  End Point T = 10   -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5Re(z)Im(z)This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.  There’s no deadline for submission.  Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/journals/   The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors.  MORE RESOURCES Book publication information: http://www.iiste.org/book/ Recent conferences:  http://www.iiste.org/conference/ IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar   

Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations

https://www.iiste.org/Journals/index.php/JNSR/article/download/7418/7552

Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations

Abstract

Similar works

Full text

Available Versions

International Institute for Science, Technology and Education (IISTE): E-Journals