New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations

Transcription

1 9 J. Mah. Fund. Sci., Vol. 8, No.,, 9-5 New Seven-Sep Numerical Mehod for Direc Soluion of Fourh Order Ordinary Differenial Equaions Zurni Omar & John Olusola Kuboye Deparmen of Mahemaics, School of Quaniaive Sciences, Universii Uara Malaysia,, Sinok, Kedah, Malaysia Absrac: A new numerical mehod for solving fourh order ordinary differenial equaions direcly is proposed in his paper. Inerpolaion and collocaion were employed in developing his mehod using seven seps. The use of he approimaed power series as an inerpolaion equaion was adoped in deriving he mehod. The basic properies of he new mehod such as zero-sabiliy, consisency, convergence and order are esablished. The numerical resuls indicae ha he new mehod gives beer accuracy han he eising mehods when i is applied o fourh ordinary differenial equaions. Keywords: block mehod; collocaion; direc soluion; inerpolaion; ordinary differenial equaions; seven-sep. Inroducion The general fourh order ordinary differenial equaions ODEs of he form,,,,, 3, [, ] s y f y y y y y ys s b are considered in his paper. Eq. can be solved by reducing i o is equivalen firs order sysem as menioned in [-9]. However, his approach suffers some sebacks, such as evaluaion of oo many funcions and heavy compuaion see [-]. Direc mehods of solving Eq. have been eamined by several researchers [,-]. They developed linear mulisep mehods using inerpolaion and collocaion whereby he use of he approimaed power series as a basis funcion was considered. Kayode [] developed an efficien zero-sable numerical mehod wih sep number k = and 5 for fourh order iniial value problems IVPs, which was implemened in predicor-correcor mode. Furhermore, a five-sep block mehod for solving fourh order ODEs direcly is presened in []. In addiion, in [3] and [] si-sep block mehods are developed for solving fourh order ODEs using a mulisep collocaion approach whereby collocaion poins are seleced a some grid poins. These mehods, however, have low accuracy. Received June h, 5, Revised December 5 h, 5, Acceped for publicaion February 8 h,. Copyrigh Published by ITB Journal Publisher, ISSN: , DOI:.5/j.mah.fund.sci..8..

2 Numerical Mehod for Direc Soluion of Fourh Order ODEs 95 In order o improve he accuracy of he eising mehods, his aricle proposes a new block mehod for direcly solving general fourh order IVPs of ODEs by increasing sep number k. Mehodology Le he approimae soluion o Eq. be a power series of he form k j y a j j where k is he sep number. The fourh derivaive of Eq. gives k j j j y j j j j3 a f, y, y, y, y 3 Eq. is inerpolaed a, i k5 k and Eq. 3 is collocaed a ni, i k. This gives a sysem of equaions in he form where X ni AX B a y nk5 a y nk a y nk3, B a3 ynk a y k nk, 3 k nk 5 nk5 nk 5 nk 5 nk 5... nk 5 3 k nk nk nk nk nk... nk 3 k nk3 nk 3 nk 3 nk3 nk3... nk3 3 k nk nk nk nk nk... nk A n... k k3 k k k n... k k3 k k n k nk... k k3 k k k n nk

3 9 Zurni Omar & John Olusola Kuboye By using Gaussian eliminaion, he values of, j as j k in Eq. are obained and hen subsiued ino Eq. o give a coninuous linear mulisep mehod in he form 5 k k j n j j n j j k j y y h f 5 For k = 7, we have , G where w w w w w w w w w w w w w w w w w w w w u u u u u u u u u u u v G v v v v v v v v v v v v v v v v v v v v v u u u u u u u u u u u w w w w w w w w w w w w w w w w w w w w w w w.

4 Numerical Mehod for Direc Soluion of Fourh Order ODEs 97 The values of w, u and v are w = 975, u = 335, v = 3958, for n. h Eq. 5 is evaluaed a he non-inerpolaing poins, i.e. = -, -5, and, o produce he discree schemes. The firs, second and hird derivaives of Eq. 5 are evaluaed a all he poins wihin he inerval, i.e. = -, -5, -, -3, -, -, and, o give he derivaives of he discree schemes. These schemes are combined in a mari, whereby boh he y and he f funcion are muliplied by he inverse of he coefficiens of yn j, j k. This yields a block of he form where AY AY hay h BY hby h E F EF 3 N N N N N N N yn ynk y nk y nk yn yn k y n k y nk YN, YN Y N, Y N, ynk yn y n y n y nk fn fnk y nk fn fnk Y N, FN, FN. y n fnk fn If k = 7, we obain A, A,

5 98 Zurni Omar & John Olusola Kuboye 3 A, B 3, B 8, E , E

6 Numerical Mehod for Direc Soluion of Fourh Order ODEs 99 The corresponding derivaives of Eq. are given by where y n y n 9 y 3 y n5 5 5 y n y 8 n7 7 9 n3 3 y n y n hy 8 h y h T f f f f f fn fn fn n n n n3 n T y n n fn y n f n y n 3 3 f n3 yn yn hyn h M f n y n5 5 fn5 y n f n y n7 7 f n7 f

7 Zurni Omar & John Olusola Kuboye for M fn y n fn y n f n y n 3 fn3 yn yn hn f n y n5 fn5 y n f n y n7 f n7 for N

8 Numerical Mehod for Direc Soluion of Fourh Order ODEs 3 Analysis of he Properies of he Block Mehod 3. Order of he Mehod The linear operaor associaed wih Eq. can be defined as L{ y : h} AY AY hay h BY h BY h E FN E FN 3 N N N N N Eq. 7 is epanded in Taylor series, which gives Ly [ : h] Cy Chy Ch y C h y p p p p p p The Eq. and he associaed linear operaor are said o have order p if C C C... C C C C, C. Therefore, our mehod s Eq. p p p p3 p has order p [8,8,8,8,8,8,8] T wih error consans T p p C p [,,,,,, ].. Cp h y n as he principal local runcaion error a poin n. is known 3. Zero Sabiliy of he Mehod Block Eq. is said o be zero-sable if he roos z,,..., N s of he firs characerisic polynomial z de za A saisfies z and he roo z has mulipliciy no greaer han he order of he differenial equaion, which is. Now, z de za A implies ha z z z. Hence z,,,,,,. Therefore, he new mehod s Eq. is convergen because i is zerosable and has order greaer han one. Numerical Eperimen The accuracy of he new mehod is eamined by solving he following differenial problems. iv Problem : y, y, y, y y, h. Eac soluion: 5 y

9 Zurni Omar & John Olusola Kuboye Problem : iv. y y,, y, y, 7 5. y, y, h. cos.sin Eac soluion: y iv 3 Problem 3: y 9 3 e, y y, y, y,. Eac Soluion: y e iv Problem : y y y y y y, Eac Soluion: y e The resuls generaed afer solving he above problems are shown in Tables. The following noaions are used in Tables 3 and : SPEB: Sequenial implemenaion of he -Poin Eplici Block Mehod. PPEB: Parallel implemenaion of he -Poin Eplici Block Mehod. S3PEB: Sequenial implemenaion of he 3-Poin Eplici Block Mehod. P3PEB: Parallel implemenaion of he 3-Poin Eplici Block Mehod. Table Comparison of new mehod wih [3] and [] for solving problem. Eac Soluion Compued Soluion Error in [3], Error in [], Error in New k = k = Mehod, k = E-.7E-.87E E-.E E E-.E E E-.E E E-.87E E-8.9E E E-8.7E E E E E E E E E-8.87E E-

10 Numerical Mehod for Direc Soluion of Fourh Order ODEs 3 Table Comparison of new mehod wih [] and [] for solving problem. Eac Soluion Compued Soluion Error in Error in [], Error in New [], k = 5 k = 5 Mehod k = E E E E E- 5.E E-8.893E-.755E E-7.9E-5.8E E-7.73E E E E E E-.9E-.8E E-.57E-.7373E E- 3.88E-.889E E- 5.5E- 3.97E-8 Table 3 Comparison of new mehod wih [7] for solving problem 3. h-values New Mehod - 7-Sep Mehod -3 7-Sep Mehod - 7-Sep Mehod -5 7-Sep Mehod h-values Omar In [7] Number Error in New Mehod, Error in [7] of Seps k = 7 k = 8 SPEB E-.778E- PPEB E-.778E- S3PEB E-.778E- P3PEB E-.778E- SPEB 5.E-.778E-3 PPEB 5.E-.778E- S3PEB 339.7E-.778E-3 P3PEB 339.7E-.778E- SPEB E-9.8E- PPEB E-9.8E- S3PEB E-.8E- P3PEB E-.8E- SPEB 5.95E-9.E-5 PPEB 5.95E-9.E-5 S3PEB E-.E-5 P3PEB E-.E-5 Table Comparison of new mehod wih [7] for solving problem. New Mehod - 7-Sep Mehod -3 7-Sep Mehod - 7-Sep Mehod -5 7-Sep Mehod Omar In [7] Number Error in New Mehod, Error in [7] of Seps k = 7 k = 8 SPEB E- 8.37E- PPEB E- 8.37E- S3PEB 39.7E E- P3PEB 39.7E E- SPEB E-3 8.3E-5 PPEB E-3 8.3E-5 S3PEB E- 8.3E-5 P3PEB E- 8.3E-5 SPEB E E- PPEB E E- S3PEB E E- P3PEB E E- SPEB 5.8E- 8.33E-7 PPEB 5.8E- 8.33E-7 S3PEB E E-7 P3PEB E E-7

11 Zurni Omar & John Olusola Kuboye 5 Conclusion A seven-sep block mehod for he soluion of fourh order ODEs is proposed in his paper. The new mehod was used o solve fourh order IVPs of ODEs. The numerical resuls were compared wih he eising mehods. The new mehod performed beer han he mehods in [,-], which employed 5 and seps refer o Tables and. This implies ha beer accuracy can be achieved when sep number k is increased. The accuracy of he new mehod was also found o be beer han he mehod in [7], which was developed hrough numerical inegraion using 8 seps refer o Tables 3 and. Thus, based on he numerical resuls, he new mehod ouperformed he eising mehods in erms of accuracy and should be considered as a viable alernaive for direcly solving fourh order iniial value problems. References [] Awari, Y.S. & Abada, A.A., A Class of Seven Poin Zero Sable Coninuous Block Mehod for Soluion of Second Order Ordinary Differenial Equaion, Inernaional Journal and Mahemaics and Saisics Invenion IJMSI,, pp. 7-5,. [] Anake, T.A., Awoyemi D.O., & Adesanya, A.O., One-Sep Implici Hybrid Block Mehod for he Direc Soluion of General Second Order Ordinary Differenial Equaions, IEANG Inernaional Journal of Applied Mahemaics,, IJAM,. [3] Bun, R.A. & Vasil yer., A Numerical Mehod for Solving Differenial Equaion of Any Orders, Comp. Mah. Phys., 33, pp , 99. [] Brugnano, L. & Trigiane, D., Solving Differenial Problems by Mulisep IVPs and BVP Mehods, Gordon and Breach Science Publishers, 998. [5] Lamber, J.D., Compuaional Mehods in Ordinary Differenial Equaions, John Wiley & Sons Inc [] Senu, N., Suleiman, M., Ismail, F., & Ohman, M., A Singly Diagonally Implici Runge-kua-Nysrom Mehod for Solving Oscillaory Problems, IEANG Inernaional Journal of Applied Mahemaics,, IJAM,. [7] Omar, Z., Parallel Block Mehods for Solving Higher Order Ordinary Differenial Equaion Direcly. Docor Thesis in he Faculy of Science and Environmenal Sudies Universii Pura Malaysia, Unpublished, 999. [8] Omar, Z. & Kuboye, J.O., Compuaion of an Accurae Implici Block Mehod for Solving Third Order Ordinary Differenial Equaions Direcly, Global Journal of Pure and Applied Mahemaics,, pp. 77-8, 5.

12 Numerical Mehod for Direc Soluion of Fourh Order ODEs 5 [9] Omar, Z. & Kuboye, J.O., Developing Block Mehod of Order Seven for Solving Third Order Ordinary Differenial Equaions Direcly using Mulisep Collocaion Approach. Inernaional Journal of Applied Mahemaics and Saisics, 533, pp. 5-73, 5. [] Adesanya, A.O., Momoh, A.A., Alkali, M.A., & Tahir, A., A Five Seps Block Mehod for he Soluion of fourh Order Ordinary Differenial Equaions, Inernaional Journal of Engineering Research and Applicaions IJERA,, pp ,. [] Bucher, J.C, Numerical Mehods for Ordinary Differenial Equaions, John Wiley & Sons, New York, 3. [] Kayode, S.J., An Efficien Zero Sable Mehod for Fourh Order Ordinary Differenial Equaions, In. J. Mah. Sci, pp. -, 8. d.doi.org/.55/8/3 [3] Mohammed, U., A Si Sep Block Mehod for Soluion of Fourh Order Ordinary Differenial Equaions, The Pacific Journal of Science and Technology,, pp. 59-5,. [] Olabode, B.T., A Si Sep Scheme for he Soluion of Fourh Order Ordinary Differenial Equaions, The Pacific Journal of Science and Technology,, pp. 3-8, 9.