Cubic Bezier Homotopy Function for Solving Exponential Equations

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1 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 omoopy Funcon for Solvng Eponenal Equaons S. S. Raml *,,. Mohamad Nor,a, N. S. Saharzan,b and M. A. Salm Nasr,c Faculy of Compuer and Mahemacal Scences, Unvers Teknolog MARA Malaysa, 445 Shah Alam, Selangor. School of Mahemacal Scences, Unvers Sans Malaysa, 8 Mnden, Pulau Pnang, Malaysa. * s.syuhadahraml@ymal.com, a hafz97my@yahoo.com, b syazanasaharzan@yahoo.com, c masn@msk.um.edu.my Absrac In hs sudy, he new homoopy funcon has been eended from Quadrac Bezer homoopy funcon for mplemenaon wh sngle eponenal equaons. The eended funcon s a combnaon of he Bezer curve and homoopy funcon ha was based on De Caseljau algorhm. The funcon wll nclude he arge funcon and he aulary funcon. The arge funcon s he eponenal equaon whle he aulary funcon seleced mus be conrollable and easy o be solved. Then, he eended funcon has solved by usng Newon omoopy Connuaon Mehod o compue he appromae soluons, accuracy of he appromae soluons and mamum absolue error of he soluons. The analysed resuls proved ha hese omoopy funcon show beer appromae soluons, accuracy of he appromae soluons and mamum absolue error of he soluons compared o Quadrac Bezer homoopy funcon. Copyrgh 6 Penerb Akadema Baru - All rghs reserved. Keywords: Bezer curve, homoopy funcon, newon homoopy connuaon mehod, sngle of eponenal equaon. INTRODUCTION Nowadays, a research abou concep of omoopy funcon has been an neresng and wdely used. The majory of hem ulzed and solve equaon comprse of a few classfcaons, for eample, algebrac, polynomals, rgonomerc, eponenal, and logarhmc equaon []. In 9, Lahaye sar o nroduce he homoopy mehod [], bu he concep of omoopy Connuaon mehod for solvng equaon sars o be denfed and suded n 97. Prevous sudy, Wu have made a lo of sudy abou he omoopy Connuaon mehod. Wu begn wh Modfed Chnese algorhm and sudy he convergence of Newon omoopy Connuaon mehod. Then, Wu has developed a new formula and compare by usng radonal Adoman Decomposon mehod wh Adoman omoopy Connuaon mehod. Besdes ha, Wu also solve a nonlnear equaon by usng Newon omoopy Connuaon mehod and adjusng an aulary funcon. Then, Wu has modfed he secan mehod wh he omoopy Connuaon echnque o new Secan omoopy Connuaon funcon. All he resul [-6] obaned shown ha omoopy Connuaon mehod s beer han classcal mehod.

2 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 In [8], Nor also have develop a new funcon of Quadrac Bezer homoopy funcon for solvng sysem of eponenal equaons. Therefore, we eend from he sudy o develop homoopy funcon for mplemenaon wh sngle eponenal equaon usng Newon omoopy Connuaon mehod o compue he appromae soluons, accuracy of appromae soluons and mamum absolue error of soluons.. CUBIC BEZIER OMOTOPY FUNCTION Frsly, nroduce he Bezer curve based on he De Caseljau algorhm as follow [9], P ( r = ( P r ( + P r + ( r =,..., k =,..., k r ( where P ( = P s he fnal pon n he curve. Then, he curve wll be consruc from he formula n ( defned by four pon P, P, P and P. Then he funcon become ( P = ( P + ( P + ( P +. ( ( P The dea of develop omoopy funcon based on De Caseljau Algorhm. In omoopy, here s a curve ha movng from one curve o anoher curve. Whle n De Caseljau, here s movemen of pon a curve and we consder here s smlary ha can relae beween hem [8]. Therefore, by usng mehod of De Caseljau, becomes easer and sysemac wh recursve consrucon of homoopy funcon s as follow, Fgure : The recursve consrucon of omoopy funcon. Based on he Fg., noe ha A ( = ( G( + ( B = ( ( + ( (

3 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 C ( = ( ( + F( D ( = ( A + B E ( = ( B + C hus ( = ( D + E ( Subsue he above equaon, ( = ( [( A + B] + [( B + C] = ( A + ( B + ( B + = ( [( G( + ( ] + ( [( ( + ( ] + ( [( ( + ( ] + [( ( + F( ] = ( G( + ( ( + ( ( + ( ( + ( ( + ( ( + ( ( + C F( = ( G( + ( ( + ( ( + F( (4 = B ( + B + B + B ( = B ( (5 wh = = G( for =, = (, = ( for = (, = F( for =, and B ( s a Bernsen funcon n Bezer curve whch s defned n [] B ( = ( B ( = (( B ( = (( B ( = (( B ( = =!!( (! (6 where =,, [,]

4 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6. NEWTON OMOTOPY CONTINUATION METOD The connuaon mehod does no sand alone, we need o combne wh oher mehod and hen s called omoopy Connuaon Mehod (CM. In hs sudy, he CM wll be combned wh he classcal mehod whch s Newon mehods. Therefore, he new name s Newon omoopy Connuaon mehod. Ths mehod wll use o solve he homoopy funcon. Thus, he formula of Newon omoopy Connuaon mehod s as follow, (, + =, =,,,..., k (7 D (, where ( s homoopy funcon. 4. NUMERICAL EXPERIMENT Consder he eamples of sngle eponenal equaon: Eample : Consder he followng eample of sngle eponenal equaon ha was aken from he prevous paper [] f ( = e (8 The aulary funcon s g ( = and he nal value s =. Thus, he equaon (8 wll be mplemened no CBF as follow, ( = ( ( + ( + ( e ( + ( ( (9 The comparson resuls beween Quadrac and homoopy funcon are show n Table, Table 4 and Table 7 wh dfferen numbers of eraons. Eample : Consder he followng eample of sngle eponenal equaon ha are aken from he prevous paper [] f ( = e ( The aulary funcon s g( =. 5 and he nal value s =. 5. Thus, he equaon ( wll be mplemened no CBF as follow, ( = ( (.5 + ( ( e ( + ( ( + ( The comparson resuls beween Quadrac and omoopy funcon are shows n Table, Table 5 and Table 8 wh he dfferen sep szes of. Eample : Consder he followng eample of sngle eponenal equaon ha are aken from he prevous paper [], 4

5 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 f ( = + e + ( The aulary funcon s g( = + and he nal value s =. Thus, he equaon ( wll be mplemened no CBF as follow, ( = ( ( + + ( + ( + e + ( + ( ( ( The comparson resul beween Quadrac and homoopy funcon are shows n Table, Table 6 and Table 9 wh he dfferen sep szes of. Table : Comparson of he appromae soluons beween Quadrac and omoopy Funcon for equaon (8 Sep sze, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Roo of an equaon, = Table : Comparson of he appromae soluons beween Quadrac and omoopy Funcon for equaon ( Sep sze, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Roo of an equaon, = Table : Comparson of he appromae soluons beween Quadrac and omoopy funcon for equaon ( Sep sze, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Roo of an equaon, = Table, Table and Table show he resuls of he appromae soluons for equaon (8, ( and ( respecvely. A Table he real roo of an equaon s The value of appromaon soluons a QBF column s approached o roo as he sep sze of decreased. Moreover, he appromae soluons are closer o roo of an equaon when we used CBF. Among hese wo homoopy funcons, he closes appromaons soluon o roo of an equaon s CBF wh sep szes =.. The same resuls are also appeared a Table 5

6 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 and Table where he appromae soluons show a beer mprovemen when he sep sze was geng smaller. Table 4: Comparson of he accuracy of appromae soluon beween Quadrac and Cubc Bezer homoopy funcon for equaon (8 Sep szes, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Table 5: Comparson of he accuracy of appromae soluon beween Quadrac and Cubc Bezer homoopy funcon for equaon ( Sep szes, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Table 6: Comparson of he accuracy of appromae soluon beween Quadrac and Cubc Bezer homoopy funcon for equaon ( Sep szes, Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Table 4, Table 5 and Table 6, show he resuls of he accuracy of appromae soluons for equaon (8, ( and ( respecvely. A Table 4, boh homoopy funcons show an mprovemen of he accuracy of appromae soluons when he sep szes of become smaller. Ths means, he accuracy of he appromae soluons ges beer as he value of decreased. Also from he able, CBF shows he mos precse value of he accuracy of appromae 7 soluons whch are.7756, and compared o QBF 5 whch are 6.49,., and.9. Ne he same resuls are shown n Table 5 and Table 6 where he value of he accuracy of appromae soluons approached o zero as he sep szes were geng smaller. All of he able showed he Cubc Bezer omoopy funcon are more accurae based on he accuracy of he appromae soluon ha approached o zero compared Quadrac Bezer homoopy funcon. Therefore, shows ha CBF s he more accurae funcon o solve hs knd of problem. 6

7 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 Table 7: Comparson of he mamum absolue error of soluons beween Quadrac and homoopy funcon for equaon (8 Sep szes, Error of Soluon, l Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Table 8: Comparson of he mamum absolue error of soluons beween Quadrac and homoopy funcon for equaon ( Sep szes, Error of Soluons, l Quadrac Bezer homoopy funcon, ( homoopy funcon, ( Table 9: Comparson of he mamum absolue error of soluons beween Quadrac and homoopy funcon for equaon ( Sep szes, Error of Soluons, l Quadrac Bezer homoopy funcon, ( homoopy funcon, ( The comparson resuls for mamum absolue errors a Table 7, Table 8 and 9, are dscussed. All of he ables show he mamum absolue error values of soluon become smaller as he value sep szes of decreased. I ndcaes ha he value of mamum absolue errors approached o zero as he sep szes of geng smaller. Besdes, from each able above shows he mprovemen value of CBF whch are 4.449,. and compared o QBF whch are 5.9,.9 and The mamum absolue errors of soluon go smaller and converged o zero were resuled as a good and relable mehod CONCLUSION Ths paper has developed he eended of CBF from QBF for solvng sngle eponenal equaon n order o compue he appromae soluon, he accuracy of he appromae soluon and he mamum absolue error of soluons. From he resul obaned, we can see ha he eended funcon gve a beer, accurae and precse value of solvng hs problem compared o prevous research whch s QBF. Concluded ha, when he appromae soluon 7

8 Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: Vol. 4, No.. Pages -8, 6 approached he real roo of an equaon, he accuracy of he appromae soluon ends o zero and he mamum absolue geng smaller. Thus, shows hs eended funcon s beer and more accurae han prevous funcon. REFERENCES [] Ismal, W. Zura Azan W. "Newon omoopy Connuaon Mehod for Solvng Nonlnear Equaons usng Mahemaca." Journal of Scence and Technology 7, no. (6. [] Rafq, Arf, and Muhammad Awas. "Convergence on he homoopy connuaon mehod." Inernaonal Journal of Applcaons Mahemacs and Mechancs 4, no. 6 (8: 6-7. [] Wu, Tzong-Mou. "A modfed formula of ancen Chnese algorhm by he homoopy connuaon echnque." Appled mahemacs and compuaon65, no. (5: - 5. [4] Wu, Tzong-Mou. "A sudy of convergence on he Newon-homoopy connuaon mehod." Appled Mahemacs and Compuaon 68, no. (5: [5] Wu, Tzong-Mou. "A new formula of solvng nonlnear equaons by Adoman and homoopy mehods." Appled mahemacs and compuaon 7, no. (6: [6] Wu, Tzong-Mou. "Solvng he nonlnear equaons by he Newon-homoopy connuaon mehod wh adjusable aulary homoopy funcon." Appled mahemacs and compuaon 7, no. (6: [7] Wu, Tzong-Mou. "The secan homoopy connuaon mehod." Chaos, Solons & Fracals, no. (7: [8] Nor, afzudn Mohamad, Ahmad Izan Md Ismal, and Ahmad Abdul Majd. "Quadrac bezer homoopy funcon for solvng sysem of polynomal equaons." Maemaka 9 (: [9] Farn, Gerald. Curves and surfaces for compuer-aded geomerc desgn: a praccal gude. Elsever, 4. [] Agoson, Ma K. Compuer graphcs and geomerc modelng. Vol.. London: Sprnger, 5. [] Chun, Changbum. "A new erave mehod for solvng nonlnear equaons."appled mahemacs and compuaon 78, no. (6: [] Fang, Lang, Tao Chen, L Tan, L Sun, and Bn Chen. "A modfed Newon-ype mehod wh sh-order convergence for solvng nonlnear equaons."proceda Engneerng 5 (:

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae