A MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS

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1 Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN Scence Publcatons do:1.844/jmssp Publshed Onlne 9 (1) 1 ( A MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS Kanttha Chompuvsed Department of Mathematcs and Appled Statstcs, Faculty of Scence and Technology, Nahon Ratchasma Rajabhat Unversty, Nahon Ratchasma, Thaland Receved 1-1-8, Revsed 1--16; Accepted ABSTRACT Solvng systems of nonlnear equaton s a great mportant whch arses n varous branches of scence and engneerng. In the last decades, several numercal technques were proposed to solve these problems. In ths study, we propose a modfed of teratve method whch s based on the dea of and Fed pont teraton method. The proposed method has been llustrated wth several eamples from the reference. The numercal results ndcate that ths proposed method provde the good performance of teratons. Keywords: System of Nonlnear Equaton, Newton Method, Fed Pont Iteraton Scence Publcatons 1. INTRODUCTION Solvng systems of nonlnear equatons s a great mportance, because these systems frequently arse n varous branches of pure and appled scences. The general form of a system of nonlnear equatons s Equaton 1: f 1( 1,,, n) =,f ( 1,,, n) =,f n( 1,,, n) = (1) where, each functon f can be thought of as mappng a vector = ( 1,,, n ) of the n dmensonal space R n, nto the real lne R. The system can alternatvely be represented by defnng a functonal F, mappng R n nto R n by. F( 1,,, n ) = (f 1 ( 1,,, n ),,f n ( 1,,, n )) T Usng vector notaton to represent the varables 1,,, n, a system (1) can be wrtten as the form: F() = The functons f 1, f,,f n are called the coordnate functons of F (Burden and Farres, 1). Recently, several teratve methods have been used to solve nonlnear equatons and the system of nonlnear equatons (Awawdeh, 9; Noor, 1; Cordero et al., 11; Sharma and Sharma, 11; Vahd et al., 1). 4 Wang (11) usng a thrd order famly of Newton-Le teraton method for solvng nonlnear equatons; Ozel (1) has consdered a new decomposton method for solvng the system of nonlnear equatons. Saha (1) has presented a modfed method to solvng nonlnear equatons by hybrdsng the results of and fed pont teraton method. Km et al. (1) developed a new scheme for the constructon of teratve methods for the soluton of nonlnear equatons and gvng a new class of methods from any teratve method. Furthermore, several teratve methods have been developed for solvng the system of nonlnear equatons by usng varous technques such as Newton s method, Revsed Adoman decomposton method, homotopy perturbaton method, Householder teratve method (Darvsh, 9; Noor and Waseem, 9; Hossen and Kafash, 1; Darvsh and Shn, 11; Hafz and Bahgat, 1a; 1b; Noor et al., 1). It s the purpose of ths study to ntroduce a new mprovement of by fed pont teraton method. We etend the Saha (1) method to solve systems of nonlnear equatons. Some eamples are tested and the obtaned results suggest that ths newly mprovement technque ntroduces a promsng tool and powerful mprovement for solvng a System of Nonlnear Equatons.

2 Kanttha Chompuvsed /Journal of Mathematcs and Statstcs 9 (1): 4-8, Descrpton of an Iteratve Method Consder a nonlnear equaton: f() = () We assume that the Equaton () admts a unque soluton *. In the all nown teratve formula used to fnd the real root s Equaton : +1 = - f ( ) Scence Publcatons f( ) () In fed pont teraton method () wll be rewrtten n the form: = g() (4) Equaton 4 whch s equvalent to () wll converge to a real root n the nterval D f g`() <1 for all n D provded the ntal appromaton s chosen n D. Choose the ntal appromaton then (, g ( )) s a pont on the curve Equaton 5: y = g() (5) The equaton of the tangent to the curve gven by (5) at the pont (, g ( )) s Equaton 6: y - g( ) = g ( )( - ) (6) Now we consder the lne Equaton 7: y = (7) Substtutng y = n (6) we have: - g( ) = g ( )( - ) g( ) - g ( ) [1- g ( )] = g( ) - g ( ) = 1- g ( ) whch produces the followng teraton scheme Equatons 8: g( ) - g ( ) +1 = 1- g ( ) 1.. The N-Dmensonal Case (8) The (Gautsch, 11; Sauer, 11) s commonly used for solvng such systems Equaton 9: 5 F() = (9) where, F: Ω R n R n s defned Equaton 1: +1 = - F ( ) F( ) (1) where, F ( ) s the Jacoban matr n pont. In fed pont teraton method (9) wll be rewrtten n the form = g() We rewrte Equaton 8 to solve the nonlnear system F() =, ths produces the followng teraton scheme Equatons 11: = [I - g ( )] [g( ) - g ( ) ] (11) where, I s an dentty matr. 1.. Numercal Eamples We present some eamples to llustrate the effcency of our proposed methods, we solve four systems of nonlnear equatons and one of a nonlnear boundary value problem. The followng tables show the Number of Iteratons (NI) to receve the requred soluton. For all test problems the stop crtera s F() <1 9. Eample 1 Consder the followng system of nonlnear equatons: + - = - = 1 1 * * * T T The eact solutons are = ( 1, ) = (1,1). To solve ths system, we set = (.1,) T as an ntal value. The results are presented n Table 1. Eample Consder the followng system of nonlnear equatons (Hossen and Kafash, 1): = = The eact solutons are * * * T T 1 = (, ) = (1,1) = (.57779, ) T To solve ths system, we set = (.5,.5) as an ntal value. The results are presented n Table.

3 Kanttha Chompuvsed /Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 Table 1. Numercal results for Eample NI X 1 X X 1 X Table. Numercal results for Eample NI X 1 X X 1 X Table. Numercal results for Eample NI X 1 X X X 1 X X Table 4. Numercal results for Eample 4 Number of teratons Method m = 5 m = 75 m = Table 5. Numercal results for Eample 5 Number of teratons Method M = 5 M = 75 M = Scence Publcatons 6 Eample Consder the followng system of nonlnear equatons (Awawdeh, 9): 1 - cos( ) -.5 = 1-81( +.1) + sn +1.6 = -1 1π - e + + = The eact solutons = (,, ) = (.5,, ). To solve are * * * * T T 1

4 Kanttha Chompuvsed /Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ths system, we set = (5, 4, ) T as an ntal value. The results are presented n Table. Eample 4 Consder the followng system of nonlnear equatons (Darvsh and Shn, 11): Scence Publcatons - cos( -1) =, = 1,,,m The eact solutons = (,,, ) = (1,1,,1). To solve ths system, are * * * * T T 1 m we set = (.5,.5,,.5) T as an ntal value. The results are presented n Table 4. Eample 5 Consder the nonlnear boundary value problem (Noor and Waseem, 9): y = -(y) - y + ln, 1, y(1) =, y() = ln Whose eact solutons s y = In. We consder the followng partton of the nterval: 1 = 1, n =, j = + jh, h =, j = 1,,,m -1 m Let us defne now: y = y( ) =, y m = ln, y = f( ), = 1,,,m -1 If we dscretze the problem by usng the second order fnte dfferences method defned by the numercal formulas: y - y y = h y+1 - y + y-1 =, = 1,,,m -1 h +1-1, = 1,,,m -1, y Then, we obtan a (m-1) (m-1) system of nonlnear equatons: 4y + y + 4y (h - ) - 4h ln =,4(y + y ) (y - y ) + 4y (h - ) - 4h ln =, =,,m -, +1-1 ( ) ( ) ( ) 4 In + y + In y + 4y h 4h In = m m m 1 m 1 we tae X wth () y = ln, = 1,,,m -1, as a 1 startng pont. The results are presented n Table 5. 7.CONCLUSION In ths study, we have demonstrated the applcablty of the modfed method for the system of nonlnear equatons wth the help of some concrete eamples. The results show that: the proposed problem can be solved by the proposed method.. REFERENCES Awawdeh, F., 9. On new teratve method for solvng systems of nonlnear equatons. Numercal Algorthms, 54: DOI: 1.17/s Burden, R.L. and J.D. Farres, 1. Numercal Analyss. 9th Edn., Cengage Learnng, Boston, MA., ISBN-1: , pp: 87. Cordero, A., J.L. Hueso, E. Martnez, J.R. Torregrosa, 11. Effcent hgh-order methods based on golden rato for nonlnear systems. Appled Math. Comput., 17: DOI: 1.116/j.amc Darvsh, M.T. and B.C. Shn, 11. Hgh-order newtonrylov methods to solve systems of nonlnear equatons. J. KSIAM., 15: 19-. Darvsh, M.T., 9. A two-step hgh order newton-le method for solvng systems of nonlnear equatons. Int. J. Pure Appled Math., 57: Gautsch, W., 11. Numercal Analyss. nd Edn., Sprnger, Boston, ISBN-1: , pp: 588. Hafz, M.A. and M.S.M. Bahgat, 1a. An effcent two-step teratve method for solvng system of nonlnear equatons. J. Math. Res., 4: 8-4. DOI: 1.559/jmr.v4n4p8 Hafz, M.A. and M.S.M. Bahgat, 1b. Modfed of householder teratve method for solvng nonlnear systems. J. Math. Comput. Sc. Hossen, M.M. and B. Kafash, 1. An effcent algorthm for solvng system of nonlnear equatons. Appled Math. Sc., 4: Km, Y.L., C. Chun and W. Km., 1. Some thrdorder curvature based methods for solvng nonlnear equatons. Stud. Nonlnear Sc., 1: Noor, M.A. and M. Waseem, 9. Some teratve methods for solvng a system of nonlnear equatons. Comput. Math. Appl., 57: DOI: 1.116/j.camwa Noor, M.A., 1. Iteratve methods for nonlnear equatons usng homotopy perturbaton technque. Appled Math. Inform. Sc., 4: 7-5.

5 Kanttha Chompuvsed /Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 Noor, M.A., M. Waseem, K.I. Noor and E. Al-Sad, 1. Varatonal teraton technque for solvng a system of nonlnear equatons. Optm Lett., DOI: 1.17/s Ozel, M., 1. A new decomposton method for solvng system of nonlnear equatons. J. Appled Math. Comput., 15: Saha, S., 1. A modfed method for solvng nonlnear equatons. Int. J. Comput. Sc. Intell. Comput., : Sauer, T., 11. Numercal Analyss. nd Edn., Prentce Hall, US., ISBN-1: , pp: 646. Sharma, J.R. and R. Sharma, 11. Some thrd order methods for solvng systems of nonlnear equatons. World Acad. Sc. Eng. Technol., 6: Vahd, A.R., S.H. Javad and S.M. Khorasan, 1. Solvng system of nonlnear equatons by restarted adoman s method. Appled Math. Comput., 6: Wang, P., 11. A thrd-order famly of newton-le teraton methods for solvng nonlnear equatons. J. Num. Math. Stochast., : Scence Publcatons 8