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Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method. (English) Zbl 07042327
Summary: A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.

MSC:
65 Numerical analysis
93 Systems theory; control
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[1] Ortega, J.M.; Rheinbodt, W.C.; ; Iterative Solution of Nonlinear Equations in Several Variables: London, UK 1970; .
[2] Traub, J.F.; ; Iterative Methods for the Solution of Equations: Englewood Cliffs, NJ, USA 1964; . · Zbl 0121.11204
[3] Burden, R.L.; Faires, J.D.; ; Numerical Analysis: Bostan, MA, USA 2001; .
[4] Abbasbandy, S.; Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method; Appl. Math. Comput.: 2003; Volume 145 ,887-893. · Zbl 1032.65048
[5] Abbasbandy, S.; Tan, Y.; Liao, S.J.; Newton-homotopy analysis method for nonlinear equations; Appl. Math. Comput.: 2007; Volume 188 ,1794-1800. · Zbl 1119.65032
[6] Chun, C.; Iterative methods improving Newton’s method by the decomposition method; Comput. Math. Appl.: 2005; Volume 50 ,1559-1568. · Zbl 1086.65048
[7] Shah, F.A.; Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations; Chin. J. Math.: 2014; Volume 2014 ,787591.
[8] Amat, S.; Busquier, S.; Grau, A.; Sanchez, M.G.; Maximum efficiency for a family of Newton-like methods with frozen derivatives and some applications; Appl. Math. Comput.: 2013; Volume 219 ,7954-7963. · Zbl 1288.65069
[9] Ullah, M.Z.; Soleymani, F.; Al-Fhaid, A.S.; Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs; Numer. Algorithm.: 2014; Volume 67 ,223-242. · Zbl 1316.65053
[10] Ahmad, F.; Tohidi, E.; Carrasco, J.A.; A parameterized multi-step Newton method for solving systems of nonlinear equations; Numer. Algorithm.: 2015; Volume 71 ,1017-1398.
[11] Ullah, M.Z.; Serra-Capizzano, S.; Ahmad, F.; An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs; Appl. Math. Comput.: 2015; Volume 250 ,249-259. · Zbl 1328.65156
[12] Ahmad, F.; Tohidi, E.; Ullah, M.Z.; Carrasco, J.A.; Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs; Comput. Math. Appl.: 2015; Volume 70 ,624-636.
[13] Poincare, H.; Second complement a l’Analysis Situs; Proc. Lond. Math. Soc.: 1900; Volume 32 ,277-308. · JFM 31.0477.10
[14] Leykin, A.; Verschelde, J.; Zhao, A.; Newton’s method with deflation for isolated singularities of polynomial systems; Theor. Comp. Sci.: 2006; Volume 359 ,111-122. · Zbl 1106.65046
[15] Liao, S.J.; The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems; Ph.D. Thesis: Shanghai, China 1992; .
[16] Liao, S.J.; On the homotopy analysis method for nonlinear problems; Appl. Math. Comput.: 2004; Volume 147 ,499-513. · Zbl 1086.35005
[17] Liao, S.J.; Campo, A.; Analytic solutions of the temperature distribution in Blasius viscous flow problems; J. Fluid Mech.: 2002; Volume 453 ,411-425. · Zbl 1007.76014
[18] Liao, S.J.; Tan, Y.; A general approach to obtain series solutions of nonlinear differential equations; Stud. Appl. Math.: 2007; Volume 119 ,297-354.
[19] Mogan, A.P.; Sommese, A.J.; A homotopy for solving general polynomial systems that respects m-homogeneous structures; Appl. Math. Comput.: 1987; Volume 24 ,101-113. · Zbl 0635.65057
[20] Pakdemirli, M.; Boyac, H.; Generation of root finding algorithms via perturbation theory and some formulas; Appl. Math. Comput.: 2007; Volume 184 ,783-788. · Zbl 1115.65056
[21] Noor, M.A.; Some iterative methods for solving nonlinear equations using homotopy perturbation method; Int. J. Comput. Math.: 2010; Volume 87 ,141-149. · Zbl 1182.65079
[22] Wu, Y.; Cheung, K.F.; Two-parameter homotopy method for nonlinear equations; Numer. Algorithm.: 2010; Volume 53 ,555-572. · Zbl 1193.65076
[23] Bhrawy, A.H.; An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system; Appl. Math. Comput.: 2014; Volume 247 ,30-46. · Zbl 1339.65188
[24] Dehghan, M.; Izadi, F.F.; The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves; Math. Comput. Model.: 2011; Volume 53 ,1865-1877. · Zbl 1219.65106
[25] Doha, E.H.; Bhrawy, A.H.; Ezz-Eldien, S.S.; Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations; Appl. Math. Comput.: 2011; Volume 35 ,5662-5672. · Zbl 1228.65126
[26] Doha, E.H.; Bhrawy, A.H.; Hafez, R.M.; On shifted Jacobi spectral method for high-order multi-point boundary value problems; Commun. Nonlinear Sci. Numer. Simul.: 2010; Volume 17 ,3802-3810. · Zbl 1251.65112
[27] Tohidi, E.; Noghabi, S.L.; An efficient Legendre Pseudospectral Method for Solving Nonlinear Quasi Bang-Bang Optimal Control Problems; J. Appl. Math. Stat. Inform.: 2012; Volume 8 ,73-85. · Zbl 1277.65104
[28] Allgower, E.L.; Bohmer, K.; Potra, F.A.; Rheinboldt, W.C.; A mesh-independence principle for operator equations and their discretizations; SIAM J. Numer. Anal.: 1986; Volume 23 ,160-169. · Zbl 0591.65043
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