Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

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Authors
Malik Zaka Ullah
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
 Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy.
Fayyaz Ahmad
 Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy.
 Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte d'Urgell 187, 08036 Barcelona, Spain.
 UCERD Islamabad, Pakistan.
Ali Saleh Alshomrani
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
A. K. Alzahrani
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Metib Said Alghamdi
 Mathematics Department, Faculty of science, Jazan University, P. O. Box 218, Jazan, Saudi Arabia.
Shamshad Ahmad
 Department of Heat and Mass Transfer Technological Center, Technical University of Catalonia, Colom 11, 08222 Terrassa, Spain.
Shahid Ahmad
 Department of Mathematics, Government College University Lahore, Lahore, Pakistan.
Abstract
Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A
frozen Jacobian multistep iterative method is presented. We divide the multistep iterative method into two
parts namely base method and multistep part. The convergence order of the constructed frozen Jacobian
iterative method is three, and we design the base method in a way that we can maximize the convergence
order in the multistep part. In the multistep part, we utilize a single evaluation of the function, solve four
systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence
order per multistep is four. Hence, the general formula for the convergence order is \(3 + 4(m  2)\) for
\(m \geq 2\) and \(m\) is the number of multisteps. In a single instance of the iterative method, we employ only
single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper
because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The
claimed convergence order is verified by computing the computational order of convergence for a system of
nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving
many nonlinear initial and boundary value problems.
Share and Cite
ISRP Style
Malik Zaka Ullah, Fayyaz Ahmad, Ali Saleh Alshomrani, A. K. Alzahrani, Metib Said Alghamdi, Shamshad Ahmad, Shahid Ahmad, Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 60216033
AMA Style
Ullah Malik Zaka, Ahmad Fayyaz, Alshomrani Ali Saleh, Alzahrani A. K., Alghamdi Metib Said, Ahmad Shamshad, Ahmad Shahid, Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs. J. Nonlinear Sci. Appl. (2016); 9(12):60216033
Chicago/Turabian Style
Ullah, Malik Zaka, Ahmad, Fayyaz, Alshomrani, Ali Saleh, Alzahrani, A. K., Alghamdi, Metib Said, Ahmad, Shamshad, Ahmad, Shahid. "Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 60216033
Keywords
 Frozen Jacobian iterative methods
 multistep iterative methods
 systems of nonlinear equations
 nonlinear initial value problems
 nonlinear boundary value problems.
MSC
 65L05
 65H10
 65L06
 34B15
 34A34
References

[1]
F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multistep Newton method for solving systems of nonlinear equations, Numer. Algorithms, 71 (2016), 631653

[2]
F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multistep Jarrattlike method for solving systems of nonlinear equations: application to PDEs and ODEs, Comput. Math. Appl., 70 (2015), 624636

[3]
E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A. S. AlFhaid, An efficient higherorder quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 2013 (2013), 11 pages

[4]
A. H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput., 247 (2014), 3046

[5]
W. H. Bi, H. M. Ren, Q. B. Wu, Threestep iterative methods with eighthorder convergence for solving nonlinear equations, J. Comput. Appl. Math., 225 (2009), 105112

[6]
A. Cordero, J. L. Hueso, E. Martínez, J. R. Torregrosa, A modified NewtonJarratt's composition, Numer. Algorithms, 55 (2010), 8799

[7]
M. Davies, B. Dawson, On the global convergence of Halley's iteration formula, Numer. Math., 24 (1975), 133135

[8]
M. Dehghan, F. FakharIzadi, The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Math. Comput. Modelling, 53 (2011), 18651877

[9]
Y. H. Geum, Y. I. Kim, A multiparameter family of threestep eighthorder iterative methods locating a simple root, Appl. Math. Comput., 215 (2010), 33753382

[10]
E. Halley,, A new exact and easy method of finding the roots of equations generally and without any previous reduction, Philos. Trans. Roy. Soc. London, 18 (1964), 136148

[11]
H. T. Kung, J. F. Traub, Optimal order of onepoint and multipoint iteration, J. Assoc. Comput. Mach., 21 (1974), 643651

[12]
H. Montazeri, F. Soleymani, S. Shateyi, S. S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations, J. Appl. Math., 2012 (2012), 15 pages

[13]
F. Soleymani, On a new class of optimal eighthorder derivativefree methods, Acta Univ. Sapientiae Math., 3 (2011), 169180

[14]
F. Soleymani, T. Lotfi, P. Bakhtiari, A multistep class of iterative methods for nonlinear systems, Optim. Lett., 8 (2014), 10011015

[15]
E. Tohidi, S. Lotfi Noghabi, An efficient Legendre pseudospectral method for solving nonlinear quasi bangbang optimal control problems, J. Appl. Math. Stat. Inform., 8 (2012), 7384

[16]
M. Z. Ullah, A. S. AlFhaid, F. Ahmad, Fourpoint optimal sixteenthorder iterative method for solving nonlinear equations, J. Appl. Math., 2013 (2013), 5 pages

[17]
M. Z. Ullah, S. SerraCapizzano, F. Ahmad, An efficient multistep iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Appl. Math. Comput., 250 (2015), 249259

[18]
M. Z. Ullah, F. Soleymani, A. S. AlFhaid, Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numer. Algorithms, 67 (2014), 223242

[19]
X. Wang, L. P. Liu, Modified Ostrowski's method with eighthorder convergence and high efficiency index, Appl. Math. Lett., 23 (2010), 549554