An eighth order frozen Jacobian iterative method for solving nonlinear IVPs and BVPs
Authors
Dina Abdullah Alrehaili
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Dalal Adnan AlMaturi
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Salem AlAidarous
 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Fayyaz Ahmad
 Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, Como 22100, Italy.
Abstract
A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in
solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs).
In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four
function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is
a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion.
The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems.
The discretization of IVPs and BVPs is employed by using JacobiGaussLobatto collocation (JGLC) method. A comparison of
JGLC methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF
are shown by solving eleven IVPs and BVPs problems.
Keywords
 Frozen Jacobian iterative methods
 systems of nonlinear equations
 nonlinear initialboundary value problems
 JacobiGaussLobatto quadrature
 collocation method.
References

[1]
F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multistep Newton method for solving systems of nonlinear equations, Numer. Algorithms, 71 (2016), 631–653.

[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), 624–636.

[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), 30–46.

[5]
A. H. Bhrawy, E. H. Doha, M. A. Abdelkawy, R. A. Van Gorder, JacobiGaussLobatto collocation method for solving nonlinear reactiondiffusion equations subject to Dirichlet boundary conditions, Appl. Math. Model., 40 (2016), 1703–1716.

[6]
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), 1865–1877.

[7]
E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, R. A. Van Gorder, JacobiGaussLobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrdinger equations, J. Comput. Phys., 261 (2014), 244–255.

[8]
E. H. Doha, A. H. Bhrawy, S. S. EzzEldien, Efficient Chebyshev spectral methods for solving multiterm fractional orders differential equations , Appl. Math. Model., 35 (2011), 5662–5672.

[9]
E. H. Doha, A. H. Bhrawy, R. M. Hafez, On shifted Jacobi spectral method for highorder multipoint boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3802–3810.

[10]
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.

[11]
J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New YorkLondon (1970)

[12]
U. Qasim, Z. Ali, F. Ahmad, F. Ahmad, S. SerraCapizzano, M. Z. Ullah, M. Asma, Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method, Algorithms (Basel), 9 (2016), 17 pages.

[13]
S. Qasim, Z. Ali, F. Ahmad, S. SerraCapizzano, M. Z. Ullah, A. Mahmood, Solving systems of nonlinear equations when the nonlinearity is expensive, Comput. Math. Appl., 71 (2016), 1464–1478.

[14]
J. Shen, T. Tang, L.L. Wang, Spectral methods, Algorithms, analysis and applications, Springer Series in Computational Mathematics, Springer, Heidelberg (2011)

[15]
F. Soleymani, T. Lotfi, P. Bakhtiari, A multistep class of iterative methods for nonlinear systems, Optim. Lett., 8 (2014), 1001–1015.

[16]
G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, American Mathematical Society, New York (1939)

[17]
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), 73–85.

[18]
J. F. Traub, Iterative methods for the solution of equations, PrenticeHall Series in Automatic Computation PrenticeHall, Inc., Englewood Cliffs, N.J. (1964)

[19]
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), 249–259.

[20]
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), 223–242.