Volume 17, Issue 3, pp 378-399
Publication Date: 2017-07-19
Dina Abdullah Alrehaili - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Dalal Adnan Al-Maturi - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Salem Al-Aidarous - 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.
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 Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C 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.
Frozen Jacobian iterative methods, systems of nonlinear equations, nonlinear initial-boundary value problems, Jacobi-Gauss-Lobatto quadrature, collocation method.
 F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multi-step Newton method for solving systems of nonlinear equations, Numer. Algorithms, 71 (2016), 631–653.
 F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: application to PDEs and ODEs, Comput. Math. Appl., 70 (2015), 624–636.
 E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A. S. Al-Fhaid, An efficient higher-order quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 2013 (2013), 11 pages.
 A. H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput., 247 (2014), 30–46.
 A. H. Bhrawy, E. H. Doha, M. A. Abdelkawy, R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for solving nonlinear reaction-diffusion equations subject to Dirichlet boundary conditions, Appl. Math. Model., 40 (2016), 1703–1716.
 M. Dehghan, F. Fakhar-Izadi, 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.
 E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrdinger equations, J. Comput. Phys., 261 (2014), 244–255.
 E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35 (2011), 5662–5672.
 E. H. Doha, A. H. Bhrawy, R. M. Hafez, On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3802–3810.
 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.
 J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, (1970).
 U. Qasim, Z. Ali, F. Ahmad, F. Ahmad, S. Serra-Capizzano, 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.
 S. Qasim, Z. Ali, F. Ahmad, S. Serra-Capizzano, M. Z. Ullah, A. Mahmood, Solving systems of nonlinear equations when the nonlinearity is expensive, Comput. Math. Appl., 71 (2016), 1464–1478.
 J. Shen, T. Tang, L.-L. Wang, Spectral methods, Algorithms, analysis and applications, Springer Series in Computational Mathematics, Springer, Heidelberg, (2011).
 F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems, Optim. Lett., 8 (2014), 1001–1015.
 G. Szeg¨ o, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, American Mathematical Society, New York, (1939).
 E. Tohidi, S. Lotfi Noghabi, An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems, J. Appl. Math. Stat. Inform., 8 (2012), 73–85.
 J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall Series in Automatic Computation Prentice-Hall, Inc., Englewood Cliffs, N.J., (1964).
 M. Z. Ullah, S. Serra-Capizzano, F. Ahmad, An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Appl. Math. Comput., 250 (2015), 249–259.
 M. Z. Ullah, F. Soleymani, A. S. Al-Fhaid, Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numer. Algorithms, 67 (2014), 223–242.