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.
MSC
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