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

Volume 9, Issue 12, pp 6021--6033
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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 multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step 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 multi-step part. In the multi-step 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 multi-step 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 multi-steps. 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, 6021--6033

##### 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):6021--6033

##### 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): 6021--6033

### Keywords

• Frozen Jacobian iterative methods
• multi-step iterative methods
• systems of nonlinear equations
• nonlinear initial value problems
• nonlinear boundary value problems.

•  65L05
•  65H10
•  65L06
•  34B15
•  34A34

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