Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs
In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BV...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2017/9407656 |
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| author | Fayyaz Ahmad Shafiq Ur Rehman Malik Zaka Ullah Hani Moaiteq Aljahdali Shahid Ahmad Ali Saleh Alshomrani Juan A. Carrasco Shamshad Ahmad Sivanandam Sivasankaran |
| author_facet | Fayyaz Ahmad Shafiq Ur Rehman Malik Zaka Ullah Hani Moaiteq Aljahdali Shahid Ahmad Ali Saleh Alshomrani Juan A. Carrasco Shamshad Ahmad Sivanandam Sivasankaran |
| author_sort | Fayyaz Ahmad |
| collection | DOAJ |
| description | In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is 5m-11, where m is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems. |
| format | Article |
| id | doaj-art-c230e0803ffe443b8a112ad6249a03c1 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-c230e0803ffe443b8a112ad6249a03c12025-02-03T01:23:18ZengWileyComplexity1076-27871099-05262017-01-01201710.1155/2017/94076569407656Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPsFayyaz Ahmad0Shafiq Ur Rehman1Malik Zaka Ullah2Hani Moaiteq Aljahdali3Shahid Ahmad4Ali Saleh Alshomrani5Juan A. Carrasco6Shamshad Ahmad7Sivanandam Sivasankaran8Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Eduard Maristanu 10, 08019 Barcelona, SpainDepartment of Mathematics, University of Engineering and Technology, Lahore, PakistanDipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, 22100 Como, ItalyDepartment of Information Systems, Faculty of Computing and Information Technology (Rabigh), King Abdulaziz University, Rabigh 21911, Saudi ArabiaDepartment of Mathematics, Government College University Lahore, Lahore, PakistanDepartment of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartament d’Enginyeria Electronica, Universitat Politècnica de Catalunya, Diagonal 647, Planta 9, 08028 Barcelona, SpainHeat and Mass Transfer Technological Center, Universitat Politècnica de Catalunya, Colom 11, 08222 Terrassa, SpainInstitute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, MalaysiaIn this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is 5m-11, where m is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.http://dx.doi.org/10.1155/2017/9407656 |
| spellingShingle | Fayyaz Ahmad Shafiq Ur Rehman Malik Zaka Ullah Hani Moaiteq Aljahdali Shahid Ahmad Ali Saleh Alshomrani Juan A. Carrasco Shamshad Ahmad Sivanandam Sivasankaran Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs Complexity |
| title | Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs |
| title_full | Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs |
| title_fullStr | Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs |
| title_full_unstemmed | Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs |
| title_short | Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs |
| title_sort | frozen jacobian multistep iterative method for solving nonlinear ivps and bvps |
| url | http://dx.doi.org/10.1155/2017/9407656 |
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