Approximate Solution of the System of Nonlinear Integral Equations

Integral equations are used as mathematical models for many physical situations and applied mathematics. The numerical solutions of such integral equations have been highly studied by many authors. In this thesis we deal with the system of nonlinear integral equations (NIEs) of the form ()()()(,)()0...

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Main Author: Hazaimeh, Oday Shafiq
Format: Thesis
Language:English
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Published: 2010
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Online Access:http://psasir.upm.edu.my/id/eprint/21118/1/IPM_2010_18_IR.pdf
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spelling my-upm-ir.211182013-05-27T08:15:08Z Approximate Solution of the System of Nonlinear Integral Equations 2010-08 Hazaimeh, Oday Shafiq Integral equations are used as mathematical models for many physical situations and applied mathematics. The numerical solutions of such integral equations have been highly studied by many authors. In this thesis we deal with the system of nonlinear integral equations (NIEs) of the form ()()()(,)()0,2(,)()().tnyttnytxtHtxdnKtxdft   (1) where 00, ()ttTytt , and the given functions 0[0,][,](,),(,)tHtKtC , 0[,]()tftC . The aim of the work is to find the unknown functions 0011[,],(),()ttxtCytC in (1). To this end, we introduce the operator function 12()((),())0,0,((),())PXPXPXXxtyt , (2) and hence (1) can be expressed in the operator form 1()2()((),())()(,)(),((),())()(,)().tnyttnytPxtytxtHtxdPxtytftKtxd We solve (2) by the modified Newton-Kantorovich method 0)())((000XPXXXP , ))(),((000tytxX . (3) Substituting the first derivatives in (3), we have 000010000()00()10000()0()()(,)()()(,())(())()(,)()(),(,)()()(,())(())()(,)()().tnnyttnyttnnyttnytxtHtnxxdHtytxytytHtxdxtKtnxxdKtytxytytKtxdft (4) where )()()(01txtxtx , )()()(01tytyty . Solving (4) in terms of (),()xtyt we obtain 11(),()xtyt , by continuing this process, we arrive to the sequence of approximate solutions (),()mmxtyt from 0011101()10()00011()()(,)()()(),(,)()()()1()(,())(())(,)()()mtnmmmyttnmmytmntnmmytxtnKtxxdFtnHtxxdxtytHtytxytHtxdxt (5) where )()()(1txtxtxmmm and )()()(1tytytymmm , m=2, 3… In discretization process the modified trapezoidal rule is applied for Eq. (5). In this thesis we have proved the existence and the uniqueness of the solution of Eq. (1). Moreover, the rate of convergence of modified Newton-Kontorovich method for Eq. (2) is established. Finally, FORTRAN code is developed to obtain numerical results which are in line with the theoretical findings Nonlinear integral equations 2010-08 Thesis http://psasir.upm.edu.my/id/eprint/21118/ http://psasir.upm.edu.my/id/eprint/21118/1/IPM_2010_18_IR.pdf application/pdf en public masters Universiti Putra Malaysia Nonlinear integral equations Institute for Mathematical Research English
institution Universiti Putra Malaysia
collection PSAS Institutional Repository
language English
English
topic Nonlinear integral equations


spellingShingle Nonlinear integral equations


Hazaimeh, Oday Shafiq
Approximate Solution of the System of Nonlinear Integral Equations
description Integral equations are used as mathematical models for many physical situations and applied mathematics. The numerical solutions of such integral equations have been highly studied by many authors. In this thesis we deal with the system of nonlinear integral equations (NIEs) of the form ()()()(,)()0,2(,)()().tnyttnytxtHtxdnKtxdft   (1) where 00, ()ttTytt , and the given functions 0[0,][,](,),(,)tHtKtC , 0[,]()tftC . The aim of the work is to find the unknown functions 0011[,],(),()ttxtCytC in (1). To this end, we introduce the operator function 12()((),())0,0,((),())PXPXPXXxtyt , (2) and hence (1) can be expressed in the operator form 1()2()((),())()(,)(),((),())()(,)().tnyttnytPxtytxtHtxdPxtytftKtxd We solve (2) by the modified Newton-Kantorovich method 0)())((000XPXXXP , ))(),((000tytxX . (3) Substituting the first derivatives in (3), we have 000010000()00()10000()0()()(,)()()(,())(())()(,)()(),(,)()()(,())(())()(,)()().tnnyttnyttnnyttnytxtHtnxxdHtytxytytHtxdxtKtnxxdKtytxytytKtxdft (4) where )()()(01txtxtx , )()()(01tytyty . Solving (4) in terms of (),()xtyt we obtain 11(),()xtyt , by continuing this process, we arrive to the sequence of approximate solutions (),()mmxtyt from 0011101()10()00011()()(,)()()(),(,)()()()1()(,())(())(,)()()mtnmmmyttnmmytmntnmmytxtnKtxxdFtnHtxxdxtytHtytxytHtxdxt (5) where )()()(1txtxtxmmm and )()()(1tytytymmm , m=2, 3… In discretization process the modified trapezoidal rule is applied for Eq. (5). In this thesis we have proved the existence and the uniqueness of the solution of Eq. (1). Moreover, the rate of convergence of modified Newton-Kontorovich method for Eq. (2) is established. Finally, FORTRAN code is developed to obtain numerical results which are in line with the theoretical findings
format Thesis
qualification_level Master's degree
author Hazaimeh, Oday Shafiq
author_facet Hazaimeh, Oday Shafiq
author_sort Hazaimeh, Oday Shafiq
title Approximate Solution of the System of Nonlinear Integral Equations
title_short Approximate Solution of the System of Nonlinear Integral Equations
title_full Approximate Solution of the System of Nonlinear Integral Equations
title_fullStr Approximate Solution of the System of Nonlinear Integral Equations
title_full_unstemmed Approximate Solution of the System of Nonlinear Integral Equations
title_sort approximate solution of the system of nonlinear integral equations
granting_institution Universiti Putra Malaysia
granting_department Institute for Mathematical Research
publishDate 2010
url http://psasir.upm.edu.my/id/eprint/21118/1/IPM_2010_18_IR.pdf
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