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 ()()()(,)()0,2(,)()().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

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