|
چکیده
|
In this thesis, nonlinear two-dimensional Volterra-Urysohn integral equations are investigated. Under Lipschitz-type conditions imposed on the kernels, the existence and uniqueness of solutions to these integral equations are established by means of the Banach fixed-point theorem. Two numerical discretization schemes namely, the Euler method and the trapezoidal rule, are analyzed for approximating the solutions, and it is rigorously demonstrated that the Euler method exhibits first-order convergence, whereas the trapezoidal rule achieves second-order convergence. To support the convergence analysis of the trapezoidal scheme, a novel discrete Gronwall inequality is introduced. The theoretical framework is further complemented by a spectral-type numerical method based on Lucas polynomial expansions, which is presented in detail in Chapter Two. The proposed primary methods, along with a benchmark method from the literature, are implemented in a series of numerical experiments. The obtained results corroborate the theoretically derived convergence rates and demonstrate the superior efficiency of the proposed approaches in comparison with the reference method.
|