علي الدبيسي

This thesis develops novel three-level algorithms for the numerical solution of second-order hyperbolic equations with mixed boundary conditions. The proposed schemes combine non-polynomial cubic splines for spatial discretization with finite-difference techniques in the time domain. A rigorous stability analysis demonstrates that, through suitable parameter selec‎tion, several existing methods for both homogeneous an‎d non-homogeneous problems appear as special cases of the present formulation. Two new families of highly accurate schemes with convergence orders of O(k^2+ h^2 )an‎d O(k^2+ h^4 ) are derived. Extensive numerical experiments confirm the robustness, precision, an‎d computational efficiency of the developed approaches, verifying their effectiveness in solving complex boundary-value problems. Furthermore, the thesis introduces a high-order numerical framework based on sextic spline functions for solving systems of second-order boundaryvalue problems frequently encountered in obstacle, unilateral, an‎d contact models. The proposed formulation attains significantly higher accuracy than conventional collocation, finite-difference, an‎d low-order spline methods. By leveraging the intrinsic smoothness an‎d flexibility of the sextic spline basis, the method captures delicate solution features an‎d efficiently han‎dles complex boundary constraints with remarkable precision. This thesis is based on the references [25,26].

اسپلاين درجه ششم , اسپلاين مكعبي غيرچندجمله‌اي , روش‌هاي تفاضل متناهي

Sextic spline , Non-polynomial cubic , Finite difference methods

Ali Dabisi

Dr. Rashidinia