The College of Education for Pure Sciences in the Department of Mathematics has researched a master's thesis on
(Hilbert space kernel cloning method to solve nonlinear integral-differential equations)
The thesis submitted by the researcher (Wafaa Kamel Nashmi) included
Integral-differential equations are a class of mathematical equations that contain derivatives and integrals. These equations have wide applications in various fields such as physics, engineering, biology, and others. This type of equations is difficult to find an analytical solution for.
Therefore, in this thesis, we aim to find approximate solutions to nonlinear integral-differential equations, as well as nonlinear Volterra-Fredholm integral-differential equations, and weak nonlinear single-kernel Volterra integral-differential equations.
The Hilbert space kernel cloning method was applied to nonlinear integro-differential equations, and combined with Taylor series to solve nonlinear Volterra-Fredholm integro-differential equations and weakly nonlinear single-kernel Volterra integro-differential equations. Taylor series was replaced by the nonlinear part of these equations to transform them into equivalent equations and ordinary differential equations. The solution methodology is based on generating an orthogonal basis for the functions, which contributed to formulating the numerical solutions in the form of a finite series. The results showed that the obtained approximate solutions are close to the exact solution, which confirms the effectiveness of the method as a reliable and practical tool for solving these equations. In the theoretical aspect, error analysis and convergence were discussed by proving some theorems.