The College of Education for Pure Sciences at the University of Basrah in the Department of Mathematics discussed a PhD thesis on (Nonlinear Ritz approximation for some nonlinear differential equations using the developed Libanov-Schmidt method)
The thesis presented by the researcher (Hadeel Ghazi Abd Ali) included this thesis studying the nonlinear Ritz approximation for some nonlinear differential equations with boundary conditions using one of the most important methods in modern mathematics, which is the developed Libanov-Schmidt method.
First: We presented the modified Lyapunov-Schmidt method in the case of inhomogeneous problems when the zero space dimension is equal to two. Then find the non-linear Ritz approximation of the Friedholm function defined by the non-linear inhomogeneous differential equations such as the Kamassa-Holm equation and the Benjamin Bona Mahoney equation. Also finding the caustic (bifurcation group) and discussing the bifurcation of the critical points to the key functions of the Friedholm function defined by the Kamassa-Holm equation and the Benjamin Bona Mahoney equation. More than that, finding the spread of critical points using the investigation of the type of critical points in all the region using the program. Maple Classification of critical points into a key function with a companion dimension of 121 by calculating the geometric description of the caustic (bifurcation group). Furthermore, the plane curves for this function were derived and plotted in the parameter space. Also finding contour lines related to the field of the Kamassa-Holm equation and the Benjamin-Buna-Mahoney equations using a program. Mathematica finally obtains approximate solutions for the nonlinear wave equation. Additionally, find the principal function corresponding to the function of this equation.