The University of Basra discussed a master's thesis in the College of Education for Pure Sciences (Branching solutions for some nonlinear differential equations with a single coefficient)
The thesis presented by the student (Marwa Abdul-Aali Jassim), where we presented in this study two examples, the first equation was a non-linear differential equation of the fourth order, where the corresponding branching equation was studied, which appeared as a non-linear equation from one algebraic equation, as well as we found the branching scheme of the equation The branching corresponding to the nonlinear differential equation, which includes the number of solutions and their distribution in the parameter space, and then we studied the cases of symmetry and asymmetry of the above equation, and we found the geometric drawing for both cases and the number of solutions, and we considered the real solutions for the cases of symmetry and asymmetry as solutions for a dynamic system, after that we found the points in The phase portrait corresponding to the equation, for each region of the parameter space.
In the case of the second example, it was a third-order nonlinear partial differential equation that was converted into a second-order nonlinear ordinary equation and by the method of reduction we obtained the corresponding branching equation, where we extracted the branching equation for it using the Libnov-Schmidt method, we got one nonlinear algebraic equation , and we presented the branching diagram of the equation, which includes a number of solutions.
The purpose of the message
We study the branching solutions of nonlinear ordinary differential equations, using the Libnov-Schmidt reduction method.
