טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentReem Daniel
SubjectThe Generalized Goursat Problem with General Operators
DepartmentDepartment of Mathematics
Supervisor Professor Emeritus Boris Paneah


Abstract

The standard Goursat problem in a rectangle in the xy-plane is a second order hyperbolic partial differential equation (PDE) where the data is given on two vertical sides of the rectangle: x=0, y=0.


In the generalized Goursat problem the data is given on two arbitrary smooth curves

,   whose graphs are included in the rectangle and intersect only at the origin.




In the generalized Goursat problem with general operators the hyperbolic equation is

     ,

 where is Lipschitzian with respect to the uniform norm.


A very wide class of equations can be put in the above form, including standard and

non-standard equations as the following two examples show:

1.       ,

    where   is continuous and satisfies the Lipschitz condition with respect

     to its latter three components.

 2.      ,
  where .


 In this work it is proved that the problem has a unique solution.


An exact formulation of the problem is given in Chapter 1. Also, a physical application and a survey of what is known about the problem in the literature are given.


Chapter 2 is devoted to the preparation stage in which the problem is transformed to an equivalent integral equation.

In Chapter 3 the Lipschitz function case (example 1 above)  is discussed. A local (in a small rectangle) uniqueness and existence theorem is proved first for a special type of rectangles called proper rectangles, and then extended to the whole original rectangle.


In Chapter 4 the general case is discussed. In order to solve this case some terminology is required. The main concept  developed there is the concept of  restrictable operators.


Chapter 5 is short and only one theorem is proved there. This theorem is about existence and sometimes uniqueness of a solution to the problem in the domain

   .