M.Sc Thesis

M.Sc StudentReem Daniel
SubjectThe Generalized Goursat Problem with General Operators
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Boris Paneah (Deceased)


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