M.Sc Thesis | |

M.Sc Student | Reem Daniel |
---|---|

Subject | The Generalized Goursat Problem with General Operators |

Department | Department 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

.