CHAPTER 1INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

1.1 INTRODUCTION A partial differential equation (PDE) can be defined as an equation that consists of one or more partial derivatives of an unknown function that depends on two or more variables (often time t) and one or several variables in space. The order of the highest derivative is called the order of the PDE.

A PDE in the dependent variable u and independent variables x, y, z and t may be written in the form

[1.1.1]

In this form F is a function of the indicated quantities and at least one of the partial derivatives.

Some examples of PDEs are:

[1.1.2]

[1.1.3]

[1.1.4]

[1.1.5]

[1.1.6]

[1.1.7]

[1.1.8]

[1.1.9]

[1.1.10]

[1.1.11]

[1.1.12]

[1.1.13]

A solution of a PDE in some region R of the space of the independent variables is a function that has all the partial derivatives appearing in the PDE in some domain D containing R and satisfies the PDE everywhere in R. In general the totality of solutions is very large.

Example: Verify that the following functions are solutions of

1.2.3.4.

In each case we can fine appropriate values of c that will satisfy the PDE..The unique solution of a PDE corresponding to a given physical problem may be obtained by the use of additional conditions arising from the problem.

Example: ,

whose solution is ………………….

ASSIGNMENT 1

1. Verify that each of the following functions is a solution of the one-

dimensional PDE with suitable c.

a.b.c.d.

2. Verify that each of the following functions is a solution of the one-

dimensional PDE with suitable c.

a.b.

c.

d.

3. Verify that each of the following functions is a solution of the two-

dimensional PDE .

a.b.

c.

d.