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CHAPTER 1

PARTIAL DIFFERENTIAL EQUATION

DEFINITION

A partial differential equation (PDE) is an equation which contains one or more partial derivatives.

A solution of a PDE is a function which satisfies the equation.

EXAMPLE :

is a solution of

………………..(A)

ORDER OF PDE

The order of PDE is defined to be the order of the highest derivative in the equation.

Example : Equation (A) is second order.

DEFINITION

A partial differential equation is said to be linear if it is of the first degree in the field variable and its partial derivatives otherwise non-linear.

EXAMPLE :

and equation A are linear, while is non-linear.

The equation is homogeneous if every term in the equation contains either the field variable or one of its derivatives, Otherwise it is inhomogeneous non-homogeneous.

EXAMPLE :

Equation A is homogeneous whereas is inhomogeneous.

Three well-known PDE's :

(Wave equation)

(Heat-conduction or diffusion equation)

(Laplace equation)

where and k are physical constants.

If u and v are solutions of PDE, then Au + Bv is also a solution of that equation where A and B are constants.

To get a particular or unique solution of PDE we need

(i) Boundary conditions

Three types :

(a) Cauchy condition : given on boundary C.

(b) Dirichlet condition : u given on C.

(c) Neumann condition : given on C.

(ii) Initial conditions

SECOND ORDER GENERAL PDE :

……………..(B)

Here discriminant .

For, ; equation (B) is hyperbolic PDE : Wave equation

For, ; equation (B) is parabolic PDE : Heat equation

For, ; equation (B) is elliptic PDE : Laplace equation

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