Mathematical Physics Vol 1

7.2 Formation of partial differential equations

335

This is a partial differential equation of the second order, whose general solution is

u = x 2 y − 1 2 xy 2 + F ( x )+ G ( y ) . In the special case, when F ( x ) = 2sin x , G ( y ) = 3 y 4 − 5, we obtain the particular solution u = x 2 y − 1 2 xy 2 + 2sin x + 3 y 4 − 5 .

When solving partial differential equations, special conditions are often imposed, which the solution should satisfy. These conditions can be initial or boundary. They will be discussed in more detail later.

Monge notation

We will observe mainly second order partial differential equations, in which the unknown function u depend of two variables x and y . In that case, we will use the following, so called Monge 3 notation

∂ u ∂ x

∂ u ∂ y

p = u x =

; q = u y =

(7.2)

∂ 2 u ∂ x 2

∂ 2 u ∂ x ∂ y

∂ 2 u ∂ y 2

r = u xx =

; s = u xy =

; t = u yy =

Using this notation, the general first order partial equation can be expressed in the form

F ( x , y , u , p , q )= 0 ,

(7.3)

and the general second order partial equation as

F ( x , y , u , p , q , r , s , t )= 0 .

(7.4)

7.2 Formation of partial differential equations

Partial differential equations are formed in one of the following ways • by elimination of variable functions, • by elimination of constants, and • by a mathematical description of a problem (in geometry, mechanics, physics, technology, etc.). Let us demonstrate this on several examples.

3 Gaspard Monge, comte de Péluse, 1746-1818, French mathematician, founder of the École polytechnique and one of the founders of the École normale. He offered a new approach to infinitesimal geometry, and was the author of a new method for geometric integration. He also obtained significant results in analytical geometry in the space.