Mathematical Physics Vol 1

Chapter 7. Partial differential equations

342

where u = u ( x 1 , x 2 ,..., x n ) is the unknown function, and the coefficients a i , b and c are, in the general case, functions of the form

a i = a i ( x 1 , x 2 ,..., x n ) , i = 1 ,..., n , b = b ( x 1 , x 2 ,..., x n ) , c = c ( x 1 , x 2 ,..., x n ) .

A quasilinear first order partial differential equation has the form

n ∑ i = 1

∂ u ∂ x i

a i

+ b = 0 .

(7.15)

In this case, coefficients next to respective derivatives are functions of the form

a i = a i ( x 1 , x 2 ,..., x n , u ) , i = 1 ,..., n , b = b ( x 1 , x 2 ,..., x n , u ) .

Thus, the equation is linear with respect to the first derivatives, but may be nonlinear with respect to the unknown function u .

7.3.1 On solutions for PDE To integrate (solve) the equation (7.14), i.e. (7.15), means to find all functions u ( x 1 ,..., x n ) that, together with their partial derivatives, identically satisfy the initial equation, for arbitrary values of the independent variables x 1 ,..., x n . The solution of equations (7.14), i.e. (7.15), is also called the integral surface (in case of a function with two variables). Complete, general, singular, mixed and particular solution Integrals (solutions) of partial differential equations can depend on • arbitrary constants (complete solution or complete integral), or • arbitrary functions (general solution or general integral), or • both arbitrary constants and arbitrary functions (mixed solution or mixed integral). In addition, the following solutions also occur • particular solution , which is obtained from the complete solution by substituting the initial conditions and • singular solution or singular integral. 5 Cauchy problem The notion of the Cauchy problem , in the theory of first order partial equations, implies the determination of the general solution that passes through some predetermined curve. For example, find that solution of the partial equation

∂ f ∂ x

= ψ ( x , y , f ) ,

which for x = x 0 becomes f = ϕ ( y ) . Thus, the task is to determine the surface ( f ( x , y ) ) that passes through the given curve ( ϕ ( y ) ),

which lies in a plane parallel to the yOz plane ( x = x 0 ). 5 This and other solutions will be discussed later in more detail.