Canonical form partial differential equations examples
Answers
Answer:
Step-by-step explanation:
The most general case of second-order linear partial differential equation (PDE) in two independent variables is given by
A
∂
2u
∂x
2
+ B
∂
2u
∂x∂y
+ C
∂
2u
∂y
2
+ D
∂u
∂x
+ E
∂u
∂y
+ Fu = G (1)
where the coefficients A, B, and C are functions of x and y and do not vanish simultaneously,
because in that case the second-order PDE degenerates to one of first order. Further, the
coefficients D, E, and F are also assumed to be functions of x and y. We shall assume that the
function u(x, y) and the coefficients are twice continuously differentiable in some domain Ω.
The classification of second-order PDE depends on the form of the leading part of the
equation consisting of the second order terms. So, for simplicity of notation, we combine the
lower order terms and rewrite the above equation in the following form
A(x, y)
∂
2u
∂x
2
+ B(x, y)
∂
2u
∂x∂y
+ C(x, y)
∂
2u
∂y
2
= Φ
x, y,u,
∂u
∂x
,
∂u
∂y
(2a)
or using the short-hand notations for partial derivatives,
A(x, y)uxx + B(x, y)uxy + C(x, y)uyy = Φ(x, y,u,ux,uy) (2b)
As we shall see, there are fundamentally three types of PDEs – hyperbolic, parabolic, and
elliptic PDEs. From the physical point of view, these PDEs respectively represents the wave
propagation, the time-dependent diffusion processes, and the steady state or equilibrium processes. Thus, hyperbolic equations model the transport of some physical quantity, such as
fluids or waves. Parabolic problems describe evolutionary phenomena that lead to a steady
state described by an elliptic equation. And elliptic equations are associated to a special state
of a system, in principle corresponding to the minimum of the energy.
Mathematically, these classification of second-order PDEs is based upon the possibility of
reducing equation (2) by coordinate transformation to canonical or standard form at a point. It
may be noted that, for the purposes of classification, it is not necessary to restrict consideration