ABSTRACT
A first-order partial differential equation with n-independent variables has the general form
F (
x1,x2, . . . ,xn,w, ∂w ∂x1
, ∂w ∂x2
, . . . , ∂w ∂xn
) = 0
where w = w(x1,x2, . . . ,xn) is the unknown function and F(. . .) is a given function. A second-order nonlinear partial differential equation with two independent vari-
ables x and y has the general form
F (
x,y,w, ∂w ∂x ,
∂w ∂y ,
∂ 2w ∂x2 ,
∂ 2w ∂x∂y ,
∂ 2w ∂y2
) = 0
where w = w(x,y) is the unknown function and F(. . . ) is a given function. A second-order semi-linear partial differential equation with two independent vari-
ables x and y has the form
a(x,y) ∂ 2w ∂x2 + 2b(x,y)
∂ 2w ∂x∂y + c(x,y)
∂ 2w ∂y2 = F
( x,y,w,
∂w ∂x ,
∂w ∂x
)
Given a point (x,y), the semi-linear partial differential equation above is said to be
parabolic if b2−ac = 0 hyperbolic if b2−ac > 0
Due to the complexity of partial differential equations there are many different
methods for solving them and a generalized list of the most popular methods include the following:
• The finite difference method is perhaps the easiest known technique for numerically solving PDEs and so it is often the first method chosen. The basic idea is to have functions be represented by their values at certain grid points and also have any partial derivatives be approximated through differences in these values. One disadvantage of this method is that it becomes quite complex when solving PDEs on irregular domains. In addition, it is not always easy to follow through and find solutions to the difference equations that result, evaluate their stability or establish their convergence especially for PDEs with variable coefficients or PDEs which are non-linear.
