Page 285 - Process Modelling and Simulation With Finite Element Methods

272 Process Modelling and Simulation with Finite Element Methods

Fredholm integral equations of the second kind:
h
(7.17)
a
Volterra integral equations of the first kind:

(7.18)
a
Volterra integral equations of the second kind:

(7.19)
a
In all four cases, g is the unknown function. K(x,t), called the kernel, and f(x) are
assumed to be known. When f(x)=O, the equation is said to be homogeneous.
One might reasonably ask why we bother with integral equations. The
answer is the theme of this chapter - integral equations are fundamentally
nonlocal. Some physical phenomena are inherently nonlocal in character, so
their description leads to integral or integro-differential equations. For instance,
Shaqfeh [ 181 derived a theory for transport properties of composite materials
that naturally leads to a nonlocal description of effective properties. Many
systems are “elliptical” in nature - the boundary data diffuses everywhere, say
steady state heat transfer or mass transfer - which results in the solution at a
point depending on the solution everywhere. Such nonlocal systems can be
conveniently described in terms of a Green’s function, which then leads to an
integral equation description for inhomogeneous systems. Finally, some
processes are conveniently described in a phase space (Fourier space, Laplace
space, size, volume or mass distribution) that involve nonlinear coupling of the
variables in phase space. When described in physical space, these phase space
couplings manifest as convolution integrals which are both nonlocal and
nonlinear. Rarely, transform methods, for instance the Abel transform, through a
clever change of variables, permits the restatement of an integral equation as an
equivalent differential equation, at least for smooth functions. Howison et al.
[19] give an example that was cited with regard to film drying in Chapter 6.
Otherwise, either discretization or power series expansion are the preferred
analysis techniques.
It is not the intention of this chapter to teach integral equation theory. An
introduction worth reading is given in Arfken’s book [20] and a thorough
grounding can be found in Stakgold [211 or Lovitt [22]. Here we intend only to