Importance of integral equations

Question

Differential equations are at the heart of applied mathematics - they are used to great success in fields from physics to economics. Certainly, they are very useful in modelling a wide range of phenomena.

Integral equations, on the other hand, do not receive such attention. While I have seen some integral equations crop up in physics (Boltzmann equation or the tautochrone problem) or biology (population dynamics), their importance pales in comparison to differential equations.

Why is it that differential equations are so much more popular than integral ones? Or am I just ignorant of the matter and there actually are many examples of integral equations in applied mathematics?

It also seems that when an integral equation appears, one immediately wants to reduce it to a differential equation. So examples, where this is not possible or not done for different reasons would be welcome.

Answer 1

One important point is that differential equations encode local behaviour of a system, while integral equations typically endcode global behaviour. Local behaviour is often easier to model and to grasp intuitively. In many cases, it can also be described by much simpler formulae.

More specifically:

• Let us consider the simple example where $p(t)$ describes the population of a species which reproduces without any resource limit. It is very intuitive to make the assumption that the growth of the population will be proportional to the size of the population, i.e., one has $$ (*) \quad \begin{cases} \dot p(t) & = c p(t), \\ p(0) & = p_0 \end{cases} $$ where $c$ is a constant, and $p_0$ is the initial size of the population. The reason why this behaviour is easy to model is that we have an intuitive understanding of growth, which is a local (with respect to time) quantity (and modelled by a derivative).

  The integral equation $$ (**) \qquad p(t) = p_0 + c \int_0^{t} p(s) \, ds $$ is mathematically equivalent to $(*)$, but its intuitive meaning is more difficult to understand, since it involves the behaviour of the population over time intervals rather than only at single instances in time.

• The local character of differential equation is reflected by the fact that initial and boundary conditions can be taken into account separately. In the initial value problem $(*)$, the initial condition $p(0) = p_0$ is separated from the differential equation, and has a clear intuitive meaning. The equivalent integral equations $(**)$ on the other hand, encodes both the dynamical behaviour of $p(t)$ and the initial condition in the same equation, which makes it more difficult to distinguish between the two effects.

• These phenomena get even more pronounced when one consides partial differential equations. For instance, the heat equation is very easy to heuristically derive locally. The behaviour at the boundary (fixed temperature = Dirichlet boundary conditions, thermal isolation = Neumann boundary conditions) can then be taken into account separately.

  Reformulating the equation as an integral equations (which, for homogeneous boundary conditions, essentially comes down to computing the resolvent of the Laplace operator with the given boundary conditions) means that ones has to include the boundary conditions in the integral equation. By corollary, such an integral formulation would also need to take the geometry of the domain into account, which can be arbitrarily complicated.

  On a related note, this also explains why it is impossible to explicitly compute the integral kernel of the resolvent (= Green function) of the Laplace operator on any but the most simple domains.

Answer 2

In physics, the predominance of differential over integral equations is not that obvious. Any system with a "memory", where the response at a certain time depends on the state at earlier times, requires an integral representation.

Electromagnetism is one area where actually it is the integral equations (Gauss, Ampère, Faraday) that appeared before the differential equations (Maxwell), and still today, the integral form of the equations is typically taught first.