# Importance of integral equations

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.

• A big subfield in PDEs is a that of non-local equations: in time and especially in space. These equations are integro-differential. web.ma.utexas.edu/mediawiki/index.php/… Nov 1, 2021 at 21:11
• In cases where both differential and integral forms of the equations exist, there are math.SE posts arguing that the integral form is what you really care about or is more fundamental because it corresponds to a conservation law. The integral form can sometimes be more useful for solving or for numerical computations. Nov 2, 2021 at 15:08
• Why the downvote? Nov 3, 2021 at 17:25
• @TimothyChow The point about fundamental conservation laws being expressed as integral equations is a really good one. Nov 5, 2021 at 18:35
• I accidentally came across the rendering equation in 3D graphics - quite an important integral equation, it seems. Dec 15, 2021 at 11:54

## 2 Answers

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.

• This was well said. Nov 2, 2021 at 18:35
• Nice answer, but I think you forgot the $c$ in $(**)$. Nov 3, 2021 at 19:10
• @MichaelSeifert: Thanks you! Corrected. Nov 3, 2021 at 20:31

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.

• Do you have any specific examples of systems with a memory? :-) Nov 3, 2021 at 11:23
• for example, the frictional force in a visco-elastic medium is given by the integral $F(t)=\int_{-\infty}^{t}\gamma(t-t')\dot{x}(t')\,dt'$, with memory kernel $\gamma(t)$; Newton's equation of motion is then an integral equation, rather than a differential equation. Nov 3, 2021 at 11:55
• Anothe example of memory: dynamics of an inertial spherical passive particle in fluid flow: arxiv.org/abs/1310.2450 Nov 6, 2021 at 3:30
• Seems like there is quite a lot of integral equations connected to fluid dynamics... Nov 7, 2021 at 12:11