# Source term in a PDE

I am studying PDE theory at the moment and many times I find equations with a function, say $F(x,t)$, on the RHS that is refered to as a source term. Now, I would like to understand what is the intuition behind this. I know most of the time these functions can be almost whatever you want them to be. But what is the idea behind introducing them? Can someone explain me, perhaps with a simple example, the physics idea behind it. I have no background in physics.

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In the case of Poisson's equation, there is a concrete meaning, the function you are solving for is the gravitational (or electrostatic) potential. The right hand side is the source whose mass or electric charge generates the potential. –  Willie Wong Mar 13 '12 at 15:16

Another example than Poisson's equation of electrostatics (mentioned in the comments) is heat conduction. This is a special case of general diffusion phenomena and modelled by the inhomogeneous heat equation $$\partial_t u - \Delta u = f$$ $$u(0,x) = u_0(x)$$ where $u(t,x)$ is the temperature at time $t$ at the place $x$.

The $u_0$ is the initial condition, that is, it describes the temperature at every place at our initial time (which we have chosen to be $t=0$).

The $f$ is the source term and $f(t,x)$ describes if the medium is heated ($f(t,x)>0$) or cooled ($f(t,x)<0$) at time $t$ and place $x$.

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The word "source" can be a bit misleading because its precise interpretation depends on the physics behind the considered equation. My background is in solid mechanics, so I can give you a specific example from mechanics. The standard wave equation can be shown to describe small amplitude deflections $u$ of a thin elastic membrane of an infinite extent: $$T\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)=\rho\frac{\partial^2 u}{\partial t^2}.$$ Physically, this equation states the Newton second law (conservation of momentum). The terms on the left hand side are associated with the elastic restoring force, whereas the term on the right hand side is associated with the inertia. This equation, essentially, assumes that restoring force and inertia are the only forces acting on the membrane. The only way to produce non-trivial solutions (waves) on such a membrane is by using non-zero initial conditions.

Alternatively, you can imagine "sources", external forces that act on the membrane. They can be introduced into wave equation explicitly via an additional (inhomogeneous) term $p(x,y,t)$: $$T\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)=\rho\frac{\partial^2 u}{\partial t^2}-p(x,y,t).$$ Obviously, function $p(x,y,t)$ represents very general form of external forcing. When people say "source", they often imply a forcing that is somehow structured/localised in space and/or in time. For example, one can imagine a time-harmonic plane wave source at $x=-\infty$, which would be represented by the forcing function $p(x,y,t)=P\exp(i(kx-\omega t))$.

Very useful in applications are generalised source functions, representing idealised point loading situations. Source function $p(x,y,t)=P\delta(x-x_0)\delta(y-y_0)\delta(t)$ may represent an instantaneous point loading. Another useful function is a time-harmonic point source $p(x,y,t)=P\delta(x-x_0)\delta(y-y_0)\exp(i\omega t)$ ($\delta$ denotes the Dirac delta function). You can think of this source as the model of a transducer situated at point $(x_0,y_0)$ that vibrates at a fixed frequency $\omega$ and is observed in the far field, i.e. the size of the transducer must be small relative to the distance to the observer. If such source is acting on your membrane for sufficiently long time, you can imagine that the effects of the initial conditions will die out and the only remaining disturbances will be stationary, i.e. independent of time. This can be written more explicitly by seeking $u(x,y,t)=u(x,y)\exp(i\omega t)$, which transforms the original hyperbolic wave equation into the elliptic Helmholtz equation: $$T\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)+\rho\omega^2 u=-P\delta(x-x_0)\delta(y-y_0).$$ The replies by Willie Wong and Dirk prove examples of sources for parabolic equations. So, the concept of a source can be meaningful for any type of PDE.

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