a source term
If a homogeneous PDE describes a system left to its own devices, a source term is the hand reaching in from outside to push it. It is the part of the equation that does not depend on the unknown u — a known function of position and time that adds, removes, or supplies whatever quantity u measures. In a heat problem the source is a heater or a chemical reaction releasing energy; in an electrostatics problem it is the charge density; for a vibrating string it is an external driving force.
In symbols, a source term is the f on the right of L[u] = f, or equivalently any term in the equation that contains no u and no derivatives of u. The heat equation with a source, u_t = k u_xx + Q(x, t), says the temperature changes both because heat diffuses (the u_xx term) and because the source Q deposits or removes energy at each point and time. Poisson's equation Laplacian u = -rho/epsilon makes the charge density rho the source of the electric potential. The source need not be constant — it can switch on and off, be concentrated at a point, or vary smoothly across the whole domain.
Source terms matter because they are how the outside world enters the model, and because an idealised source — a point source, a sudden unit impulse, mathematically a Dirac delta — is the key to a powerful solution strategy. If you can find u's response to a single point source (the fundamental solution or Green's function), then by superposition you can build the response to any distributed source by adding up the responses to all its little pieces. So the humble source term is not just a forcing: it is the doorway to the whole Green's-function method of solving inhomogeneous linear PDEs.
A point heat source switched on at the origin makes the right-hand side a Dirac delta: u_t - k u_xx = delta(x) delta(t). The response to this single idealised source is the heat kernel; superposing such responses solves u_t - k u_xx = Q(x, t) for any source Q.
A source can be smooth or idealised to a point delta; the delta response is the building block for all sources.
A Dirac delta source is not an ordinary function — it is a distribution, an idealisation of a unit amount concentrated at a point. Treating it as if it were a normal function leads to nonsense; the rigorous handling lives in the theory of distributions and Green's functions.