Application: Differential Equations

From ODE Superposition to PDE Convolution¶

In ODEs, if $L$ is a linear operator and we can solve $Ly_i = f_i$ for individual sources $f_i$, then linearity gives $L(\sum c_i y_i) = \sum c_i f_i$. This is the superposition principle for finite sums.

For PDEs, we replace the finite sum with an integral. Any source $f$ can be decomposed as a continuous superposition of point sources:

This is just the sifting property of the delta, but read physically it says: $f$ is a superposition of impulses $\delta(\cdot - y)$, each weighted by $f(y)$. If $E$ is the response to a single impulse at the origin (i.e. $LE = \delta$), then by translation $E(\cdot - y)$ is the response to an impulse at $y$. Superposing all these responses:

Since $L$ acts on $x$ and the integral is over $y$, linearity gives

So $u = E * f$ solves $Lu = f$. Convolution with the fundamental solution is the superposition principle, generalized from sums to integrals.

Fundamental Solutions¶

The equation $LE = \delta$ is understood in the distributional sense: $\langle LE, \varphi \rangle = \langle \delta, \varphi \rangle = \varphi(0)$ for all $\varphi \in \mathcal{D}(\mathbb{R}^d)$. By the definition of distributional differentiation, this becomes $\langle E, L^*\varphi \rangle = \varphi(0)$, where $L^*$ is the formal adjoint of $L$.

Why distributions are essential here. The equation $LE = \delta$ has no meaning in classical analysis: the right-hand side is not a function. But even the left-hand side is misleading classically. Consider the Laplacian in $\mathbb{R}^3$: the function $E(x) = -1/(4\pi|x|)$ satisfies $\Delta E(x) = 0$ for every $x \neq 0$. Pointwise, the Laplacian cannot “see” the singularity at the origin. But as a distribution, $E$ acts on test functions via integration, and the singularity contributes. When we compute $\langle \Delta E, \varphi \rangle = \langle E, \Delta\varphi \rangle$ and integrate by parts (excising a ball $B_\varepsilon(0)$, applying Green’s identity, taking $\varepsilon \to 0$), the boundary terms on the small sphere do not vanish: they converge to $\varphi(0)$. That is where the $\delta$ comes from. The distributional derivative detects a contribution at the singularity that pointwise differentiation misses entirely.

The Malgrange-Ehrenpreis Theorem¶

A natural question: does every linear constant-coefficient operator have a fundamental solution? The answer is yes.

This is a deep existence result, due independently to Malgrange and Ehrenpreis in the 1950s.

Structure of the general solution¶

The fundamental solution $E$ is a particular solution of $Lu = \delta$, just as in ODE theory. The general solution has the familiar form: general = homogeneous + particular.

This is the exact analogue of the ODE theorem: the general solution of a nonhomogeneous linear equation is a particular solution plus the general solution of the homogeneous equation. Distribution theory provides the particular solution via convolution; the homogeneous solutions depend on the specific operator $L$ and on boundary or initial conditions.

Solving PDEs via Convolution¶

With a fundamental solution in hand, the superposition argument from the beginning of this section becomes rigorous.

This is the distributional realization of the superposition principle: each point $y$ contributes an impulse response $f(y)\, E(\cdot - y)$, and the integral over all $y$ assembles the solution.