A derivation of the
one-dimensional diffusion equation

Branko Ćurgus

Mathematical facts used in the derivation

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• The Fundamental Theorem of Calculus: Let $f$ be a continuous real-valued function defined on an open interval which includes a real number $x$. Then \[ \lim_{\Delta x \to 0} \frac{1}{\Delta x} \int_{x}^{x+\Delta x} \!\!\! f(\xi)\, d\xi = f(x). \]
• Leibniz's Rule for differentiation under the integral sign: Let $F$ be a function of two independent variables $x$ and $t$ which is defined on a rectangle $(a,b)\times(c,d)$. Assume that the functions $F$ and $\displaystyle \frac{\partial F}{\partial t}$ are continuous. Let $x_1, x_2 \in (a,b)$. Then \[ \frac{d}{d t} \int_{x_1}^{x_2} F(\xi,t) d\xi = \int_{x_1}^{x_2} \frac{\partial F}{\partial t}(\xi,t) d\xi, \quad \text{for all} \quad t \in (c,d). \]

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Place the cursor over the image to see the diffusion of the dye.

The physical quantities at play in diffusion of dye

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• Since we are deriving the one-dimensional diffusion equation, we assume that at the fixed time the dye concentration and other relevant quantities depend only on the horizontal position which we denote by \(x\). The animation above is constructed in such a way that at each fixed time quantities related to dye are constant across each vertical cross section of the container.
• Density of Dye: The density of the dye at position $x$ and time $t$ is denoted by $u(x,t)$. Although our model involves only one spatial variable, $u(x,t)$ represents mass of dye per unit volume, measured, for example, in $\mathrm{g}/\mathrm{cm}^3$.
• Dye Flux: The dye flux at position $x$ and time $t$ is denoted by $\phi(x,t)$. It measures the signed rate at which dye mass passes through a cross section perpendicular to the $x$-axis, per unit area. Positive values indicate flow to the right, and negative values indicate flow to the left. Its units are units of mass per unit of area per unit of time, for example $\mathrm{g}/(\mathrm{cm}^2\,\mathrm{s})$.

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The physical laws that govern diffusion of dye

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• Conservation of Dye: We assume that dye is neither created nor destroyed. The rate of change of the amount of dye in a fixed region therefore equals the total inflow minus the total outflow. For the portion of the container between $x_1$ and $x_2$, this gives \[ \frac{d}{dt}\int_{x_1}^{x_2}u(\xi,t)A\,d\xi = A\phi(x_1,t)-A\phi(x_2,t). \] Here $A>0$ denotes the cross-sectional area of the container, assumed to be constant, and no dye passes through the side walls. Both sides of this equation have units of mass per unit of time. Dividing by $A$, we obtain \[ \frac{d}{dt}\int_{x_1}^{x_2}u(\xi,t)\,d\xi = -\bigl(\phi(x_2,t)-\phi(x_1,t)\bigr). \] Both sides of this last equation have units of mass per unit of area per unit of time.
• Fick's Law: The dye flows from regions of higher density to regions of lower density. More precisely, the dye flux is proportional to the spatial rate of change of dye density, with the opposite sign: \[ \phi(x,t) = -k\, \frac{\partial u}{\partial x}(x,t), \quad k > 0. \] Since $\phi$ has units of $\mathrm{g}/(\mathrm{cm}^2\,\mathrm{s})$ and $\partial u/\partial x$ has units of $\mathrm{g}/\mathrm{cm}^4$, the diffusion coefficient $k$ must have units of $\mathrm{cm}^2/\mathrm{s}$. In general, the units of $k$ are units of area per unit of time. The above equality assumes that the dye is diffusing in a uniform medium. If the medium is nonuniform, then $k$ might depend on $x$: \[ \phi(x,t) = -k(x)\, \frac{\partial u}{\partial x}(x,t), \quad k(x) > 0. \] The units of $k(x)$ remain the same.

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Putting things together

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We assume that the derivatives used below exist and are continuous.

• For the interval $[x,x+\Delta x]$, where $\Delta x>0$, Leibniz's Rule and the Conservation of Dye Law give:

  $\displaystyle \frac{d}{dt}\int_x^{x+\Delta x}u(\xi,t)\,d\xi $	$\displaystyle =\int_x^{x+\Delta x}\frac{\partial u}{\partial t}(\xi,t)\,d\xi $	by Leibniz's Rule
  $\displaystyle \frac{d}{dt}\int_x^{x+\Delta x}u(\xi,t)\,d\xi $	$\displaystyle =-\bigl(\phi(x+\Delta x,t)-\phi(x,t)\bigr) $	by the Conservation of Dye Law

  We conclude that \[ \int_x^{x+\Delta x}\frac{\partial u}{\partial t}(\xi,t)\,d\xi =-\bigl(\phi(x+\Delta x,t)-\phi(x,t)\bigr). \]
• Dividing both sides by $\Delta x$, we obtain \[ \frac{1}{\Delta x} \int_x^{x+\Delta x}\frac{\partial u}{\partial t}(\xi,t)\,d\xi =-\frac{\phi(x+\Delta x,t)-\phi(x,t)}{\Delta x}. \] Letting $\Delta x\to 0^+$ and using the Fundamental Theorem of Calculus on the left and the definition of the partial derivative on the right, we obtain \[ \frac{\partial u}{\partial t}(x,t) =-\frac{\partial\phi}{\partial x}(x,t). \]
• Substituting Fick's Law into the preceding equation, we obtain \[ \frac{\partial u}{\partial t}(x,t) =\frac{\partial}{\partial x} \left(k(x)\frac{\partial u}{\partial x}(x,t)\right). \] This is the Diffusion Equation. In a uniform medium, $k$ is constant, and the equation simplifies to \[ \frac{\partial u}{\partial t}(x,t) =k\,\frac{\partial^2 u}{\partial x^2}(x,t). \]

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Place the cursor over the image to see the diffusion of the dye.