Sections 13.1 – 13.4
What PDEs Are: Foundations and Conditions
Definition · Functions of Several Variables · Partial Derivatives · Initial and Boundary Conditions
Section 13.1
What a Partial Differential Equation Is
An ordinary differential equation (ODE) involves an unknown function of a single independent variable and its ordinary derivatives with respect to that variable. Everything studied in Chapters 1 through 12 falls into this category. The unknown function $y(t)$ or $y(x)$ depends on one variable, and derivatives such as $\frac{dy}{dt}$ or $\frac{d^2y}{dx^2}$ are taken with respect to that single variable.
A partial differential equation (PDE) involves an unknown function of two or more independent variables and its partial derivatives with respect to those variables. The unknown might be $u(x, t)$, depending on position $x$ and time $t$. Derivatives are taken with respect to each variable separately, holding the others fixed. This distinction is not merely notational — it reflects a genuine difference in the kind of system being described.
Partial Derivative Notation
If $u$ is a function of $x$ and $t$, the partial derivative of $u$ with respect to $t$ (holding $x$ fixed) is written as $\frac{\partial u}{\partial t}$ or $u_t$. The partial derivative with respect to $x$ (holding $t$ fixed) is $\frac{\partial u}{\partial x}$ or $u_x$. Second partial derivatives are $u_{tt}$, $u_{xx}$, and $u_{xt}$. Both notations are standard and used throughout this chapter.
The Three Classical PDEs
| Name | Equation | Describes |
|---|
| Heat Equation | $u_t = k\, u_{xx}$, $k > 0$ | Temperature distribution in a rod; diffusion of concentration |
| Wave Equation | $u_{tt} = c^2 u_{xx}$, $c > 0$ | Displacement of a vibrating string; propagation of sound or light |
| Laplace's Equation | $u_{xx} + u_{yy} = 0$ | Steady-state temperature; electric potential; ideal fluid flow |
All three are second-order linear PDEs. A PDE is linear if the unknown function and its partial derivatives appear only to the first power and are not multiplied together. Solutions to PDEs are functions of multiple variables, and verifying a solution requires computing the appropriate partial derivatives and confirming that the equation is satisfied.
Example 13.1.1
Verifying a Solution to the Heat Equation
- ProblemVerify that $u(x, t) = e^{-k\pi^2 t} \sin(\pi x)$ satisfies the heat equation $u_t = k\, u_{xx}$.
- ThinkCompute $u_t$ and $u_{xx}$ separately, then check whether $u_t = k\,u_{xx}$. Treat $x$ as constant when differentiating with respect to $t$, and $t$ as constant when differentiating with respect to $x$.
- Solve
$$u_t = \frac{\partial}{\partial t}\!\left[e^{-k\pi^2 t}\sin(\pi x)\right] = -k\pi^2 e^{-k\pi^2 t}\sin(\pi x)$$
$$u_x = e^{-k\pi^2 t}\cdot \pi\cos(\pi x), \qquad u_{xx} = -\pi^2 e^{-k\pi^2 t}\sin(\pi x)$$
$$k\, u_{xx} = k\cdot(-\pi^2 e^{-k\pi^2 t}\sin(\pi x)) = -k\pi^2 e^{-k\pi^2 t}\sin(\pi x) = u_t \checkmark$$
- InterpretThis solution describes a temperature profile shaped like a sine wave that decays exponentially in time. The amplitude decreases at a rate determined by $k\pi^2$. As $t \to \infty$, the temperature approaches zero everywhere, consistent with heat flowing out toward cold boundaries.
Answer: $u(x, t) = e^{-k\pi^2 t}\sin(\pi x)$ satisfies $u_t = k\,u_{xx}$.
- Classify each equation as ODE or PDE, state its order, and identify it as linear or nonlinear:
(a) $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} - y = 0$ (b) $u_t = 5u_{xx}$ (c) $u_{tt} = 9u_{xx} - u$ (d) $u_x u_y = u$ (e) $u_{xx}+u_{yy}+u_{zz}=0$
- Verify that $u(x, t) = e^{-4kt}\sin(2x)$ satisfies $u_t = k\,u_{xx}$.
- Verify that $u(x, t) = \cos(3x)\cos(3ct)$ satisfies $u_{tt} = c^2 u_{xx}$.
- Verify that $u(x, y) = e^x\sin(y)$ satisfies Laplace's equation $u_{xx} + u_{yy} = 0$.
- Explain in one paragraph why an ODE alone cannot describe temperature in a two-dimensional plate.
Section 13.2
Functions of Several Variables
A function $u(x, t)$ assigns a single real number to each pair $(x, t)$ in its domain. Understanding such a function visually means understanding its cross-sections. The cross-section at fixed $t = t_0$ is $u(x, t_0)$, showing how the quantity is distributed across space at a particular moment. The cross-section at fixed $x = x_0$ is $u(x_0, t)$, showing how the quantity at a particular location evolves over time.
A level curve of $u(x, t)$ is a curve in the $xt$-plane where $u$ takes a constant value. For the heat equation, level curves show isotherms — lines of equal temperature — moving through the material over time.
Example 13.2.1
Evaluating a Function of Two Variables and Reading Cross-Sections
- ProblemFor $u(x, t) = e^{-t}\sin(x)$, evaluate $u\!\left(\frac{\pi}{2}, 1\right)$, find the cross-section $u(0, t)$ for all $t$, and find the initial profile $u(x, 0)$.
- Solve
$$u\!\left(\tfrac{\pi}{2}, 1\right) = e^{-1}\sin\!\left(\tfrac{\pi}{2}\right) = \tfrac{1}{e}\cdot 1 = \tfrac{1}{e}$$
$$u(0, t) = e^{-t}\sin(0) = 0 \text{ for all } t \quad\text{(left boundary is always zero)}$$
$$u(x, 0) = e^{0}\sin(x) = \sin(x) \quad\text{(initial temperature profile)}$$
- InterpretThe cross-section $u(0,t) = 0$ tells us the left boundary is held at zero temperature for all time. The profile $u(x,0) = \sin(x)$ is the initial shape of the temperature distribution across the rod.
Answer: $u(\pi/2, 1) = 1/e$. Left boundary: $u(0,t)=0$. Initial profile: $u(x,0)=\sin(x)$.
Section 13.3
Partial Derivatives Review
The partial derivative $\frac{\partial u}{\partial x}$ measures the rate of change of $u$ with respect to $x$ while holding $t$ fixed. The partial derivative $\frac{\partial u}{\partial t}$ measures the rate of change with respect to $t$ while holding $x$ fixed. Computing partial derivatives uses all the same rules as ordinary differentiation — power rule, product rule, quotient rule, chain rule — with all other variables treated as constants.
Example 13.3.1
Computing Partial Derivatives
- ProblemFor $u(x, t) = x^2 \sin(t) + e^{3x}\cos(t)$, compute $u_x$, $u_t$, $u_{xx}$, and $u_{tt}$.
- Solve
$$u_x = 2x\sin(t) + 3e^{3x}\cos(t)$$
$$u_t = x^2\cos(t) - e^{3x}\sin(t)$$
$$u_{xx} = 2\sin(t) + 9e^{3x}\cos(t)$$
$$u_{tt} = -x^2\sin(t) - e^{3x}\cos(t)$$
Answer: As computed above. When finding $u_x$, treat $\sin(t)$ and $\cos(t)$ as constants; when finding $u_t$, treat $x^2$ and $e^{3x}$ as constants.
Example 13.3.2
Verifying a Solution to the Wave Equation
- ProblemShow that $u(x,t) = \sin(x - ct)$ satisfies $u_{tt} = c^2 u_{xx}$.
- Solve
$$u_x = \cos(x-ct),\quad u_{xx} = -\sin(x-ct)$$
$$u_t = -c\cos(x-ct),\quad u_{tt} = -c^2\sin(x-ct)$$
$$c^2 u_{xx} = c^2\cdot(-\sin(x-ct)) = -c^2\sin(x-ct) = u_{tt} \checkmark$$
- InterpretThe function $\sin(x-ct)$ is a wave moving to the right at speed $c$. At $t=0$ the wave is $\sin(x)$; at $t=1$ it is $\sin(x-c)$, the same shape shifted right by $c$. This confirms that the wave equation describes rightward-traveling waves at speed $c$.
Answer: $u(x,t)=\sin(x-ct)$ satisfies the wave equation with wave speed $c$.
Student Note
When computing partial derivatives, the most common mistake is failing to treat the other variable as a constant. When differentiating with respect to $x$, write the $t$-dependent parts as if they were numbers — because in that moment of differentiation, they are. This habit prevents nearly all partial derivative errors.
Section 13.4
Initial Conditions and Boundary Conditions
PDEs require two types of auxiliary conditions. Initial conditions specify the state of the system at the starting time $t = 0$ throughout the spatial domain. Boundary conditions specify the behavior of the system at the edges of the spatial domain for all $t \geq 0$. Together, the PDE plus appropriate initial and boundary conditions form a well-posed problem with a unique solution.
| Condition Type | Specifies | Heat Equation Meaning | Wave Equation Meaning |
|---|
| Dirichlet | Value of $u$ at boundary | Fixed endpoint temperatures: $u(0,t) = T_0$ | Clamped string endpoints: $u(0,t)=0$ |
| Neumann | Normal derivative $u_x$ at boundary | Insulated ends: $u_x(0,t)=0$ (no heat flow) | Free ends: $u_x(L,t)=0$ |
| Mixed (Robin) | $u_x + h\cdot u$ at boundary | Convective cooling: heat exchange with environment | Elastically supported end |
How Many Conditions Are Needed?
The heat equation $u_t = ku_{xx}$ has one time derivative → needs one initial condition: $u(x,0) = f(x)$ for all $x$ in the domain.
The wave equation $u_{tt} = c^2 u_{xx}$ has two time derivatives → needs two initial conditions: initial displacement $u(x,0) = f(x)$ and initial velocity $u_t(x,0) = g(x)$.
For a rod $0 \leq x \leq L$, both equations need two boundary conditions — one at each end.
Example 13.4.1
Writing a Complete Initial-Boundary Value Problem
- ProblemA metal rod of length $\pi$ is held at temperature zero at both ends. Its initial temperature distribution is $f(x) = \sin(x)$ for $0 \leq x \leq \pi$. Write the full problem for the temperature $u(x,t)$.
- ThinkThe governing equation is the heat equation. The boundary conditions describe the fixed endpoint temperatures. The initial condition describes the temperature profile at $t=0$.
- Solve
$$\text{PDE: } u_t = k\,u_{xx}, \quad 0 < x < \pi,\; t > 0$$
$$\text{Boundary conditions: } u(0,t) = 0,\quad u(\pi,t) = 0,\quad t \geq 0$$
$$\text{Initial condition: } u(x,0) = \sin(x),\quad 0 \leq x \leq \pi$$
- InterpretThe boundary conditions fix the rod's ends at zero for all time — something must actively maintain these temperatures. The initial condition gives the rod's temperature shape at the moment we begin observing.
Answer: The three-line system of PDE, boundary conditions, and initial condition above constitutes the complete problem.
Student Note
In ODE problems, conditions are given at a single point: $y(0)=1$. In PDE problems, initial conditions are given along an entire line in the domain (at $t=0$ for all relevant $x$), and boundary conditions are given along the boundary (at $x=0$ and $x=L$ for all relevant $t$). This difference in the structure of conditions is one of the first adjustments when moving from ODEs to PDEs.
- A rod of length $L$ is held at $u=0$ at $x=0$ and $u=T_0$ at $x=L$. Write the Dirichlet boundary conditions formally.
- A rod of length $L$ has insulated ends. Write the Neumann boundary conditions using partial derivative notation.
- A string of length $L$ is clamped at both ends. Write the boundary conditions for the wave equation.
- For the heat problem on $0\leq x\leq 2$ with $u(0,t)=3$ and $u(2,t)=7$, what type of boundary conditions are these? Find the steady-state solution.
- A rod of length $\pi$ is held at $u=0$ at both ends. Its initial temperature is $u(x,0)=2\sin(3x)$. Write the complete initial-boundary value problem.
Sections 13.5 – 13.10
The Three Classical Equations and Equilibrium
Heat Equation · Diffusion · Wave Equation · Vibrating Strings · Laplace's Equation · Steady State
Section 13.5
The Heat Equation as an Introduction
The heat equation arises from two physical principles: Fourier's Law of Heat Conduction, which states that heat flux is proportional to the negative temperature gradient ($q = -\kappa\, u_x$, heat flows from hot to cold), and conservation of energy, which requires that heat accumulating in a segment equals the net heat flowing in. Combining both and taking a limit gives:
$$u_t = k\,u_{xx}, \qquad k = \frac{\kappa}{\rho c_p} > 0$$
where $\kappa$ is thermal conductivity, $\rho$ is density, and $c_p$ is specific heat. The thermal diffusivity $k$ measures how quickly heat spreads through the material.
The equation says the rate at which temperature changes at a point is proportional to $u_{xx}$, the concavity of the temperature profile. If the profile is concave down ($u_{xx} < 0$), that point is hotter than its neighbors, so heat flows away and temperature decreases. If concave up ($u_{xx} > 0$), the point is cooler than its neighbors, so heat flows in. Hot spots cool down and cool spots warm up. The heat equation is the precise mathematical statement of this tendency.
Standard Problem: Rod with Zero Endpoints
$$u_t = k\,u_{xx},\quad 0 < x < L,\; t > 0$$
$$u(0,t) = 0,\quad u(L,t) = 0,\quad t \geq 0$$
$$u(x,0) = f(x),\quad 0 \leq x \leq L$$
As $t \to \infty$, all heat flows out through the cold endpoints. The solution approaches zero everywhere. If instead $u(0,t) = T_0$ and $u(L,t) = T_1$, the long-term state is the linear profile $T_0 + (T_1-T_0)x/L$.
Example 13.5.1
Interpreting a Superposition of Modes
- ProblemFor $k=1$ on $[0,\pi]$ with zero endpoints, each term $u_n = e^{-n^2 t}\sin(nx)$ satisfies the PDE and boundary conditions. Verify, and determine which mode decays most slowly.
- Solve
$$(u_n)_t = -n^2 e^{-n^2 t}\sin(nx), \quad (u_n)_{xx} = -n^2 e^{-n^2 t}\sin(nx)$$
$$\Rightarrow (u_n)_t = (u_n)_{xx} \checkmark$$
$$u_n(0,t) = e^{-n^2 t}\sin(0) = 0 \checkmark, \quad u_n(\pi,t) = e^{-n^2 t}\sin(n\pi) = 0 \checkmark$$
- InterpretThe term for $n=1$ decays as $e^{-t}$. The term for $n=2$ decays as $e^{-4t}$ — four times faster. Higher modes decay faster still. For large $t$, the fundamental mode ($n=1$) dominates. The long-term temperature profile approaches the shape $b_1\sin(x)$ scaled by the slowly decaying factor $e^{-t}$.
Answer: Each mode satisfies the PDE and boundary conditions. The $n=1$ mode decays most slowly. The rod eventually cools to zero everywhere.
Section 13.6
Diffusion Models
The heat equation describes diffusion more generally. The same equation $u_t = D\,u_{xx}$ (with diffusivity $D$) describes concentration of a dissolved substance spreading through a still fluid, the probability density of a randomly moving particle (Brownian motion), and the spread of biological populations across a landscape. Diffusion is among the most universally applicable PDEs in science.
A remarkable explicit solution on the entire real line, starting from a concentrated unit of substance at the origin, is the fundamental solution (heat kernel):
$$u(x,t) = \frac{1}{\sqrt{4\pi D t}}\, e^{-x^2/(4Dt)}$$
This is a Gaussian (bell-curve) profile centered at $x=0$ with standard deviation $\sigma = \sqrt{2Dt}$. As time increases, the bell curve spreads and flattens, conserving the total area (total amount of diffusing substance). The width grows as $\sqrt{t}$, not as $t$ — spread is proportional to the square root of time. This is characteristic of diffusion and much slower than the linear spread of a traveling wave.
Diagram Instruction — Fundamental Solution
Draw a horizontal $x$-axis and a vertical $u$-axis. Show three Gaussian curves, each centered at $x=0$. The first (small $t$) is tall and narrow. The second (moderate $t$) is shorter and wider. The third (large $t$) is shorter still and very wide. Label each curve with its time. Shade the area under each curve to show that all areas are equal, representing conservation of the diffusing substance.
Example 13.6.1
Comparing Diffusion Timescales
- ProblemA drop of dye is released in still water with $D = 10^{-9}$ m²/s. Using $\sigma = \sqrt{2Dt}$, estimate the standard deviation of the concentration profile after 1 second, 1 minute, and 1 hour.
- Solve
$$t=1\text{ s}: \quad \sigma = \sqrt{2\times 10^{-9}\times 1} \approx 4.5\times 10^{-5}\text{ m} = 0.045\text{ mm}$$
$$t=60\text{ s}: \quad \sigma = \sqrt{2\times 10^{-9}\times 60} \approx 3.5\times 10^{-4}\text{ m} = 0.35\text{ mm}$$
$$t=3600\text{ s}: \quad \sigma = \sqrt{2\times 10^{-9}\times 3600} \approx 2.7\times 10^{-3}\text{ m} = 2.7\text{ mm}$$
- InterpretPure molecular diffusion is extremely slow for macroscopic distances. After an hour, the dye has spread only about 2.7 millimeters. This is why stirring, convection, and turbulence are necessary to mix fluids rapidly in practical applications — diffusion alone is too slow.
Answer: $\sigma\approx 0.045$ mm after 1 s, $\approx 0.35$ mm after 1 min, $\approx 2.7$ mm after 1 hour.
Section 13.7
The Wave Equation as an Introduction
The wave equation arises from Newton's second law applied to a small segment of a vibrating string of linear mass density $\rho$ under tension $T$. For small displacements, the net transverse force is $T\,u_{xx}\,\Delta x$. Applying $F = ma$ and dividing by $\rho\,\Delta x$:
$$u_{tt} = \frac{T}{\rho}\,u_{xx} = c^2\,u_{xx}, \qquad c = \sqrt{T/\rho}$$
A tighter string (higher $T$) or a lighter string (lower $\rho$) produces a higher wave speed, matching physical experience.
The key structural difference from the heat equation: the heat equation has a first-order time derivative ($u_t$), while the wave equation has a second-order time derivative ($u_{tt}$). The heat equation describes an irreversible process — heat flows in one direction. The wave equation describes a reversible process — waves propagate without loss of energy and if you reversed time in a wave equation solution you would get another valid solution.
For the wave equation on the entire real line with initial conditions $u(x,0) = f(x)$ and $u_t(x,0) = g(x)$, d'Alembert's formula gives the explicit solution:
$$u(x,t) = \frac{f(x+ct)+f(x-ct)}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$$
This is a superposition of a right-traveling wave ($f(x-ct)$) and a left-traveling wave ($f(x+ct)$). When initial velocity is zero, the initial displacement splits into two half-size copies traveling in opposite directions.
Example 13.7.1
D'Alembert's Solution with Zero Initial Velocity
- ProblemA string satisfies $u_{tt} = 4u_{xx}$ (so $c=2$) with $u(x,0) = e^{-x^2}$ and $u_t(x,0)=0$. Write the solution and describe the motion.
- ThinkWith $g=0$, d'Alembert's formula simplifies to $u = \frac{f(x+ct)+f(x-ct)}{2}$. Substitute $f(x) = e^{-x^2}$ and $c=2$.
- Solve$$u(x,t) = \frac{e^{-(x+2t)^2}+e^{-(x-2t)^2}}{2}$$
- InterpretAt $t=0$, the solution is a single Gaussian centered at $x=0$. As time advances, this hump splits into two half-height Gaussians — one centered at $x=-2t$ (moving left) and one at $x=+2t$ (moving right), each with amplitude $\frac{1}{2}$. They separate at speed 2 in each direction.
Answer: $u(x,t) = \dfrac{e^{-(x+2t)^2}+e^{-(x-2t)^2}}{2}$. The initial hump splits into two half-size copies traveling in opposite directions at speed 2.
Section 13.8
Vibrating Strings as a Model
A string of length $L$ stretched between two fixed endpoints satisfies the wave equation with Dirichlet conditions: $u(0,t)=0$ and $u(L,t)=0$. The solutions that arise naturally from separation of variables are normal modes (standing waves):
$$u_n(x,t) = \sin\!\left(\frac{n\pi x}{L}\right)\!\left(A_n\cos\!\left(\frac{n\pi c}{L}t\right) + B_n\sin\!\left(\frac{n\pi c}{L}t\right)\right), \quad n = 1,2,3,\ldots$$
Each normal mode is a sinusoidal spatial pattern oscillating in time. The $n$-th mode has $n-1$ interior nodes — points that never move — and oscillates at frequency $\omega_n = \frac{n\pi c}{L}$. The lowest frequency $\omega_1 = \pi c/L$ is the fundamental frequency. Higher frequencies $\omega_n = n\omega_1$ are harmonics.
The rich sound of a musical string comes from the superposition of many harmonics simultaneously. The fundamental frequency determines perceived pitch. The mix of harmonics determines timbre (tone color). A string plucked at its midpoint produces a strong fundamental but suppresses the second harmonic, which has a node there. Plucking near the bridge emphasizes higher harmonics, producing a brighter sound.
Diagram Instruction — Normal Modes
Draw a horizontal axis from $x=0$ to $x=L$. Show three standing wave patterns stacked vertically. The first (labeled $n=1$, fundamental) shows a single arch with maximum at $x=L/2$. The second ($n=2$, second harmonic) shows two arches of opposite sign with a node at $x=L/2$. The third ($n=3$) shows three arches with nodes at $x=L/3$ and $x=2L/3$. Label nodes on each diagram.
Example 13.8.1
Fundamental Frequency of a Guitar String
- ProblemA guitar string has length $L=0.65$ m and wave speed $c=440$ m/s. Find the fundamental frequency $f_1$ and the second harmonic $f_2$ in Hz, where $f_n = \omega_n/(2\pi)$.
- Solve
$$\omega_1 = \frac{\pi\times 440}{0.65} \approx 2126 \text{ rad/s}, \qquad f_1 = \frac{2126}{2\pi} \approx 338\text{ Hz}$$
$$f_2 = 2f_1 \approx 676\text{ Hz}$$
- InterpretThe string vibrates at approximately 338 Hz (close to the musical note E4). The second harmonic at 676 Hz is an octave higher. Adjusting string tension changes $c$ and hence all frequencies — this is how string instruments are tuned.
Answer: $f_1 \approx 338$ Hz. $f_2 \approx 676$ Hz.
Section 13.9
Laplace's Equation as an Introduction
Laplace's equation in two spatial dimensions is:
$$u_{xx} + u_{yy} = 0$$
This equation has no time variable. It describes the steady-state distribution of quantities such as temperature in a two-dimensional plate (after all transients have died out), electric potential in a charge-free region, gravitational potential in a mass-free region, and velocity potential in ideal fluid flow.
Solutions to Laplace's equation are called harmonic functions. They have remarkable properties: they are infinitely smooth; they satisfy the mean value property (the value at any interior point equals the average over any circle centered at that point); and they have no interior maxima or minima — the maximum and minimum are always achieved on the boundary.
Example 13.9.1
Verifying a Harmonic Function
- ProblemVerify that $u(x,y) = x^2 - y^2$ satisfies Laplace's equation.
- Solve
$$u_x = 2x,\quad u_{xx} = 2; \qquad u_y = -2y,\quad u_{yy} = -2$$
$$u_{xx} + u_{yy} = 2 + (-2) = 0 \checkmark$$
- InterpretThe function $u = x^2 - y^2$ is harmonic. Its level curves are rectangular hyperbolas. This function could represent the electrostatic potential near a saddle-shaped electrode configuration or the velocity potential for a certain ideal fluid flow.
Answer: $u_{xx}+u_{yy} = 0$, confirming $u = x^2 - y^2$ is harmonic.
Student Note
Laplace's equation has no time variable. It is a purely spatial equation describing an equilibrium state. When you see it, the situation has already reached a steady state or the steady state is what is directly sought. Laplace's equation needs only boundary conditions — not initial conditions — because there is no time evolution to start from.
Section 13.10
Equilibrium and Steady State
A steady-state solution of a time-dependent PDE is one where $\frac{\partial u}{\partial t} = 0$. For the heat equation $u_t = k\,u_{xx}$, setting $u_t = 0$ gives $u_{xx} = 0$. In one dimension, integrating twice yields a linear profile:
$$u_{\text{steady}}(x) = Ax + B$$
The constants $A$ and $B$ are determined by the boundary conditions. The full solution decomposes as:
$$u(x,t) = u_{\text{steady}}(x) + v(x,t)$$
where $v(x,t)$ is the transient solution that decays to zero as $t\to\infty$. The transient satisfies the heat equation with homogeneous (zero) boundary conditions. This decomposition is a standard technique: solve for the steady state from boundary conditions alone, subtract it from the problem, and solve the remaining homogeneous problem for the transient. In two dimensions, the steady-state equation becomes exactly Laplace's equation $u_{xx}+u_{yy}=0$, confirming their deep connection.
Example 13.10.1
Finding the Steady-State Temperature
- ProblemA rod of length 2 has one end held at temperature 5 and the other at temperature 1. Find the steady-state temperature profile.
- ThinkSet $u_t = 0$, so $u_{xx} = 0$. The general solution is linear: $u(x) = Ax + B$. Apply boundary conditions.
- Solve
$$u(0)=B=5,\quad u(2)=2A+5=1 \;\Rightarrow\; A=-2$$
$$u_{\text{steady}}(x) = -2x + 5$$
- Check$u_{\text{steady}}(0)=5$ ✓, $u_{\text{steady}}(2)=1$ ✓, $u_{xx}=0$ ✓.
- InterpretTemperature decreases linearly from 5 at the left to 1 at the right. This represents uniform heat flow from hot end to cold end — exactly what Fourier's law predicts in equilibrium.
Answer: $u_{\text{steady}}(x) = -2x + 5$.
- A rod solution is $u(x,t) = e^{-9t}\sin(3x) + 3e^{-t}\sin(x)$ on $[0,\pi]$. (a) Which term decays faster? (b) What does the temperature look like for large $t$? (c) What are the boundary temperatures at $x=0$ and $x=\pi$?
- For $u(x,t) = \sin(2x)\cos(4t)$: (a) verify it satisfies $u_{tt}=c^2 u_{xx}$ and find $c$; (b) identify the spatial and temporal frequencies; (c) describe the physical motion.
- A rod of length 5 has $u(0,t)=0$ and $u_x(5,t)=0$. Explain physically what each boundary condition means.
- Find the steady-state temperature for a rod on $[0,\pi]$ with $u(0)=10$ and $u(\pi)=0$.
- A guitar string has length $L=0.5$ m and wave speed $c=350$ m/s. Find the fundamental frequency and the first three harmonics in Hz.
- Verify that $u(x,y)=\ln(x^2+y^2)$ satisfies Laplace's equation for $(x,y)\neq(0,0)$.
Section 13.11
Separation of Variables as an Introduction
Product Solutions · The Separation Constant · Eigenvalue Problems · Fourier Coefficients
Section 13.11
Separation of Variables as an Introduction
Separation of variables is the most important solution technique for linear PDEs with constant coefficients on bounded domains. The idea is to look for solutions of the form:
$$u(x,t) = X(x)\cdot T(t)$$
Not all solutions have this product form, but the ones that do — called product solutions or separated solutions — can be found systematically. The general solution is then built as a superposition of all product solutions, using the linearity of the equation.
Ten-Step Procedure for the Heat Equation
Separation of Variables — Heat Equation on $[0,L]$
Step 1Assume a product solution. Let $u(x,t) = X(x)T(t)$.
Step 2Substitute into the PDE. $X(x)T'(t) = k X''(x)T(t)$.
Step 3Separate variables. Divide by $kX(x)T(t)$: $\dfrac{T'}{kT} = \dfrac{X''}{X}$. Both sides must equal the same constant $-\lambda$.
Step 4Write two ODEs. $X'' + \lambda X = 0$ (spatial) and $T' + k\lambda T = 0$ (time).
Step 5Read off boundary conditions for $X$. $u(0,t)=0 \Rightarrow X(0)=0$; $u(L,t)=0 \Rightarrow X(L)=0$.
Step 6Solve the eigenvalue problem. $X''+\lambda X=0$, $X(0)=X(L)=0$. For $\lambda>0$: $X = B\sin(\sqrt{\lambda}\,x)$. Condition $X(L)=0$ requires $\sqrt{\lambda}L = n\pi$, giving eigenvalues $\lambda_n = n^2\pi^2/L^2$ and eigenfunctions $X_n(x)=\sin(n\pi x/L)$, $n=1,2,3,\ldots$
Step 7Solve the time ODE for each $\lambda_n$. $T_n(t) = e^{-k\lambda_n t} = e^{-kn^2\pi^2 t/L^2}$.
Step 8Form product solutions. $u_n(x,t) = e^{-kn^2\pi^2 t/L^2}\sin(n\pi x/L)$.
Step 9Apply superposition. $u(x,t) = \displaystyle\sum_{n=1}^{\infty} b_n e^{-kn^2\pi^2 t/L^2}\sin\!\left(\frac{n\pi x}{L}\right)$.
Step 10Apply initial condition. $u(x,0) = \sum_{n=1}^{\infty} b_n\sin(n\pi x/L) = f(x)$. This is a Fourier sine series; coefficients are $b_n = \dfrac{2}{L}\displaystyle\int_0^L f(x)\sin\!\left(\frac{n\pi x}{L}\right)dx$.
Example 13.11.1
Full Separation of Variables Solution
- ProblemSolve: $u_t = u_{xx}$, $00$; $u(0,t)=u(\pi,t)=0$; $u(x,0) = 3\sin(2x)-\sin(4x)$.
- ThinkWith $k=1$ and $L=\pi$, the general solution is $\sum b_n e^{-n^2 t}\sin(nx)$. The initial condition is already a finite sum of eigenfunctions, so only finitely many $b_n$ are nonzero.
- Solve
$$u(x,0) = \sum_{n=1}^{\infty}b_n\sin(nx) = 3\sin(2x)-\sin(4x)$$
$$\Rightarrow\quad b_2 = 3,\quad b_4 = -1,\quad \text{all other } b_n = 0$$
$$u(x,t) = 3e^{-4t}\sin(2x) - e^{-16t}\sin(4x)$$
- Check
$u_t = -12e^{-4t}\sin(2x)+16e^{-16t}\sin(4x)$
$u_{xx} = -12e^{-4t}\sin(2x)+16e^{-16t}\sin(4x)$
$u_t = u_{xx}$ ✓ $u(0,t)=0$ ✓ $u(\pi,t)=0$ ✓ $u(x,0)=3\sin(2x)-\sin(4x)$ ✓
- InterpretThe solution consists of two decaying modes. The $n=2$ mode decays as $e^{-4t}$ and the $n=4$ mode decays as $e^{-16t}$ — four times faster. For large $t$, the temperature profile approaches $3e^{-4t}\sin(2x)$.
Answer: $u(x,t) = 3e^{-4t}\sin(2x) - e^{-16t}\sin(4x)$.
Separation of Variables for the Wave Equation
The same method applies to the wave equation. Substituting $u = X(x)T(t)$ into $u_{tt}=c^2 u_{xx}$ and separating gives the same spatial eigenvalue problem $X''+\lambda X=0$ — so the spatial structure (sine functions) is identical. What differs is the time behavior:
$$T_n'' + \left(\frac{n\pi c}{L}\right)^2 T_n = 0 \quad\Rightarrow\quad T_n(t) = A_n\cos\!\left(\frac{n\pi c}{L}t\right)+B_n\sin\!\left(\frac{n\pi c}{L}t\right)$$
This is simple harmonic oscillation — not exponential decay. For the wave equation, the two initial conditions $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ determine both sets of coefficients $A_n$ and $B_n$.
Student Note — The Unifying Eigenvalue Problem
The eigenvalue problem $X''+\lambda X=0$ with $X(0)=X(L)=0$ appears in both the heat and wave equation analyses. The spatial structure — sine functions — is the same in both cases. What differs is the time behavior: exponential decay for heat, oscillation for waves. This makes physical sense: heat diffuses irreversibly, while waves oscillate persistently.
- Carry out the full separation of variables procedure for $u_t = u_{xx}$ on $00$, with $u(0,t)=u(1,t)=0$ and $u(x,0)=f(x)$. Find all eigenvalues and eigenfunctions and write the general series solution.
- For the problem in Problem 1, write the Fourier coefficient formula for $b_n$. For the specific case $f(x)=\sin(3\pi x)$, identify which coefficient is nonzero and write the exact solution.
- A string of length $\pi$ with $c=1$ satisfies the wave equation with $u(0,t)=u(\pi,t)=0$, $u(x,0)=\frac{1}{3}\sin(x)+\sin(2x)$, and $u_t(x,0)=0$. Write the complete solution and identify the oscillation frequencies of each mode present.
- For the heat equation $u_t = 2u_{xx}$ on $[0,\pi]$ with zero endpoints and $u(x,0)=\sin(x)+4\sin(2x)$, write the solution without computing integrals. At what time has the amplitude of the $n=2$ mode decayed to 1% of its initial value?
- Explain in your own words why a single product solution $u=X(x)T(t)$ is not the general solution. Under what special circumstances is a single-mode solution exact?
Sections 13.12 – 13.14
Applications, Common Mistakes, and the Road Ahead
PDEs Across Science and Engineering · Eight Common Errors · Preparing for Advanced PDE Courses
Section 13.12
PDEs in Physics, Engineering, and Finance
PDEs are the foundation of quantitative modeling across every scientific and engineering discipline. Several of the most important are surveyed here.
| Field | Equation | What It Governs |
|---|
| Electromagnetics | Maxwell's equations (system of PDEs) | Electric and magnetic fields; light as an electromagnetic wave at speed $c = 1/\sqrt{\varepsilon_0\mu_0}$ |
| Fluid Mechanics | Navier-Stokes equations (nonlinear) | Motion of viscous fluids; weather, aerodynamics, blood flow; one of the Millennium Prize Problems |
| Quantum Mechanics | Schrödinger equation: $i\hbar\psi_t = -\frac{\hbar^2}{2m}\psi_{xx}+V\psi$ | Quantum state of a particle; probability amplitudes; foundation of chemistry and materials science |
| Structural Engineering | Euler-Bernoulli beam equation (4th order in $x$) | Deflections and vibrations of beams, bridges, aircraft wings |
| Environmental Science | Advection-diffusion: $c_t = D c_{xx} - v c_x$ | Spread of pollutants in rivers, groundwater, and atmosphere |
| Mathematical Finance | Black-Scholes: $V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + rS V_S - rV = 0$ | Price of financial options; same structure as the heat equation after a change of variables |
| Biology | Reaction-diffusion: $u_t = D u_{xx} + f(u)$ | Pattern formation, epidemic spatial spread, neural signal propagation, species invasion |
MGU Library Connection
- MGU Physics pages: Maxwell's equations, the Schrödinger equation, classical mechanics
- MGU Engineering pages: beam equations, control systems, fluid dynamics
- MGU Environmental Science pages: advection-diffusion models, pollutant transport
- MGU Economics and Finance pages: Black-Scholes and option pricing models
- MGU Biology pages: reaction-diffusion, Turing patterns, epidemic modeling
- MGU Numerical Methods course: finite difference and finite element methods for PDEs
Section 13.13
Common PDE Preview Mistakes
Mistake 1 — Treating Partial Derivatives as Ordinary Derivatives
$\frac{\partial}{\partial x}[x^2 t + \sin(t)] = 2x + \cos(t)$
Correct: $\frac{\partial}{\partial x}[x^2 t + \sin(t)] = 2xt + 0 = 2xt$. When differentiating with respect to $x$, $\sin(t)$ is a constant; its $x$-derivative is zero.
Always identify which variable you are differentiating with respect to and treat every other variable as a fixed number.
Mistake 2 — Applying Separation of Variables to Nonlinear PDEs
Assuming $u = X(x)T(t)$ works for the Burgers equation $u_t + uu_x = \nu u_{xx}$
Correct: Separation of variables works because of linearity. Nonlinear PDEs do not produce a cleanly separable equation when $u = XT$ is substituted. Verify linearity first.
Before applying separation of variables, confirm the PDE is linear.
Mistake 3 — Forgetting Boundary Conditions in the Eigenvalue Problem
Using eigenfunctions $\sin(n\pi x/L)$ when the boundary conditions are actually Neumann
Correct: Neumann conditions $u_x(0,t)=u_x(L,t)=0$ produce eigenfunctions $\cos(n\pi x/L)$, not $\sin(n\pi x/L)$. Always write out boundary conditions for $X$ before solving the eigenvalue problem.
The boundary conditions determine which eigenfunctions arise. Do not assume sines without checking.
Mistake 4 — Confusing the Heat Equation with the Wave Equation
Writing one initial condition for the wave equation problem
Correct: $u_t = ku_{xx}$ (one time derivative) needs one initial condition and produces decaying solutions. $u_{tt}=c^2 u_{xx}$ (two time derivatives) needs two initial conditions and produces oscillating solutions.
Identify the order of the time derivative first. The number of initial conditions equals the order of the time derivative.
Mistake 5 — Adding a Time Variable to Laplace's Equation
Applying an initial condition to $u_{xx}+u_{yy}=0$
Correct: Laplace's equation $u_{xx}+u_{yy}=0$ has two spatial variables and no time variable. It describes a steady state. It needs only boundary conditions — not initial conditions — because there is no time evolution.
Steady state means $u_t=0$. Laplace's equation is what remains. Time does not appear.
Mistake 6 — Assuming a Finite Sum Is Always Sufficient
Writing $u(x,t)=b_1\sin(\pi x/L)e^{-\lambda_1 t}$ as the complete solution for an arbitrary $f(x)$
Correct: In general, the solution requires infinitely many terms. A single mode is exact only when the initial condition happens to be exactly one eigenfunction. Apply the Fourier coefficient formula $b_n = \frac{2}{L}\int_0^L f(x)\sin(n\pi x/L)\,dx$ to find all coefficients.
A finite sum is a special case, not the general rule. Write the full infinite series and only truncate when you can verify which terms are actually zero.
Mistake 7 — Claiming a Product Solution Is the General Solution
Saying "$u = X(x)T(t)$ is the general solution to the heat equation"
Correct: A single product solution satisfies the PDE and boundary conditions but not necessarily the initial condition. The general solution is an infinite superposition $\sum b_n X_n T_n$, with coefficients determined by the initial condition.
Product solutions are building blocks. The general solution assembles infinitely many of them.
Mistake 8 — Ignoring the Physical Meaning of Boundary Conditions
Applying $u(0,t)=0$ and $u_x(0,t)=0$ simultaneously at the same endpoint
Correct: Dirichlet ($u=0$) means the end is held at zero temperature. Neumann ($u_x=0$) means the end is insulated. On a single endpoint, you typically have one condition — not both. Always ask what the physics requires at each boundary.
Interpret each boundary condition before applying it. What is happening physically at that boundary?
Section 13.14
Preparing for Advanced Applied Mathematics
This chapter has provided a genuine introduction to partial differential equations. Students who have worked through it can recognize a PDE and identify its type and order, state the heat, wave, and Laplace equations and describe what each models, set up initial and boundary conditions for a physical problem, apply separation of variables to the heat equation with Dirichlet conditions, find eigenvalues and eigenfunctions, write the series solution and apply the initial condition, find steady-state solutions, and recognize PDEs across science, engineering, finance, and biology.
A full course in PDEs builds on everything introduced here. Major topics ahead include Fourier series (the essential tool for applying initial conditions in separation of variables), Fourier transforms for problems on unbounded domains, PDEs in two and three spatial dimensions with Bessel functions and Legendre polynomials as eigenfunctions, Sturm-Liouville theory as the general framework for eigenvalue problems, Green's functions as a unifying solution tool, and numerical methods including finite difference, finite element, and spectral methods.
Looking back across all fourteen chapters, a single unifying thread is visible. Differential equations describe how things change. First-order equations describe simple rates. Second-order equations describe acceleration and oscillation. Systems describe interaction. Laplace transforms convert differential equations into algebra. Numerical methods extend the reach of the theory. Qualitative methods describe behavior without formulas. And PDEs extend all of this to systems where quantities vary across space and time together. Every field of quantitative science and engineering is built, in part, on this foundation.
Chapter 13 — Back Matter
Chapter Summary
- Sec 13.1PDEs involve unknown functions of two or more independent variables and their partial derivatives. The three classical second-order linear PDEs are the heat equation, the wave equation, and Laplace's equation. Solutions are functions of multiple variables, verified by computing partial derivatives.
- Sec 13.2Functions of two variables are understood through cross-sections (fixing one variable) and level curves (sets where the function is constant). Cross-sections at $t=t_0$ show spatial distribution; at $x=x_0$ they show time evolution at one location.
- Sec 13.3Partial derivatives are computed by differentiating with respect to one variable while holding all others constant. All ordinary differentiation rules apply. Second partials $u_{xx}$, $u_{tt}$, and $u_{xt}$ are computed by differentiating twice.
- Sec 13.4PDEs require both initial conditions (state at $t=0$ across the domain) and boundary conditions (behavior at spatial boundaries for all time). Dirichlet conditions specify values; Neumann conditions specify normal derivatives; Robin conditions specify a linear combination of both.
- Sec 13.5The heat equation $u_t = ku_{xx}$ arises from Fourier's law and energy conservation. The concavity $u_{xx}$ drives temperature toward neighbors: hot spots cool, cool spots warm. Solutions decay to the steady-state profile determined by boundary conditions.
- Sec 13.6The diffusion equation $u_t = Du_{xx}$ also describes concentration, probability, and population spread. The fundamental solution is a Gaussian whose standard deviation grows as $\sqrt{t}$, not $t$. Pure diffusion is slow at macroscopic scales.
- Sec 13.7The wave equation $u_{tt}=c^2 u_{xx}$ arises from Newton's second law applied to a string. Unlike the heat equation, it is reversible in time. D'Alembert's solution expresses the general solution as a superposition of rightward and leftward traveling waves.
- Sec 13.8Vibrating strings with clamped endpoints produce standing waves (normal modes): sinusoidal spatial patterns oscillating at discrete frequencies $\omega_n = n\pi c/L$. Musical timbre arises from the superposition of harmonics at integer multiples of the fundamental.
- Sec 13.9Laplace's equation $u_{xx}+u_{yy}=0$ has no time variable. Its solutions (harmonic functions) describe steady-state distributions. Harmonic functions satisfy the mean value property and achieve their extrema on the boundary.
- Sec 13.10Steady-state solutions satisfy $u_t=0$, reducing to $u_{xx}=0$ (linear profile in 1D) or Laplace's equation in 2D. The full solution decomposes as steady state plus transient. The transient decays to zero, leaving the steady state as the long-time behavior.
- Sec 13.11Separation of variables assumes $u=X(x)T(t)$, reduces the PDE to two ODEs, and introduces a separation constant $-\lambda$. The spatial ODE with boundary conditions is an eigenvalue problem. Its solutions are eigenfunctions (sine functions for Dirichlet conditions). The general solution is an infinite series of product solutions with Fourier coefficients from the initial condition.
- Sec 13.12PDEs appear across physics (Maxwell, Schrödinger), engineering (Navier-Stokes, beam equation), environment (advection-diffusion), finance (Black-Scholes), and biology (reaction-diffusion). The mathematical structures introduced here underpin all of these fields.
- Sec 13.13Eight common PDE mistakes: treating partial as ordinary derivatives; applying separation to nonlinear PDEs; ignoring boundary conditions in eigenvalue problems; confusing heat and wave equations; adding time to Laplace's equation; assuming finite sums; claiming product solutions are general; ignoring physical meaning of boundary conditions.
- Sec 13.14This chapter opens the door to a full PDE course. The road ahead includes Fourier series, Fourier transforms, higher-dimensional problems, Sturm-Liouville theory, Green's functions, and numerical methods for PDEs. The unifying thread of the entire textbook — differential equations as the mathematics of change — reaches its broadest expression in PDEs.
Key Terms Review
partial differential equation
An equation involving an unknown function of two or more independent variables and its partial derivatives.
partial derivative
The derivative of a function with respect to one variable, holding all other variables constant.
heat equation
$u_t = ku_{xx}$. Describes temperature in a rod over time. Also describes diffusion of any spreading quantity.
wave equation
$u_{tt} = c^2 u_{xx}$. Describes displacement of a vibrating string or propagation of a wave at speed $c$.
Laplace's equation
$u_{xx}+u_{yy}=0$. Describes steady-state temperature, electric potential, and other equilibrium distributions in two dimensions.
Dirichlet condition
A boundary condition specifying the value of $u$ at the boundary. Fixed temperature or clamped endpoint.
Neumann condition
A boundary condition specifying the normal derivative $u_x$ at the boundary. Insulated end (heat) or free end (wave).
mixed (Robin) condition
A boundary condition specifying a linear combination of $u$ and $u_x$ at the boundary. Arises in convective heat transfer.
initial condition
A condition specifying the state of the system at $t=0$ across the entire spatial domain.
separation of variables
A method for solving linear PDEs by seeking product solutions $u=X(x)T(t)$, reducing the PDE to two ODEs.
eigenvalue
A special value of the separation constant $\lambda$ for which the spatial ODE with boundary conditions has a nontrivial solution.
eigenfunction
A nontrivial solution $X_n(x)$ of the eigenvalue problem corresponding to eigenvalue $\lambda_n$.
steady-state solution
A time-independent solution satisfying the PDE and boundary conditions, found by setting $u_t=0$.
transient solution
The part of the full solution that decays to zero over time, leaving only the steady-state distribution.
harmonic function
A function satisfying Laplace's equation. Harmonic functions satisfy the mean value property and achieve extrema on the boundary.
d'Alembert's solution
The explicit formula for the wave equation on the real line: $u = \frac{f(x+ct)+f(x-ct)}{2}+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$.
Fourier sine series
A representation of a function on $[0,L]$ as an infinite sum $\sum b_n\sin(n\pi x/L)$, used in separation of variables.
superposition principle
For linear PDEs, any linear combination of solutions is also a solution. The general solution is an infinite series of product solutions.
fundamental mode
The $n=1$ mode in a standing wave expansion. It has the lowest frequency and decays (or oscillates) most slowly.
fundamental solution
The Gaussian solution $\frac{1}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}$ to the diffusion equation on the real line starting from a concentrated source.
PDE Classification Practice
Classify each as ODE or PDE, state its order, and identify it as linear or nonlinear.
- $u_t = 5u_{xx}$
- $u_{tt} = 9u_{xx} - u$
- $u_x u_y = u$
- $u_{xx}+u_{yy}+u_{zz}=0$
- $u_t + u\,u_x = \nu u_{xx}$
- $u_{xx}+2u_{xy}+u_{yy}=0$
- Verify $u=\cos(3x)\cos(3ct)$ satisfies $u_{tt}=c^2u_{xx}$.
- Verify $u(x,y)=\ln(x^2+y^2)$ satisfies $u_{xx}+u_{yy}=0$ for $(x,y)\neq(0,0)$.
- A rod of length $L$ has insulated ends. Write the Neumann boundary conditions and explain physically what they mean.
- A string of length $L$ has a fixed left end and a free right end (condition $u_x(L,t)=0$). Write both boundary conditions and explain the physics of the free end.
- For the heat problem on $[0,5]$ with $u(0,t)=0$ and $u_x(5,t)=0$, find the steady-state solution.
- A rod of length $\pi$ has $u(0,t)=0$ and $u(\pi,t)=0$, with initial temperature $u(x,0)=2\sin(3x)$. Write the complete problem and explain why the solution is a single decaying mode.
Heat and Wave Interpretation
- A rod solution is $u(x,t) = e^{-9t}\sin(3x)+3e^{-t}\sin(x)$ on $[0,\pi]$. (a) Which term decays faster? (b) For large $t$, what does the temperature profile look like? (c) What are the boundary temperatures?
- For $u(x,t)=\sin(2x)\cos(4t)$: (a) verify it satisfies $u_{tt}=c^2 u_{xx}$ and find $c$; (b) identify spatial and temporal frequencies; (c) describe the physical motion.
- For the fundamental solution of the diffusion equation with $D=1$: (a) verify $u_t=u_{xx}$; (b) find $u(0,t)$ and describe how the peak concentration changes; (c) at what time does $\sigma=1$?
- Explain physically why the heat equation solution always approaches a steady state while the wave equation solution continues oscillating.
Error Analysis
Each item shows a student's work. Identify the error and write the correct reasoning or result.
Error Analysis 1
$\frac{\partial}{\partial x}[x^2 t + \sin(t)] = 2x + \cos(t)$
Identify both errors. What is the correct partial derivative with respect to $x$?
Error Analysis 2
A student writes one initial condition $u(x,0)=f(x)$ for the wave equation and cannot match the initial velocity. "The wave equation only needs one initial condition, like the heat equation."
Explain the error. How many initial conditions does the wave equation need, and why?
Error Analysis 3
A student solving the Laplace equation $u_{xx}+u_{yy}=0$ on a square writes: "I need to specify the initial value $u(x,y,0)$ throughout the interior."
Identify the fundamental conceptual error. What conditions are actually needed, and why?
Error Analysis 4
$u(x,t) = b_1\sin(\pi x)e^{-\pi^2 t}$ is written as the complete solution to the heat equation with $u(x,0)=\sin(\pi x)+\sin(2\pi x)$.
What is missing? Write the correct complete solution.
Error Analysis 5
A student verifies $u=\sin(x)\cos(t)$ satisfies $u_t = u_{xx}$ and concludes "this function doesn't work for the heat equation" after finding $u_t\neq u_{xx}$, then stops.
Is the student's conclusion correct? Which PDE does $\sin(x)\cos(t)$ actually satisfy?
Chapter 13 Checkpoint
Chapter 13 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 13
Checkpoint Problem 1: Partial Derivatives and Verification
Let $u(x,t) = e^{-2t}\sin(x)+e^{-8t}\sin(2x)$.
- Compute $u_t$, $u_x$, and $u_{xx}$.
- Verify that $u$ satisfies the heat equation $u_t = 2u_{xx}$.
- Determine the boundary values: find $u(0,t)$ and $u(\pi,t)$ for all $t\geq 0$.
- Find the initial temperature profile $u(x,0)$.
- Describe the long-term behavior as $t\to\infty$. Which mode decays faster?
Checkpoint Problem 2: Setting Up a PDE Problem
A copper rod of length $L=\pi$ m has thermal diffusivity $k=1.17\times 10^{-4}$ m²/s. Both ends are held at $0°$C. The initial temperature profile is $u(x,0) = 50\sin(x)-20\sin(3x)$.
- Write the full initial-boundary value problem (PDE, boundary conditions, initial condition).
- State the type of boundary conditions.
- Write the solution using separation of variables (no integrals needed — the initial condition is already in eigenfunction form).
- At what time $t$ has the amplitude of the $n=3$ mode decayed to 10% of its initial value?
- For large $t$, what does the temperature profile look like?
Checkpoint Problem 3: Separation of Variables
Carry out the full separation of variables procedure for $u_t = u_{xx}$, $00$, with $u(0,t)=u(1,t)=0$ and $u(x,0)=f(x)$.
- Assume $u=X(x)T(t)$, substitute into the PDE, and show how variables separate. Introduce the separation constant $-\lambda$.
- Write the two ODEs for $X$ and $T$.
- Write the boundary conditions for $X$ implied by $u(0,t)=u(1,t)=0$.
- Solve the eigenvalue problem: find all $\lambda_n$ and eigenfunctions $X_n(x)$.
- Solve the time ODE for each $\lambda_n$ and write $T_n(t)$.
- Write the general series solution $u(x,t)$.
- State the Fourier coefficient formula for $b_n$ as an integral involving $f(x)$.
- For $f(x)=\sin(3\pi x)$, identify the nonzero coefficient and write the exact solution.
Checkpoint Problem 4: Wave Equation and Vibrating String
A string of length $\pi$ with wave speed $c=2$ is clamped at both ends, with $u(x,0)=\frac{1}{3}\sin(x)+\sin(2x)$ and zero initial velocity.
- Write the full initial-boundary value problem.
- Write the separated solutions $u_n(x,t)$ for each $n$.
- Apply both initial conditions and find all coefficients.
- Write the complete solution.
- What are the frequencies of oscillation of each mode present?
- Is the full motion periodic? If so, what is the period?
Checkpoint Problem 5: Steady State and Physical Interpretation
A rod of length 4 has $u(0,t)=2$ and $u(4,t)=10$. The initial temperature is $u(x,0)=6$ (uniform).
- Find the steady-state temperature profile $u_{\text{steady}}(x)$.
- What is the steady-state temperature at $x=2$?
- Let $v(x,t)=u(x,t)-u_{\text{steady}}(x)$. Find $v(x,0)$ and the boundary conditions for $v$.
- As $t\to\infty$, what happens to $v(x,t)$? What does this imply for $u(x,t)$?
- Interpret the problem physically: describe what happens to the temperature distribution over time as the initially uniform rod is placed between two reservoirs at different temperatures.
Looking Ahead
Chapter 14: Differential Equations Capstone and Applied Modeling
Chapter 13 opened the door into partial differential equations. Chapter 14 closes the book by bringing together all fifteen chapters of this course: first-order equations, second-order equations, systems, Laplace transforms, numerical methods, qualitative analysis, and the PDE preview completed here.
The capstone chapter emphasizes choosing a model, interpreting parameters, using technology, checking assumptions, and explaining dynamic behavior in ordinary language. Students will practice the complete cycle: define variables and units, write rate equations, classify the equation, choose a solution or analysis method, solve or approximate, interpret, check, and communicate.
The capstone modeling problems draw on all five parts of the course and all major application areas: physics, engineering, biology, economics, environmental science, and data science. Students who have completed this course and this chapter have built a foundation that will serve them wherever their quantitative journey leads next.
MGU Library Connection
- Calculus Readiness Review — Appendix A of this textbook
- Integration Techniques Review — Appendix B of this textbook
- Appendix P: PDE Preview Reference — formula summary for heat, wave, and Laplace equations
- MGU Advanced PDE Course: Fourier series, Fourier transforms, higher-dimensional separation of variables
- MGU Numerical Methods Course: finite difference and finite element methods for PDEs
- MGU Physics pages: Maxwell's equations, Schrödinger equation, classical wave theory
- MGU Engineering pages: Euler-Bernoulli beam, Navier-Stokes, control systems
- MGU Finance pages: Black-Scholes equation and option pricing
- MGU Biology pages: reaction-diffusion, Turing patterns, epidemic spatial modeling