Unit 13: Partial Differential Equations and Fourier Series
One equation splits into two ordinary ones, a square wave gets built from harmonics, and heat diffuses flat while a wave keeps traveling.
12 topics · 41 questions
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10.1 What Makes an Equation Partial Instead of Ordinary
- How it worksHow does a partial derivative let an equation track change with respect to position and time separately?
- Common mistakeWhy isn't a partial differential equation just an ordinary differential equation with an extra letter added?
- Going furtherWhat happens to the complexity of a partial differential equation when a third spatial variable is added?
- Worked exampleFind whether ut = u_xx + u is an ordinary or partial differential equation and name its independent variables.
Your turn1 of 2
Classify the equation ut = 3u_xx + 2u: is it an ordinary or partial differential equation, and what are its independent variables?10.5 Separation of Variables
- How it worksHow does substituting u(x,t) = X(x)T(t) turn one partial differential equation into two separate ordinary ones?
- How it worksHow does setting the separated equation equal to a constant, called the separation constant, force both sides to match?
- Worked exampleFind the two ordinary differential equations that result from separating the heat equation ut = k u_xx with u(x,t) = X(x)T(t).
Your turn1 of 2
Separating ut = 3u_xx with u=X(x)T(t) gives T'/(3T) = X''/X = -lambda. If lambda = 4, find the coefficient of T in T' = -lambda·3·T, i.e., find -lambda·k.10.5 Why Separation of Variables Doesn't Always Work
- Common mistakeWhy isn't it true that you can always separate variables in a partial differential equation just by assuming a product form?
- Going furtherWhat happens to the separation of variables technique when the boundary conditions themselves mix x and t together?
- Ties togetherHow does the requirement of linear, homogeneous boundary conditions here connect to why superposition works for the heat equation?
Your turn1 of 2
A student tries separation of variables on the nonlinear PDE ut = u_xx + u·ux, substituting u=X(x)T(t). Dividing through, one side ends up as X''/X + X'T (still containing T), rather than a pure function of x alone. What does this signal?10.2 Building a Periodic Function from Sines and Cosines
- How it worksHow does a Fourier series add a fundamental frequency and its harmonics together, each with its own weight, to build a target function?
- Worked exampleFind the first three nonzero terms of the Fourier series for a square wave that equals 1 on (0, π) and -1 on (π, 2pi).
- Going furtherWhat happens to the approximation as more and more harmonics are added to a Fourier series for a square wave?
- Worked exampleFind the first three nonzero terms of the Fourier series for f(x) = x2 on the interval from -π to π.
Your turn1 of 2
For a square wave equal to 4 on (0,π) and -4 on (π,2pi), the Fourier series is (4A/π)[sin(x)+sin(3x)/3+sin(5x)/5+...]. Find the coefficient of the third nonzero term, sin(5x).10.2 Computing Fourier Coefficients
- How it worksHow does the orthogonality of sines and cosines over one period make the Fourier coefficient formulas work?
- Worked exampleFind the Fourier coefficients a0, an, and bn for f(x) = x on the interval from -π to π.
- Worked exampleFind the Fourier series of f(x) = the absolute value of x on the interval from -π to π.
Your turn1 of 2
For f(x) = 4x on the interval from -π to π, find the Fourier coefficient bn using bn = 2k(-1)n+1/n, evaluated at n = 4.10.3 Fourier Sine and Cosine Series
- How it worksHow does extending a function as odd or even before computing its Fourier series produce a pure sine or pure cosine expansion?
- Worked exampleFind the Fourier sine series of f(x) = 1 on the interval from 0 to π.
- Common mistakeWhy isn't the choice between an odd and an even extension of a function just a matter of personal preference?
- Worked exampleFind the Fourier cosine series of f(x) = x on the interval from 0 to π.
Your turn1 of 2
The Fourier sine series coefficient for f(x) = 4 on (0,π) is bn = (2k/(n·π))·(1-cos(n·π)). Find bn for n = 4.10.2 The Gibbs Phenomenon
- How it worksHow does the overshoot near a jump discontinuity stay at roughly the same height even as more harmonics are summed in?
- Going furtherWhat happens to the width of the overshoot region around a jump as more and more Fourier terms are included?
- Ties togetherHow does the persistent overshoot at a jump connect back to why the Fourier series only converges to the midpoint value there?
Your turn1 of 2
A function has a jump discontinuity of size 8 (the difference between the left and right limits). Using the known Gibbs overshoot fraction of approximately 0.09, find the approximate overshoot amount.10.5 The Heat Equation
- How it worksHow does the heat equation ut = k u_xx say that temperature changes fastest wherever the temperature profile is most sharply curved?
- Worked exampleFind the boundary and initial conditions for a rod of length π with both ends held at zero and initial temperature sin(x).
- Going furtherWhat happens to a rod's temperature profile after an extremely long time has passed, regardless of how it started?
- Common mistakeWhy doesn't a hot spot in the middle of a rod ever come back once the heat equation has smoothed it away?
Your turn1 of 2
For the heat equation ut = 2.24u_xx on a rod of length 4 with both ends held at zero, the fundamental mode decays as e−lambda t with lambda = k·(π/L)2. Find lambda.10.5 Solving the Heat Equation with Separation of Variables
- Worked exampleSolve ut = u_xx on 0 < x < π with u(0,t) = u(π,t) = 0 and initial condition u(x,0) = 3 sin(2x).
- Worked exampleSolve ut = u_xx on 0 < x < π with u(0,t) = u(π,t) = 0 and initial condition u(x,0) = 5 sin(x) - 2 sin(3x).
- Ties togetherHow does each term of the heat equation's series solution decaying at its own rate connect to the Fourier sine coefficients of the initial data?
Your turn1 of 2
Solve ut = u_xx on 0 < x < π with u(0,t)=u(π,t)=0 and u(x,0) = 4sin(3x). Find u(x,t).10.7 The Wave Equation
- How it worksHow does the wave equation u_tt = a2 u_xx describe acceleration in terms of the string's curvature instead of a first-time-derivative rate?
- Worked exampleFind the boundary and initial conditions for a plucked string of length π, fixed at both ends, released from rest with shape sin(x).
- Going furtherWhat happens to a vibrating string's motion over time compared to a rod cooling under the heat equation, given the same initial shape?
- Common mistakeWhy doesn't a wave equation's solution ever settle into a fixed steady shape the way the heat equation's does?
Your turn1 of 2
For the wave equation u_tt = 32 u_xx on a string of length 4 fixed at both ends, the fundamental mode oscillates at angular frequency omega = a·π/L. Find omega.10.7 Solving the Wave Equation with Separation of Variables
- Worked exampleSolve u_tt = 4 u_xx on 0 < x < π with fixed ends, initial shape u(x,0) = sin(x), and initial velocity zero.
- Going furtherWhat happens to a plucked string's motion when it's released with a nonzero initial velocity instead of from rest?
- Ties togetherHow does the wave equation's non-decaying oscillation in time connect back to the undamped free vibration case for a single mass?
Your turn1 of 2
Solve u_tt = 22 u_xx on 0 < x < π with fixed ends, initial shape u(x,0) = 4sin(4x), and initial velocity zero. Find u(x,t).10.7 Comparing the Heat and Wave Equations
- How it worksHow does a first time-derivative versus a second time-derivative decide whether a disturbance diffuses away or propagates and reflects?
- Going furtherWhat happens to information about a sharp initial disturbance under the heat equation compared to under the wave equation?
- Ties togetherHow does the contrast between diffusing and propagating tie together what the course covered about first- and second-order behavior?
Your turn1 of 2
The heat equation's n-th mode decays as e−n^2 k t; the wave equation's n-th mode oscillates as cos(a·n·t). For n = 3, k = 2, a = 3, find the heat mode's decay rate n2·k minus the wave mode's angular frequency a·n.