Laplace, Fourier and PDEs (PDE): free practice, theory and problems

1. The Laplace transform

What is it about?

The Laplace transform turns a function of time, $f(t)$, into a function of a complex variable $s$: $F(s) = \mathcal L\{f(t)\} = \int_0^\infty f(t)e^{-st}\,dt$. The key point is that it turns differential equations into algebraic equations – instead of differentiating and integrating in the time domain, you multiply and divide in the $s$-domain. This is the very tool engineers use to analyze circuits, mechanical systems and control systems without solving the differential equation directly.

Key concepts and formulas

• Linearity: $\mathcal L\{af+bg\} = aF(s)+bG(s)$.
• Derivatives: $\mathcal L\{f'\} = sF(s)-f(0)$, $\mathcal L\{f''\} = s^2F(s)-sf(0)-f'(0)$.
• Standard transforms: $\mathcal L\{1\}=1/s$, $\mathcal L\{t^n\}=n!/s^{n+1}$, $\mathcal L\{e^{at}\}=1/(s-a)$, $\mathcal L\{\sin\omega t\}=\omega/(s^2+\omega^2)$, $\mathcal L\{\cos\omega t\}=s/(s^2+\omega^2)$.
• s-shift: $\mathcal L\{e^{at}f(t)\}=F(s-a)$.
• t-shift (second shifting theorem): $\mathcal L\{f(t-a)u(t-a)\}=e^{-as}F(s)$, where $u$ is the unit step function.
• Convolution theorem: $\mathcal L\{f*g\}=F(s)G(s)$, where $(f*g)(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau$.
• Initial and final value theorems: $f(0^+)=\lim_{s\to\infty}sF(s)$, and $\lim_{t\to\infty}f(t)=\lim_{s\to0}sF(s)$ (if the limit exists).
• Dirac impulse: $\mathcal L\{\delta(t)\}=1$.

How to solve the problems

1. Transform the whole differential equation term by term, using the derivative rules together with the initial conditions.
2. Solve the resulting algebraic equation for $Y(s)$.
3. Split $Y(s)$ with partial fractions into simple terms you recognize from the table.
4. Transform back term by term to get $y(t)$.
5. Check the answer: does $y(0)$ match the initial condition?

Example

Solve $y'' + 4y = 0$, $y(0) = 2$, $y'(0) = 0$.

1. Transform: $s^2Y - 2s - 0 + 4Y = 0$.
2. Solve for $Y$: $Y(s^2+4) = 2s$, so $Y(s) = \dfrac{2s}{s^2+4}$.
3. We recognize this as the transform of $\cos 2t$: $y(t) = 2\cos 2t$.
4. Check: $y(0) = 2\cos 0 = 2$. Correct.

Common mistakes

• Forgetting the initial conditions when using $\mathcal L\{f'\}$ or $\mathcal L\{f''\}$.
• Mixing up s-shift ($e^{at}f(t)$) with t-shift ($f(t-a)u(t-a)$) – they give completely different results.
• Sign or factor errors in the partial fraction expansion.
• Assuming the final value theorem holds even when the system is unstable (poles with positive real part, or more than one pole at the origin).

Concepts in this part

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2. Fourier series

What is it about?

Any reasonable periodic function can be written as a sum of sine and cosine terms of different frequencies – a Fourier series. This is powerful because linear systems (circuits, mechanical oscillators) respond simply to each individual sine wave, so you can analyze the response to a complicated periodic signal term by term. Fourier series are the foundation of signal processing, acoustics, and solving partial differential equations with periodic or bounded boundary conditions.

Key concepts and formulas

• For a $2L$-periodic function: $f(x) = \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^\infty\left(a_n\cos\dfrac{n\pi x}{L} + b_n\sin\dfrac{n\pi x}{L}\right)$.
• Coefficients: $a_n = \dfrac1L\displaystyle\int_{-L}^{L} f(x)\cos\dfrac{n\pi x}{L}\,dx$, $b_n = \dfrac1L\displaystyle\int_{-L}^{L} f(x)\sin\dfrac{n\pi x}{L}\,dx$.
• Even function: only cosine terms ($b_n=0$). Odd function: only sine terms ($a_n=0$).
• Half-range series on $[0,L]$: extend $f$ as odd (only $b_n$, a sine series) or as even (only $a_n$, a cosine series), depending on which boundary conditions you need.
• Fundamental angular frequency: $\omega_0 = \pi/L$.
• Parseval's identity: $\dfrac1L\displaystyle\int_{-L}^{L} f(x)^2\,dx = \dfrac{a_0^2}{2} + \displaystyle\sum_{n=1}^\infty(a_n^2+b_n^2)$ – the energy of the signal is the sum of the energy in each term.

Convergence (Dirichlet's theorem): the series converges to $f(x)$ where $f$ is continuous, and to the average of the left- and right-hand limits at a jump discontinuity. Gibbs phenomenon: an overshoot of about 9% near a jump, which does not disappear even with more terms.

How to solve the problems

1. Check symmetry first: is $f$ even, odd, or neither? This decides whether you can skip $a_n$ or $b_n$.
2. Compute $a_0$, $a_n$ and $b_n$ with the integral formulas (use integration by parts where $f$ is a polynomial times sine/cosine).
3. Assemble the series, or use a known standard result (square wave, sawtooth, triangle wave) if the problem asks for it.
4. For half-range series: decide whether the problem needs a sine or a cosine expansion based on the boundary conditions.
5. Use Parseval's identity to compute power/energy without summing the whole series explicitly.

Example

A sawtooth wave has $f(x)=x$ on $(-\pi,\pi)$, periodic with period $2\pi$. Find $b_3$.

1. $f$ is odd, so $a_n=0$ for all $n$, and we only need $b_n$.
2. The standard result for this function is $b_n = \dfrac{2(-1)^{n+1}}{n}$.
3. $b_3 = \dfrac{2(-1)^4}{3} = \dfrac23 \approx 0.667$.

Common mistakes

• Forgetting to check symmetry, and computing $a_n$ for an odd function (it comes out zero, but it is wasted work – and easy to get wrong).
• Confusing $2L$-periodic with $L$-periodic in the formulas for $a_n$ and $b_n$.
• Believing the Gibbs overshoot disappears with more terms – it moves closer to the jump, but does not shrink.
• Using the wrong symmetry (odd instead of even) for a half-range expansion.

Concepts in this part

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3. Partial differential equations

What is it about?

Partial differential equations (PDEs) describe quantities that vary both in space and time – temperature in a rod, displacement of a string, potential in a region. The three classic examples are the heat equation (diffusion), the wave equation (propagating oscillations) and Laplace's equation (steady states). Separation of variables breaks a PDE into simpler ordinary differential equations, and Fourier series glue the solutions together so the boundary and initial conditions are satisfied.

Key concepts and formulas

• Heat equation: $u_t = c^2u_{xx}$ (parabolic). Wave equation: $u_{tt}=c^2u_{xx}$ (hyperbolic). Laplace's equation: $\nabla^2u=0$ (elliptic).
• General classification: for $Au_{xx}+Bu_{xy}+Cu_{yy}+\dots=0$ the PDE is elliptic if $B^2-4AC<0$, parabolic if $=0$, and hyperbolic if $>0$.
• Separation of variables: assume $u(x,t)=X(x)T(t)$. Substituting into the PDE gives two coupled ordinary differential equations with a separation constant $\lambda$.
• Eigenvalue problem with Dirichlet boundary $X(0)=X(L)=0$: eigenvalues $\lambda_n=(n\pi/L)^2$, eigenfunctions $X_n=\sin(n\pi x/L)$, $n=1,2,\dots$
• Eigenvalue problem with Neumann boundary (insulated ends) $X'(0)=X'(L)=0$: eigenfunctions $X_n=\cos(n\pi x/L)$, $n=0,1,2,\dots$ (including the trivial constant solution $\lambda_0=0$).
• Heat equation solution: $u(x,t)=\displaystyle\sum_n B_n\sin(n\pi x/L)e^{-c^2(n\pi/L)^2t}$ – each mode dies out, higher modes fastest.
• D'Alembert's solution of the wave equation: $u(x,t)=F(x-ct)+G(x+ct)$, two waves that travel their own way without changing shape.

How to solve the problems

1. Identify the type of equation (heat/wave/Laplace) and the boundary conditions (Dirichlet: value given; Neumann: derivative given).
2. Set $u=X(x)T(t)$, substitute into the PDE, and separate so that one side depends only on $x$ and the other only on $t$.
3. Solve the eigenvalue problem for $X$ with the boundary conditions – this gives the allowed $\lambda_n$ and eigenfunctions.
4. Solve the equation for $T(t)$ with the same $\lambda_n$.
5. Sum up the modes with coefficients determined by the initial condition (often via a Fourier series).

Example

A string of length $L=2$ and wave speed $c=3$ has an initial shape that is a pure mode $n=1$: $u(x,0)=5\sin(\pi x/2)$, with zero initial velocity. Find $u(1, 0.2)$.

1. With only mode 1 present, the solution is $u(x,t) = 5\sin(\pi x/2)\cos(\omega_1 t)$, where $\omega_1 = cn\pi/L = 3\pi/2$.
2. At $x=1$: $\sin(\pi/2)=1$.
3. At $t=0.2$: $\omega_1t = 3\pi/2\cdot 0.2 \approx 0.942$, and $\cos(0.942)\approx 0.588$.
4. $u(1, 0.2) \approx 5\cdot 1\cdot 0.588 \approx 2.94$.

Common mistakes

• Using Dirichlet eigenfunctions (sine) when the boundary condition is actually Neumann (insulated end, cosine).
• Forgetting that the general solution of the wave equation has two arbitrary functions (a forward and a backward wave), not one.
• Mixing up the sign of the separation constant $\lambda$ – the wrong sign gives growing solutions instead of decaying or oscillating ones.
• Believing Laplace's equation has a time-dependent solution – it describes a steady state.

Concepts in this part

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