PDE Applications Practice Questions

Module II: Applications of Partial Differential Equations
Practice Questions with Solutions

Question 1 – Classification of Second-Order PDEs

Classify the following PDE and reduce it to canonical form:

\frac{\partial^2 u}{\partial x^2} - 2\frac{\partial^2 u}{\partial x\partial y} - 3\frac{\partial^2 u}{\partial y^2} = 0

Solution

Here $A=1$ , $B=-2$ , $C=-3$ .

Discriminant:

\Delta = B^2 - 4AC = 4 - 4(1)(-3) = 4 + 12 = 16 > 0

Type: Hyperbolic

Characteristic equations:

\frac{dy}{dx} = \frac{B \pm \sqrt{\Delta}}{2A} = \frac{-2 \pm 4}{2}

\frac{dy}{dx} = 1 \quad\text{and}\quad \frac{dy}{dx} = -3

Integrals:

\xi = y - x, \qquad \eta = y + 3x

In these variables the equation reduces to the canonical form:

\frac{\partial^2 u}{\partial\xi\partial\eta} = 0

Question 2 – Separation of Variables (Wave Equation)

Solve the one-dimensional wave equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L,\ t > 0

subject to

u(0,t)=0,\quad u(L,t)=0,\quad u(x,0)=f(x),\quad u_t(x,0)=0

Solution

Assume $u(x,t)=X(x)T(t)$ .

Substituting:

\frac{T''}{c^2 T} = \frac{X''}{X} = -\lambda

Boundary conditions $\Rightarrow X(0)=X(L)=0$ give

X_n = \sin\left(\frac{n\pi x}{L}\right), \quad \lambda_n = \left(\frac{n\pi}{L}\right)^2, \ n=1,2,3,\dots

Time equation:

T_n'' + c^2\lambda_n T_n = 0 \implies T_n = A_n\cos\left(\frac{n\pi c t}{L}\right) + B_n\sin\left(\frac{n\pi c t}{L}\right)

Initial velocity $u_t(x,0)=0 \Rightarrow B_n=0$ .

Thus

u(x,t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right)\cos\left(\frac{n\pi c t}{L}\right)

where

A_n = \frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx

Question 3 – Heat Conduction Equation

Solve

\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi,\ t>0

with

u(0,t)=0,\quad u(\pi,t)=0,\quad u(x,0)=x(\pi-x)

Solution

Assume $u=X(x)T(t)$ .

\frac{T'}{kT} = \frac{X''}{X} = -\lambda

$X(0)=X(\pi)=0$ gives eigenvalues $\lambda_n=n^2$ and .

Time part:

T_n = A_n e^{-kn^2 t}

General solution:

u(x,t) = \sum_{n=1}^{\infty} A_n\sin(nx)\,e^{-kn^2 t}

Initial condition:

x(\pi-x) = \sum A_n\sin(nx)

A_n = \frac{2}{\pi}\int_0^\pi x(\pi-x)\sin(nx)\,dx = \frac{4(1-(-1)^n)}{n^3\pi}

(Only odd $n$ survive: $A_n = \dfrac{8}{n^3\pi}$ for odd.)

Question 4 – Laplace Equation (Cartesian)

Solve

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 0<x<a,\ 0<y<b

with boundary conditions

u(0,y)=0,\quad u(a,y)=0,\quad u(x,0)=0,\quad u(x,b)=f(x)

Solution

Assume $u=X(x)Y(y)$ .

Separation gives

\frac{X''}{X} = -\frac{Y''}{Y} = -\lambda

$X(0)=X(a)=0$ and $Y(0)=0$ yield

X_n = \sin\left(\frac{n\pi x}{a}\right), \quad Y_n = \sinh\left(\frac{n\pi y}{a}\right)

Solution:

u(x,y) = \sum_{n=1}^{\infty} A_n\sin\left(\frac{n\pi x}{a}\right)\sinh\left(\frac{n\pi y}{a}\right)

Coefficient from $u(x,b)=f(x)$ :

A_n = \frac{2}{a\sinh(n\pi b/a)}\int_0^a f(x)\sin\left(\frac{n\pi x}{a}\right)dx

Question 5 – Laplace Equation (Polar)

Solve Laplace’s equation in polar coordinates inside the unit disk

\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2} = 0, \quad r<1

with $u(1,\theta)=f(\theta)$ (bounded at origin).

Solution

Separation $u=R(r)\Theta(\theta)$ with periodicity in $\theta$ gives

u(r,\theta) = a_0 + \sum_{n=1}^{\infty} r^n (a_n\cos n\theta + b_n\sin n\theta)

Fourier coefficients:

\begin{aligned} a_0 &= \frac{1}{2\pi}\int_0^{2\pi} f(\theta)\,d\theta \\ a_n &= \frac{1}{\pi}\int_0^{2\pi} f(\theta)\cos n\theta\,d\theta \\ b_n &= \frac{1}{\pi}\int_0^{2\pi} f(\theta)\sin n\theta\,d\theta \end{aligned}

Question 6 – Transmission Line Equation

Show that the lossless transmission line equations reduce to the wave equation

\frac{\partial^2 V}{\partial x^2} = LC\frac{\partial^2 V}{\partial t^2}

Solution

From Kirchhoff’s laws:

\frac{\partial V}{\partial x} = -L\frac{\partial I}{\partial t}, \qquad \frac{\partial I}{\partial x} = -C\frac{\partial V}{\partial t}

(when $R=0$ , $G=0$ )

Differentiate the first with respect to $x$ and the second with respect to $t$ :

\frac{\partial^2 V}{\partial x^2} = -L\frac{\partial^2 I}{\partial x\partial t} = -L\left(-C\frac{\partial^2 V}{\partial t^2}\right) = LC\frac{\partial^2 V}{\partial t^2}

Hence the voltage (and similarly the current) satisfies the one-dimensional wave equation with speed $c=1/\sqrt{LC}$ .

Question 7 – Two-Dimensional Wave Equation

Write the general solution of the two-dimensional wave equation

\frac{\partial^2 u}{\partial t^2} = c^2\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)

by separation of variables in a rectangle $0\le x\le a$ , $0\le y\le b$ with zero boundary conditions.

Solution

u(x,y,t) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right) \Bigl[A_{mn}\cos(\omega_{mn}t)+B_{mn}\sin(\omega_{mn}t)\Bigr]

where

\omega_{mn} = c\pi\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}

Practice Tips

Always compute the discriminant first for classification.
For separation of variables, write boundary conditions clearly before solving the eigenvalue problem.
Remember the difference between finite and infinite domain solutions (Fourier series vs d’Alembert).
For transmission lines, start from the two first-order equations and eliminate one variable.