Applications of PDEs | Wave, Heat & Laplace Equations

Module II: Applications of Partial Differential Equations – All Important Formulas

General form:

A\,u_{xx} + B\,u_{xy} + C\,u_{yy} + D\,u_x + E\,u_y + F\,u = G

Discriminant

\Delta = B^2 - 4AC

Condition	Type	Canonical Form
$\Delta > 0$	Hyperbolic	$u_{\xi\eta} = \cdots$

Assume a product solution:

u(x,t) = X(x)\,T(t) \quad\text{or}\quad u(x,y) = X(x)\,Y(y)

Substitute into the PDE and separate the variables to obtain ordinary differential equations.

One-dimensional

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Two-dimensional

\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)

General form

\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u

D’Alembert’s Solution (1-D infinite string)

u(x,t) = \frac{1}{2}\bigl[f(x+ct)+f(x-ct)\bigr] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds

where $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ .

One-dimensional

\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}

Two-dimensional

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

General form

\frac{\partial u}{\partial t} = k \nabla^2 u

Solution by Separation of Variables (1-D, finite rod)
For boundary conditions $u(0,t)=u(L,t)=0$ :

u(x,t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{L}\right) e^{-k(n\pi/L)^2 t}

where

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

Cartesian coordinates

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

Polar coordinates

\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

Solution by Separation of Variables (Cartesian)
Assuming $u(x,y)=X(x)Y(y)$ :

\frac{X''}{X} = -\frac{Y''}{Y} = \lambda \quad (\text{separation constant})

Typical solutions involve sinh/sin or exponential terms depending on boundary conditions.

General solution in polar coordinates (axisymmetric or Fourier series)

u(r,\theta) = a_0 + b_0\ln r + \sum_{n=1}^{\infty}\bigl(a_n r^n + b_n r^{-n}\bigr)\cos n\theta + \sum_{n=1}^{\infty}\bigl(c_n r^n + d_n r^{-n}\bigr)\sin n\theta

From Kirchhoff’s laws, the voltage $V(x,t)$ and current $I(x,t)$ satisfy:

\frac{\partial V}{\partial x} = -L\frac{\partial I}{\partial t} - RI

\frac{\partial I}{\partial x} = -C\frac{\partial V}{\partial t} - GV

Eliminating one variable yields the Telegraph Equation:

\frac{\partial^2 V}{\partial x^2} = LC\frac{\partial^2 V}{\partial t^2} + (RC+LG)\frac{\partial V}{\partial t} + RG\,V

Special Cases

Lossless line ( $R=0$ , $G=0$ ): reduces to wave equation $\frac{\partial^2 V}{\partial x^2} = LC\frac{\partial^2 V}{\partial t^2}$

These are the complete standard formulas required for Module II.