Method of Separation of Variables

SEPERATION OF VARIABLES

Let second order PDE is given.

And $u$ be a function of two independent variables $x$ & $t$.

We assume that all required solution is separable and of form

$u(x, t) = X(x) \cdot T(t) \quad \text{..... (1)}$

Using this equation (1) we reduce the PDE in form of

$f(X, X', X'', ..... ) = g(T, T', T'', .....) \quad \text{..... (2)}$

Which is separable in $X$ & $T$.

From equation (2)

$f(X, X', .....) = k \quad \text{..... (3)}$

$g(T, T', .....) = k \quad \text{..... (4)}$

Equation (3) & (4) are ODE and can be solved by known methods.

Solution of One-Dimensional Wave and Heat Equation

One-Dimensional Wave Equation

Equation: $\frac{\partial^2 u}{\partial t^2} = C^2 \frac{\partial^2 u}{\partial x^2}$

• $C$ Wave speed, a constant that represents how fast the wave propagates through the medium.
• $x$ Spatial coordinate, representing the position along the one-dimensional medium.
• $t$ Time variable.

Boundary Conditions:

1. $u(0, t) = 0$ (Dirichlet boundary condition)
2. $u(L, t) = 0$ (Dirichlet boundary condition)

Initial Conditions:

1. $u(x, 0) = f(x)$ (initial displacement)
2. $\frac{\partial u}{\partial t}(x, 0) = g(x)$ (initial velocity)

One-Dimensional Heat Equation

Equation: $\frac{\partial u}{\partial t} = C^2 \frac{\partial^2 u}{\partial x^2}$

• $C$ Thermal diffusivity, a constant that indicates how quickly heat diffuses through the medium.
• $x$ Spatial coordinate, representing the position along the one-dimensional medium.
• $t$ Time variable.

Boundary Conditions:

1. $u(0, t) = 0$ (Dirichlet boundary condition)
2. $u(L, t) = 0$ (Dirichlet boundary condition)

Initial Condition:

1. $u(x, 0) = f(x)$ (initial temperature distribution)

These conditions define the behavior of the solutions in a specified domain over time.

Boundary and Initial Conditions

• $L$: Length of the domain for both equations.
• $f(x)$: Initial displacement (for the wave equation) or initial temperature distribution (for the heat equation) as a function of position $x$.
• $g(x)$: Initial velocity as a function of position $x$ (only relevant for the wave equation).

Two-Dimensional Laplace's Equation

The two-dimensional Laplace's equation is given by:

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

• $u(x, y)$: A scalar potential function, which can represent various physical quantities such as electric potential, fluid velocity potential, or temperature.
• $x$: The spatial coordinate in one direction (e.g., horizontal).
• $y$: The spatial coordinate in the perpendicular direction (e.g., vertical).