Partial Differential Equations and Separation of Variables

Partial Differential Equations: Basic Definitions and Terminologies

• Definition of Partial Differential Equations (PDE):

  • Let $u = u(x, y)$ where $x$ and $y$ are the independent variables.

  • A PDE is an equation containing at least one partial derivative of the function $u$.

• Notations for Partial Derivatives:

  • First-order derivative with respect to $x$: $u_x = \frac{\partial u}{\partial x}$.

  • Second-order derivative with respect to $x$: $u_{xx} = \frac{\partial^2 u}{\partial x^2}$.

  • First-order derivative with respect to $y$: $u_y = \frac{\partial u}{\partial y}$.

  • Mixed second-order derivative: $u_{xy} = \frac{\partial^2 u}{\partial y \partial x} = \frac{\partial}{\partial y} (\frac{\partial u}{\partial x})$.

• Standard Examples of PDEs:

  • 1-Dimensional Wave Equation: $\frac{\partial^2 u}{\partial t^2} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$, where $\alpha$ is a constant.

  • 1-Dimensional Heat Equation: $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$, where $\alpha$ is a constant.

  • 2-Dimensional Laplace Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$.

  • 2-Dimensional Wave Equation: $\frac{\partial^2 u}{\partial t^2} = \alpha^2 (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2})$, where $\alpha$ is a constant.

  • 3-Dimensional Heat Equation: $\frac{\partial u}{\partial t} = \alpha^2 (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2})$, where $\alpha$ is a constant.

Method of Separation of Variables (MSV)

• Introduction to MSV:

  • MSV is a standard method used to solve PDE problems by reducing them into a system of Ordinary Differential Equations (ODEs).

  • Core Assumption: Seeks a solution in the form $u(x, t) = X(x)T(t)$, where $X(x)$ is a function solely of $x$ and $T(t)$ is a function solely of $t$.

• MSV Process applied to the Heat Equation:

  • Consider the equation $\frac{\partial u}{\partial t} = k^2 \frac{\partial^2 u}{\partial x^2}$.

  • Assume $u(x, t) = X(x)T(t)$.

  • Calculate partial derivatives relative to the assumption:

    • $u_t = X(x)T'(t)$

    • $u_{xx} = X''(x)T(t)$

  • Substitute these into the heat equation: $X T' = k^2 X'' T$.

  • Separation of Variables: Rearrange the equation so each side depends on only one variable: $\frac{T'}{k^2 T} = \frac{X''}{X}$.

  • Introduction of Separation Constant: Since a function of $t$ can only equal a function of $x$ if both are equal to a constant, set the ratio equal to $\lambda$: $\frac{1}{k^2} \frac{T'}{T} = \frac{X''}{X} = \lambda$.

  • Resulting ODEs:

    1. $X'' - \lambda X = 0$

    2. $T' - k^2 \lambda T = 0$

• Exercises: Reduction to ODEs:

  • Wave Equation: $\frac{\partial^2 u}{\partial t^2} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ reduces to $X'' - \lambda X = 0$ and $T'' - \alpha^2 \lambda T = 0$.

  • Laplace Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ reduces to $X'' - \lambda X = 0$ and $Y'' + \lambda Y = 0$.

Solving Heat Equation Using MSV

• Problem Definition:

  • Heat Equation: $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ for $0 < x < L$, $t > 0$.

  • Initial Condition (IC): $u(x, 0) = f(x)$, $0 < x < L$.

• Common Types of Boundary Conditions (BCs):

  • Zero Endpoints (Dirichlet): $u(0, t) = 0$, $u(L, t) = 0$ for $t > 0$.

  • Insulated Endpoints (Neumann): $u_x(0, t) = 0$, $u_x(L, t) = 0$ for $t > 0$.

  • Mixed Endpoints: Example: $u(0, t) = 0$ and $u_x(L, t) = 0$.

• General Five-Step Solution Process:

  • Step 1: Reduce the PDE to two ODEs using Separation of Variables.

  • Step 2: Form the BCs for the ODEs by applying the given PDE boundary conditions to $X(x)$.

  • Step 3: Consider three cases for the separation constant $\lambda$:

    • Case 1: $\lambda = 0$.

    • Case 2: $\lambda > 0$ (Let $\lambda = p^2$).

    • Case 3: $\lambda < 0$ (Let $\lambda = -p^2$).

  • Step 4: Determine which case provides a non-trivial solution and sum the solutions (Superposition Principle).

  • Step 5: Apply the Initial Condition ($u(x, 0) = f(x)$) to solve for constants/coefficients.

• Example 1 (Zero Endpoints):

  • Solve $\frac{\partial u}{\partial t} = 4 \frac{\partial^2 u}{\partial x^2}$ where $0 < x < 1$, $t > 0$.

  • BCs: $u(0, t) = 0$, $u(1, t) = 0$.

  • IC: $u(x, 0) = x^4$.

• Example 2 (Insulated Endpoints):

  • Solve $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ with $u_x(0, t) = 0$ and $u_x(L, t) = 0$.

  • IC: $u(x, 0) = x$, $0 < x < L$.

Wave for Infinite Length: D’Alembert Method

• Concept:

  • Used for wave equations with infinite boundaries (or boundaries far enough away that they don't influence wave length).

  • Governing Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ for $-\infty < x < \infty$, $t > 0$.

  • Subject to:

    • Initial displacement: $u(x, 0) = f(x)$.

    • Initial velocity: $u_t(x, 0) = g(x)$.

• D’Alembert Formula:

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

• Example 3:

  • Solve $\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0$ (indicating $c = 1$).

  • Given: $u(x,0) = \sin(\pi x)$, $\frac{\partial u}{\partial t}(x, 0) = 0.5$.

• Example 4:

  • Show the solution for $\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}$ with displacement $u(x, 0) = \sin(2x) + \sin(x)$ and velocity $u_t(x, 0) = 0$ is:

  • $u(x, t) = \sin(2x) \cos(2t) + \sin(x) \cos(t)$.

Solving Wave Equations Using MSV

• Elastic String Model:

  • Motion of a string of length $L$ described by: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ for $0 < x < L$, $t > 0$.

  • Boundary Conditions: Fixed ends: $u(0, t) = 0$, $u(L, t) = 0$.

  • Initial Conditions:

    • Displacement: $u(x, 0) = f(x)$.

    • Velocity: $u_t(x, 0) = g(x)$.

• Example 5:

  • PDE: $\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$ ($c = 2$).

  • Interval: $0 < x < 2$ ($L = 2$).

  • BCs: $u(0, t) = 0$, $u(2, t) = 0$.

  • ICs: $u(x, 0) = 2 - x$, $u_t(x, 0) = 1$.

Solving Laplace's Equations Using MSV

• Equation Properties:

  • Laplace Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ on domain $0 < x < a$, $0 < y < b$.

  • Time Independence: There is no dependence on time, only spatial variables $x, y$.

  • Represents Steady State Situations:

    • Steady state temperature distributions.

    • Steady state stress distributions.

    • Steady state potential distributions.

• Example 6 (Square Plate):

  • Domain: Bounded by $x = 0, x = a, y = 0, y = a$.

  • Boundary Conditions:

    • $u(0, y) = 0$, $u(a, y) = 0$ for $0 < y < a$.

    • $u(x, 0) = 0$.

    • $u(x, a) = u_0 (\sin(\frac{\pi x}{a}) + 2 \sin(\frac{2\pi x}{a}))$ where $u_0$ is a constant.

  • Objective: Determine potential distribution $u(x, y)$.

  • Required Proof: Show temperature at the plate center $(\frac{a}{2}, \frac{a}{2})$ is:

    • $u(\frac{a}{2}, \frac{a}{2}) = \frac{u_0 \sinh(\frac{\pi}{2})}{\sinh(\pi)}$ (Simplified from the result of the separation of variables steps).