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# Partial Differential Equations ## Examples of Important Equations ### The Heat Equation $\frac{\partial u}{\partial t} = k \nabla^2 u$ ### The Wave Equation $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$ ### Laplace's Equation $\nabla^2 u = 0$ ### Poisson's Equation $\nabla^2 u = f$ ##...

# Partial Differential Equations ## Examples of Important Equations ### The Heat Equation $\frac{\partial u}{\partial t} = k \nabla^2 u$ ### The Wave Equation $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$ ### Laplace's Equation $\nabla^2 u = 0$ ### Poisson's Equation $\nabla^2 u = f$ ### Schrödinger's Equation $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi$ ## Types of Partial Differential Equations **Elliptic:** Solutions are generally very smooth. Example: Laplace's Equation. **Parabolic:** Solutions smooth out as time increases. Example: The Heat Equation. **Hyperbolic:** Solutions can have sharp edges that propagate at a finite speed. Example: The Wave Equation. ## Example: The 1-D Wave Equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ We can rewrite this equation as $\left(\frac{\partial}{\partial t} - c \frac{\partial}{\partial x}\right) \left(\frac{\partial}{\partial t} + c \frac{\partial}{\partial x}\right) u = 0$ Let $\xi = x + ct$ and $\eta = x - ct$. Then $\frac{\partial}{\partial \xi} = \frac{\partial x}{\partial \xi} \frac{\partial}{\partial x} + \frac{\partial t}{\partial \xi} \frac{\partial}{\partial t} = \frac{\partial}{\partial x} + \frac{1}{c} \frac{\partial}{\partial t}$ $\frac{\partial}{\partial \eta} = \frac{\partial x}{\partial \eta} \frac{\partial}{\partial x} + \frac{\partial t}{\partial \eta} \frac{\partial}{\partial t} = \frac{\partial}{\partial x} - \frac{1}{c} \frac{\partial}{\partial t}$ Therefore, $\frac{\partial}{\partial \xi} = \frac{1}{2} \left(\frac{\partial}{\partial x} + \frac{1}{c} \frac{\partial}{\partial t}\right)$ $\frac{\partial}{\partial \eta} = \frac{1}{2} \left(\frac{\partial}{\partial x} - \frac{1}{c} \frac{\partial}{\partial t}\right)$ and $\frac{\partial}{\partial x} = \frac{\partial}{\partial \xi} + \frac{\partial}{\partial \eta}$ $\frac{\partial}{\partial t} = c \left(\frac{\partial}{\partial \xi} - \frac{\partial}{\partial \eta}\right)$ Substituting into the wave equation, we get $\frac{\partial^2 u}{\partial \xi \partial \eta} = 0$ The general solution is $u = f(\xi) + g(\eta) = f(x + ct) + g(x - ct)$ ## Example: Separation of Variables Consider the heat equation $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ Let's guess that the solution is of the form $u(x, t) = X(x) T(t)$ Substituting into the heat equation, we get $X(x) T'(t) = k X''(x) T(t)$ Dividing both sides by $X(x) T(t)$, we get $\frac{T'(t)}{T(t)} = k \frac{X''(x)}{X(x)}$ Since the left side depends only on $t$ and the right side depends only on $x$, both sides must be equal to a constant, which we will call $-\lambda$. $\frac{T'(t)}{T(t)} = k \frac{X''(x)}{X(x)} = -\lambda$ Therefore, we have two ordinary differential equations: $T'(t) = -\lambda T(t)$ $X''(x) = -\frac{\lambda}{k} X(x)$