partial-differential-equations-by-evans · Problem 13

Nonlinear First-Order PDE

Mathematics — Calculus · heat-equation-boundary-value-formula

Problem

Given

g : [0,\infty) \to \mathbb{R}

, with

g(0) = 0

, derive the formula

u(x,t) = \frac{x}{\sqrt{4\pi}} \int_0^t \frac{1}{(t-s)^{3/2}} e^{\frac{-x^2}{4(t-s)}} g(s)\, ds

for a solution of the initial/boundary-value problem

\begin{cases} u_t - u_{xx} = 0 & \text{in } \mathbb{R}_+ \times (0,\infty) \\ u = 0 & \text{on } \mathbb{R}_+ \times \{t=0\}, \\ u = g & \text{on } \{x=0\} \times [0,\infty). \end{cases}

(Hint: Let

v(x,t) := u(x,t) - g(t)

and extend

v

\{x < 0\}

by odd reflection.)