course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Methods |
IB |
53 |
|
Paper 4, Section II, 17C |
2015 |
Describe the method of characteristics to construct solutions for 1st-order, homogeneous, linear partial differential equations
with initial data prescribed on a curve
Consider the partial differential equation (here the two independent variables are time
with initial data
(i) Calculate the characteristic curves of this equation and show that
(ii) Let $\tilde{x}{0}$ denote the $x$ value of a characteristic at time $t=0$ and thus label the characteristic curves. Let $\tilde{x}$ denote the $x$ value at time $t$ of a characteristic with given $\tilde{x}{0}$. Show that $\partial \tilde{x} / \partial \tilde{x}{0}$ becomes a non-monotonic function of $\tilde{x}{0}$ (at fixed