[微積] 關於偏微分方程
有人會這兩題嗎?
1.The Inviscid Burgers’Equation can be stated as
PD u/ PD t + u* PDu/ PD x = 0
Consider the initial condition
u(x,0) = 1/(1+x 平方)
At what time does a shock form?
Hint:write the solution in the form u(x,t)=f(x-u(x,t)t),
then find the values of t for which a unique real solution may be found.
2. Consider the quasi-liner wave equation of the form
PD u/ PD t + u(1+x 平方) 3/2次方 PD u/ PD x = x/ 庚號(1+x平方)
with the initial condition u(x,0) = 1. Solve for u(x,t).
Hite: the following derivative will be useful in finding the general solution
in terms of X0 and t.
d/dx(x/更號1+x平方) = 1/(1+x平方)3/2次方
Furthermore, the solution will involve the cosh and sinh functions.
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