- #1
docnet
Gold Member
- 696
- 348
- Homework Statement
- solve the modified transport equation using the method of characteristics.
- Relevant Equations
- ##\partial_t u + <b, Du> + cu=0##
##u(0,x)=f##
Hi all, I
Fix $$(t,x) ∈ (0,\infty) \times R^n$$and consider auxillary function
$$w(s)=u(t+s,x+sb)$$
Then, $$\partial_s w(s)=(\partial_tu)(t+s,x+sb)\frac{d}{ds}(t+s)+<Du(t+s,x+sb)\frac{d}{ds}(x+sb)>$$
$$=(\partial_tu)(t+s,x+sb)+<b,Du(t+s,x+sb)>$$
$$=-cu(t+s,x+sb)$$
$$\partial_sw(s)=-cu(t+s,x+sb)$$
by Fundamental theorem of calculus,
$$u(t,x)-f(x-tb)=u(t,x)-u(0,x-tb)$$
$$=w(0)-w(-t)$$
$$=\int^0_{-t}\partial_sw(s)ds$$
$$=\int^0_{-t}-cu(t+s,x+sb)ds$$
$$s_o=s+t$$
$$=\int^t_{0}cu(s_o,x+(s_o-t)b)ds_o$$
$$u(t,x)=f(x-tb)+\int^t_{0}cu(s_o,x+(s_o-t)b)ds_o$$