Solve PDE Method of Characteristics help


by lackrange
Tags: characteristics, differential eq, pde
lackrange
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#1
Sep20-11, 05:54 PM
P: 20
The problem: Solve for u(x,y,z) such that

[tex] xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y) [/tex]

So I write

[itex]\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .[/itex]

Thus [tex]u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0 [/tex]
but from here I can't figure out what to do, there are several ways I can write s....I have only done the method of characteristics before with two variables, and those are pretty much the only examples I can find. Can someone help please?
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Sep20-11, 06:19 PM
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Quote Quote by lackrange View Post
The problem: Solve for u(x,y,z) such that

[tex] xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y) [/tex]

So I write

[itex]\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .[/itex]

Thus [tex]u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0 [/tex]
but from here I can't figure out what to do, there are several ways I can write s....I have only done the method of characteristics before with two variables, and those are pretty much the only examples I can find. Can someone help please?
If you have done the method of characteristiced with two variables it pretty much generalizes to three variables directly- write out the characteristics in terms of x, y, and z and use those to define new variables.

If [itex]x= x_0e^s[/itex] and [itex]y= y_0e^{2s}[/itex], you have [itex]s= ln(x/x_0)= (1/2)ln(y/y_0)[/itex] so that ln(x/x_0)- ln(y^{1/2}/y_0^{1/2})= ln((y_0^{1/2}/x_0)(x/y^{1/2}= 0 which then gives [itex]x/y^{1/2}= constant[/itex]. Each curve with different constant is a characteristic. With [itex]z= s+ z_0[/itex] that gives also [itex]s= z-z_0= ln(x/x_0)[/itex], have [itex]x/x_0= e^{z-z_0}= e^{-z_0}e^z[/itex] so [itex]x= (constant)e^z[/itex] or [itex]xe^{-z}= constant[/itex]. Putting [itex]s= z-z_0= ln(y^{1/2}/y_0^{1/2})[/itex], we get [itex]y^{1/2}e^{-z}= constant[/itex].

We want to use those "characteristics" or "characteristic curves" as axes: let [itex]u= x/y^{1/2}[/itex], [itex]v= xe^{-z}[/itex], and [itex]w= y^{1/2}e^{z}[/itex].


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