PDEMOC6

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$u_x + xu_y -u_z = u\,$ , $u(x,y,1)=x+y\,$ ,

Let the initial curve be parameterized by $\Gamma(s_1,s_2,1)=s_1+s_2\,$ .

The characteristics are:

$\frac{dx_1}{dt}=1, \frac{dx_2}{dt}=x, \frac{dx_3}{dt}=-1, \frac{dz}{dt}=z\,$

$\frac{dx_1}{dt}=1\,$

$x_1(s,t) = t + c_1(s)\,$

$x_1(s,0) = c_1(s) = s_1\,$

$x_1 = t+s_1\,$

$s_1 = x_1-t\,$

$\frac{dx_2}{dt}=x\,$

$x_2(s,t) = \frac{1}{2}t^2 + c_1(s)t + c_2(s)\,$

$x_2(s,0) = c_2(s) = s_2\,$

$x_2 = \frac{1}{2}t^2 + s_1 t + s_2\,$

$s_2 = x_2 - \frac{1}{2}t^2-(x_1-t)t\,$

$\frac{dx_3}{dt}=-1\,$

$x_3(s,0) = c_3(s)=1\,$

$x_3 = 1-t\,$

$t=1-x_3\,$

$\frac{dz}{dt}=z\,$

$\ln z = t+c_4(s)\,$

$z(s,t) = c_5(s)e^t\,$

$z(s,0) = c_5(s) = s_1 + s_2\,$

$z=(s_1+s_2)e^t\,$

Use these equations to get:

$u(x,y)=z=\left[x_1 + x_2 + (x_3-1)\left[\frac{1}{2}(x_3+1)+x_1\right]\right] e^{1-x_3}\,$

Main Page : Partial Differential Equations : Method of Characteristics

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