I am really stuck in the following problem
consider the following semi linear PDE: $$xu_x+yu_y = 4u$$ where $u(x,y)$ lies on unit circle given by $x^2+y^2$=1. Then find the value of $u(2,2)$.
Any help would be appericiated...
cauchy problem for semilinear PDE $xu_x+yu_y = 4u$
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pde
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0When you have a Cauchy problem, you are usually given a PDE and the values of an unknown function on a certain CURVE. What your curve, then? – 2017-02-09
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0What does it mean "$u(x,y)$ lies on unit circle"? Does it mean that $u=0$ on these points? If affirmative, $u$ is identically zero. – 2017-02-09
1 Answers
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This problem can be solved by the method of characteristics. Consider a curve $(x(t), y(t))$ where $$\dot{x} = x$$ $$\dot{y} = y$$ By the multivariable chain rule, $$D_t u = xu_x + yu_y = 4u$$ Thus, $$u(x(t), y(t)) = u(x(0), y(0))e^{4t}$$ We can also solve for $x(t), y(t)$ to get $$x(t) = x(0)e^t, \quad y(t) = y(0)e^t$$ so, assuming the initial data is given on the unit circle (I'll let you work out the details): $$u(x,y) = u\left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right)\left(x^2+y^2\right)^2.$$
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0How did you simplified $xu_x+yu_y = 4u$.? – 2017-02-09
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0@MyGlasses That comes from assuming $u$ satisfies the PDE $xu_x + yu_y = 4u$ – 2017-02-09
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0and $u(2,2)=$.? – 2017-02-09
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0@MyGlasses Plug in $x = y = 2$. I get $u(2,2) = 64u\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$. – 2017-02-09