Problem 3 (8 points each) Let's see a simple example of the method of characteristics in...

 

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Problem 3 (8 points each) Let's see a simple example of the method of characteristics in higher dimension. Let w ER be a constant, and let u = u(x, y, t) satisfy the PDE Uz - xUz + YUy = 0 u(x, y,0) = f(x,y) (a) Given arbitrary initial conditions E, CER, solve the characteristic equations d [3]=5158] [*]=[] dr to find the characteristics = (E. C, T) and y = y(§. 5,7). Sketch a few of the tra- jectories in the zy-plane. Then, use the multivariable chain rule, along with t = 7, to find the PDE and initial conditions satisfied by U, where U(E.S.T) = u(x(E, S, T). y(E. C. T), t(E, S. T)) Applied PDE, Section B Spring 2023 (b) Use the PDE you found in (a) to find the solution u(x, y, t) in terms of the initial data f(x, y). Then, suppose f(x,y) = x² + y². Sketch a few level sets of u(x, y,0) and u(x, y, 1) in the ry-plane, and describe the shape of the level sets (i.e. are they circles, squares, etc...) Problem 3 (8 points each) Let's see a simple example of the method of characteristics in higher dimension. Let w ER be a constant, and let u = u(x, y, t) satisfy the PDE Uz - xUz + YUy = 0 u(x, y,0) = f(x,y) (a) Given arbitrary initial conditions E, CER, solve the characteristic equations d [3]=5158] [*]=[] dr to find the characteristics = (E. C, T) and y = y(§. 5,7). Sketch a few of the tra- jectories in the zy-plane. Then, use the multivariable chain rule, along with t = 7, to find the PDE and initial conditions satisfied by U, where U(E.S.T) = u(x(E, S, T). y(E. C. T), t(E, S. T)) Applied PDE, Section B Spring 2023 (b) Use the PDE you found in (a) to find the solution u(x, y, t) in terms of the initial data f(x, y). Then, suppose f(x,y) = x² + y². Sketch a few level sets of u(x, y,0) and u(x, y, 1) in the ry-plane, and describe the shape of the level sets (i.e. are they circles, squares, etc...)