Hello, I am preparing for my PDE final, and I have questions relating to two problems:

My usual approach to solving PDE problems in our course (like the heat and wave equation) has been to separate the variables and then look for common solutions. However, this doesn't appear to work here. How do I approach a problem like this? I see that it's an inviscid form of burger's equation, but I can't understand how to solve that.Solve the initial value problem for u = u(x,t)

U_t + UU_x = 0

-inf < x < inf for t > 0 and u(x,0) = tanh x for -inf < x < inf

And for the second question:

For this problem, I'm able to separate the variables, but I am unsure how to form a power series solution to this problem. Thank you for any help you can provide!Find a series solution to the boundary value problem

U_xx + U_yy = 0, 0 < x < 1, 0 < y < 4

U_y(x,0) = f(x), U_y(x,4) = 0, 0 < x < 1

U_x(0,y) = 0, U_x(1,y) = 0, 0 < y < 4