PDE's

I would like see how these problems are solved. Thank you. 5. Establish the orthogonality of the trigonomietric system over the interval [-п , п]. Answer: This is the special case p = п of exercise 6(b). 6(b) Trigonomertic systems of arbitrary period. Let p > 0 and consider the trigonometric system 1, cos(п/p)x. cos2(п/p)x, cos3(п/p)x, … , cosm(п/p)x, …, sin(п/p)x, sin2(п/p)x, sin3(п/p)x, …, sinm(п/p)x, … State and prove the orthogonality relations for this system on the interval [-p, p] 7. Sums of periodic functions. Show that if f1, f2, …, fn, … are T-periodic functions, then a1f1 +... click for more

Sobolev Spaces

See attached question. I think the answer is alpha < n/p - m, but I would like a detailed explanation.

Solve

1. solve X*U_x + Y*U_y =0 answer is suppose to be U(X,Y)=f(Y/X) 2. solve U_x + U_y =1 answer unknown

Sobolev Spaces

See attached problem.

Interpolation Inequality

Please see attached problem.