See attached question. I think the answer is alpha < n/p - m, but I would like a detailed explanation.
Partial Differential Equations and the Lagrange Method
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
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Interpolation Inequality and Sobolev Spaces
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Wave Equations and Energy Density
For a solution of the wave equation with p=T=C=1 the energy density is defined as e=1/2 (U_t ^2 + U_x ^2) and the momentum density as p=U_t*U_x Show that de/dt=dp/dx and dp/dt=de/dx Show that both e(x,t) and p(x,t) also satisfy the wave equation http://tosio.math.toronto.edu/pdewiki/index.php/2006APM346Midterm1 It's ...continues
Trigonometric Systems and Inner Products; Two-Dimensional Laplace Equations
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Show that S(x,y,t)=S(x,t)S(y,t) satisfies the diffusion equation. S_t = k(S_xx + S_yy)
Wave Equations and Periodic Differentiable Functions
3. Solve the wave equation, ∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞ With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1) 4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coeffici ...continues
A solution of the wave equation using D'Alembert's solution
Solve the wave equation subject to the initial conditions u(x,0)=sin(x)/(x^2+1), du/dt(x,0)=x/(x^2+1)
The Fourier coefficients of a derivative
Let f be a 2 pi periodic, differentiable function with Fourier coefficients a_n and b_n. Let (a_n)*, (b_n)* be the Fourier coefficients of f'. a) Show that (a_0)*=0 b) Use integration by parts to find a formula for the Fourier coefficients of f' in terms of the Fourier coefficients of f. (The attachment contains the ...continues
