please post full solution for 5 stars! This question relates to finding partial
ID: 2841998 • Letter: P
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please post full solution for 5 stars!
This question relates to finding partial derivatives. Burger's Equation is a partial differential equation, used for describing wave processes in acoustics and hydrodynamics, Verify that is a solution, where lambda and alpha are arbitrary constants. It can be shown that the radially symmetric temperature distribution in a spherical metal ball is given by the solution to the partial differential equation where the constant alpha is the thermal diffusivity. Verify that satisfies the PDE for the temperature distribution in the ball.Explanation / Answer
(a) i'll note lambda = L, alpha=a for simplier notation here.
dw/dt = -2L / (x+Lt+a)^2
dw/dx = -2/(x+Lt+a)^2
dw/dx^2 = 4/(x+Lt+a)^3
dw/dx^2 + w dw/dx
= 4/(x+Lt+a)^3 -2(L+2/(x+Lt+a))/(x+Lt+a)^2
= (4 -2(L(x+Lt+a)+2)/(x+Lt+a)^3
= -2L(x+Lt+a)/(x+Lt+a)^3
= -2L/(x+Lt+a)^2
= dw/dt
(b) Here i'll ignore the index of sum for simplier notation sum = sum_{n=1..+inf}
So
du/dt = 2 sum sin(n*P*ir)/r * (-a*n^2Pi^2) * e ^(-a*n^2*Pi^2*t)
du/dr = 2 sum( (n*Pi cos(n*Pi*r) r - sin(n*Pi*r) ) /r^2 e(-a*n^2*Pi^2*t)
so r^2 du /dr = 2 sum( n*Pi*cos(n*Pi*r)*r - sin(n*Pi*r) ) e(-a*n^2*Pi^2*t)
And d/dr(r^2 du/dr)
= 2 sum (-n^2*Pi^2 sin(n*Pi*r)*r+n*Pi*cos(n*Pi*r)-n*Pi*cos(n*Pi*r))e^(-a*n^2*Pi^2*t)
= 2 sum (-n^2*Pi^2 sin(n*Pi*r)*r)e^(-an^2Pi^2t)
Finally a/r^2 * d/dr(r^2 du/dr)
= 2 sum -a*n^2*Pi^2*sin(n*Pi*r) /r e^(-a*n^2*Pi^2*t)
= du/dt
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