Laplace equation Boundary condition

Question

By using the method of separation of variable I'm trying to solve the laplace equation

u_(xx) + u_(yy) = 0

in the unit disc with the boundary condition

u(x, y) = x^2*y for all x and y such that x^2 + y^2 = 1

Answer 1

u_xx + u_yy = 0

 

 

Let u(x,y) = X(x)Y(y), then

 

YX_xx + XY_yy = 0  or  YX_xx = -XY_yy.

 

 

Separating variables, Y_yy / Y = -X_xx / X = k².

 

We get Y_yy = k²Y  and   X_xx = -k²X

 

Y_k(y) = A_kcosh(ky) + B_ksinh(ky)

 

X_k(x) = C_kcos(kx) + D_ksin(kx)

 

 

u(x,y) = ΣX_k(x)Y_k(y)

 

= Σ[C_kcos(kx) + D_ksin(kx)][A_kcosh(ky) + B_ksinh(ky)]

 

 

On the boundary, x = ±√(1 - y²)   and   u(x,y) = x²y = (1 - y²)y.

 

Since x²y is even in x and odd in y, u(x,y) = ∑E_kcos(kx)sinh(ky).

 

 

 

Find the constants--E_k--for

 

(1 - y²)y = ∑E_kcos(k√(1 - y²))sinh(ky)