Consider the Laplace equation ^2u/(x^2) + ^2u/(y^2) = 0 that models the steady-s

Question

Consider the Laplace equation

^2u/(x^2) + ^2u/(y^2) = 0

that models the steady-state heat distribution u(x, y) in a rectangular plate 0 x a and 0 y b. In physics, the Fourier’s law of heat conduction states that the heat flux density (heat energy that flows through a surface per unit time) is proportional to the temperature gradient. From multivariable calculus, we know that the gradient vector is given by

u = (u/x, u/y )

Suppose that the plate is held at zero degrees at y = 0 and y = b, and that it is insulated at x = 0 so that the heat flux there is zero. These three boundary conditions can be expressed mathematically as:

u(x, 0) = u(x, b) = ux(0, y) = 0 .

Given a heat flux profile at x = a such that u_x(a, y) = f(y), solve the Laplace equation using separation of variables.

Explanation / Answer

^2u/(x^2) + ^2u/(y^2)

Here t=0 because state is steady

Using seperation of variable

U(x,y)=X(x)Y(y)

CONSIDER Y*^2X/(x^2) + X*^2Y/(y^2) =-K^2

(1/X)*^2X/(x^2)=-K^2

(1/Y)*^2Y/(x^2)=-K^2

Equation1:^2X/(x^2)+K^2 =0

Equation2:^2Y/(x^2)+K^2 =0

On solving equation 1 and 2 we get

D^2+K^2=0 so D=+iK and -iK

X=AcosKx+BsinKx

y=CcosKy+DsinKy

So on applying boundary condition we get U(x,y)=A*C*sink*sinKy

Where A and C are constants

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