Boundary Value Problem , triangular plate

[jc2009]

This exercise deal with the temperature u(x,y,t) in a homogeneous and thin plate. We assume that the top and bottom of the plate are insulated and the material has diffusivity k. Write the BVP .

Problem: The plate is triangular , picture this as a right triangle with this coordinates, (0,0) ,
(0,5) , ( 10,0) , with the hypotenuse(slanted) side being insulated the vertical side with 0 degrees and the horizontal side with 50 degrees.

THe initial temperature is 100 degrees throughout.

Solution: what i did first is to get the equation of the slanted side which is y = -(1/2)x + 5
or 2y + x - 10 = 0 i don't know if this helps at all.

$$u_{x}(x,0,t) = 50$$ ; 0<x<10
$$u(0,y,t) = 0$$ ; 0<y<5
now for the slanted side i don't know if this is right
$$u(x,y,t) = 2y + x - 10 = 0$$

any help/hints would be appreciated.

NOTE: the use of u(x,t) and confuses me , sometimes i see that they use u_x for the vertical side or BVP problems and sometimes they use u_x for the horizontal . can you help me to clarify this notation issue?

 

[gtoguy97]

Answer: The boundary value problem (BVP) for the temperature u(x,y,t) in the homogeneous and thin plate is as follows: 1. u_x(x, 0, t) = 50, 0 < x < 102. u_x(0, y, t) = 0, 0 < y < 53. u(x, y, t) = -(1/2)x + 5, 0 < x < 10, 0 < y < 54. u(x, y, 0) = 100, 0 < x < 10, 0 < y < 55. u_t(x, y, t) = k*u_xx(x, y, t) + k*u_yy(x, y, t), 0 < x < 10, 0 < y < 5The notation used is as follows:u_x: partial derivative of u with respect to xu_xx: second partial derivative of u with respect to xu_t: partial derivative of u with respect to tk: diffusivitiy of the material