Boundary Value Problem , triangular plate

In summary, the problem deals with finding the temperature distribution u(x,y,t) in a homogeneous and thin plate with insulated top and bottom, and a triangular shape with the hypotenuse side insulated and the other two sides at 0 and 50 degrees. The initial temperature is 100 degrees throughout. The BVP for this problem involves four boundary conditions and one governing equation, with the notation u_x used for both the vertical and horizontal sides. The use of u_x and u_t may vary in different problems and further clarification may be needed.
  • #1
jc2009
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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.

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

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?
 
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  • #2
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
 

Related to Boundary Value Problem , triangular plate

1. What is a boundary value problem in relation to a triangular plate?

A boundary value problem is a mathematical analysis tool used to determine the behavior of a system, such as a triangular plate, based on the values at its boundaries. In this case, the boundaries refer to the edges of the triangular plate.

2. How is a boundary value problem different from an initial value problem?

A boundary value problem involves finding a solution that satisfies given conditions at the boundaries of a system, while an initial value problem involves finding a solution that satisfies conditions at a single starting point. In other words, boundary value problems are concerned with the behavior of a system as a whole, while initial value problems focus on the behavior at a specific point.

3. What are the key components of a boundary value problem for a triangular plate?

The key components of a boundary value problem for a triangular plate include the geometry of the plate (e.g. length, width, angles), the material properties of the plate (e.g. elasticity, density), and the boundary conditions (e.g. forces, displacements) at each edge of the plate.

4. Why are boundary value problems important in the study of triangular plates?

Boundary value problems are important in the study of triangular plates because they allow us to determine the behavior of the plate under various conditions. This information can be used to design stronger and more efficient triangular plates for various applications, such as in construction or aerospace engineering.

5. Are there any real-world applications for solving boundary value problems for triangular plates?

Yes, there are several real-world applications for solving boundary value problems for triangular plates. Some examples include designing bridges, determining the stability of aircraft wings, and analyzing the behavior of mechanical components in machinery. Boundary value problems are also used in fields such as geology and meteorology to study the behavior of natural systems.

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