# Heat Equation: Boundary Value Problem

• JonathanT
In summary, the conversation discusses the difficulty of solving a boundary condition in a heat equation problem with a non-time-independent boundary. The solution involves introducing a change of variables and finding a function that satisfies the boundary conditions.
JonathanT
http://img821.imageshack.us/img821/7901/heatp.png

I'm having difficulty with the boundary conditions on this problem. I don't need a solution or a step by step. I've just never solved a boundary condition like this.

Its the u(pi,t) = cos(t) that is giving me difficulty

I tried getting a steady state solution for this. However, I end up with

v(x) = (x/pi)*Cos(t)

which makes no sense because v(x) should not be dependent on 't.'

I can't make it homogeneous in order to solve it by separation of variables. Any advice would be greatly appreciated.

Last edited by a moderator:
Your "steady state" is correct. You shouldn't expect a time-independent steady state for a problem like this since you have time-dependent forcing from the boundary. To see that, imagine the wave equation equivalent. If you have a string with some initial condition, but you keep shaking one end of the string indefinitely, you don't expect it to have a true steady state, right?

Anyway, the trick to solving a heat equation with inhomogeneous boundary conditions is to introduce a change of variables that allows you to make it homogeneous; namely,

v(x,t) = u(x,t) - u0(x,t)

where u0(x,t) is any function that satisfies the boundary conditions, i.e. u0(0,t) = 0, u0(pi,t) = cos(t). Can you think of any function that would accomplish this?

Hmmm, I guess I got confused by v(x) having cos(t) in it. My book literally says "find a function v(x), independent of 't.'

I'll keep working it. I'll return if I need help. Or if I figure it out.

It would certainly be nice if v had no time dependence, but now it just becomes an inhomogeneous heat equation with homogeneous boundary conditions

## What is the Heat Equation?

The heat equation is a mathematical formula that describes how heat is distributed and changes over time in a given region. It is a partial differential equation that is commonly used in physics and engineering to model heat transfer.

## What is a Boundary Value Problem?

A boundary value problem is a type of mathematical problem that involves finding a solution to a differential equation subject to specified boundary conditions. In the case of the heat equation, the boundary conditions refer to the temperature at the boundaries of the region in which heat is being transferred.

## What are some applications of the Heat Equation?

The heat equation has many practical applications, including predicting the temperature distribution in various systems and materials, such as buildings, engines, and electronic devices. It is also used in fields like meteorology, oceanography, and geology to study heat transfer in the Earth's atmosphere, oceans, and crust.

## What are the main assumptions of the Heat Equation?

The heat equation assumes that the material being studied is homogeneous, meaning that its properties do not vary throughout the region. It also assumes a steady state, meaning that the temperature does not change with time. Additionally, the heat equation assumes that there are no internal heat sources or sinks and that thermal conductivity is constant.

## How is the Heat Equation solved?

The heat equation can be solved using various mathematical methods, such as separation of variables, Fourier series, and numerical methods. The specific method used depends on the boundary conditions and the complexity of the problem. In some cases, analytical solutions may be possible, while in others, numerical approximations are used.

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