How can the Heat Equation be solved for a periodic heating scenario?

In summary, the problem involves a semi-infinite bar with periodic heating at one end and a temperature of zero at the other end. The heat equation is used to solve for the temperature distribution, with the final solution being T(x,t)= T_0exp(αx)cos(ωt − x sqrtω), where alpha is a constant to be determined. The variables are separated and solved for the t dependence first, but there is difficulty in finding the appropriate approach for solving the problem.
  • #1
TobyDarkeness
38
0
thanks allot they worked out fine, just another quick question if could help.
A semi-infinite bar 0<x<infinity is subject to periodic heating at x=0; the temperature at x=0 is T_0cos[tex]\omega[/tex]t and is zero at x=infinity. By solving the heat equation show that

T(x,t)= T_0exp([tex]\alpha[/tex]x)cos([tex]\omega[/tex]t-x[tex]\sqrt{\omega}[/tex])

where alpha is a constant to be determined.

I know we have to separate the variables and solve the t dependence first, but its not really working. Any advice on how to tackle this question appropriately.
 
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  • #2
my attempt so far

∂T/∂t= 1/2*(∂^2T/∂x^2)

T(x,t)=X(x)T(t)

∂/∂t*[X(x)T(t)]=1/2*[(∂^2)/(∂x^2)]*(X(x)T(t))

X(x)*[∂T(t)/∂t]=1/2*T(t)*[∂^2X(x)]/[∂x^2]

dividing through by 1/[X(x)T(t)]


1/[T(t)]*[∂T(t)/∂t]=1/2*[1/X(x)]*(∂^2 X(x))/∂x^2


2/T(t)*∂T(t)/∂t=1/X(x)*[(∂^2X(x))/(∂x^2)]

boundary conditions
T(x,t) =T_0exp(αx)cos(ωt − x sqrtω)

T(0,t)=T_0cos(ωt)

T(infinity,0)=0
 

1. What is the heat equation?

The heat equation is a mathematical equation that describes the distribution of heat in a given system over time. It is a partial differential equation that can be solved using various mathematical methods.

2. What are the applications of the heat equation?

The heat equation has numerous applications in fields such as physics, engineering, and mathematics. It is commonly used to model heat transfer in various systems, including heat conduction in solid objects, heat convection in fluids, and heat radiation in electromagnetic systems.

3. What are the key parameters in the heat equation?

The key parameters in the heat equation are the thermal conductivity, the specific heat capacity, and the density of the material. These parameters determine the rate at which heat is transferred in a given system and are essential in solving the equation.

4. What are the different methods for solving the heat equation?

There are several methods for solving the heat equation, including analytical methods, numerical methods, and approximate methods. Analytical solutions involve finding a closed-form solution using mathematical techniques, while numerical methods use computer algorithms to approximate the solution. Approximate methods, such as the separation of variables method, provide an approximate solution to the equation.

5. How is the heat equation used in real-world problems?

The heat equation is used in real-world problems to predict the temperature distribution in various systems and to design efficient heat transfer systems. It is also used in studying the behavior of materials under different thermal conditions and in analyzing the thermal stability of systems.

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