How to Solve the Inhomogeneous Heat Equation for a Cylindrical Rod?

In summary, the equation given is the heat equation for a cylindrical rod with a constant rate of heat production. The task is to find an ordinary differential equation for θ(r), but the method of separation of variables does not work with the given constant.
  • #1
mumaga
16
0
inhomogeneuos heat equation!

Homework Statement


∂θ/∂t= D∇2θ + K, the mensioned equation is the heat equation for a cylindrical rod , and the requaired is to find the ordinary differential equation for θ(r) .where the radius of the rod is R , and K is constant ( correspond to a constant rate pf heat production)



Homework Equations





The Attempt at a Solution


i use separation of variables to obtain the required for a homogeneous heat equation , but with the constant the method didn't work out.
thanks for your time.
 
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  • #2
Welcome to PF!

mumaga said:
∂θ/∂t= D∇2θ + K, the mensioned equation is the heat equation for a cylindrical rod , and the requaired is to find the ordinary differential equation for θ(r) .where the radius of the rod is R , and K is constant ( correspond to a constant rate pf heat production)

Hi mumaga! Welcome to PF! :smile:

(I assume you mean ∂θ/∂t= C∇2θ + K, where C and K are constants, and θ depends only on t and r.)

See http://en.wikipedia.org/wiki/Heat_equation#Homogeneous_heat_equation :smile:
 

FAQ: How to Solve the Inhomogeneous Heat Equation for a Cylindrical Rod?

What is the inhomogeneous heat equation?

The inhomogeneous heat equation is a partial differential equation that describes the distribution of heat in a non-uniform medium. It takes into account external sources of heat, such as heat generation or heat flow, in addition to the diffusion of heat within the medium.

How is the inhomogeneous heat equation different from the homogeneous heat equation?

The homogeneous heat equation only considers the diffusion of heat within a medium without any external heat sources. The inhomogeneous heat equation, on the other hand, includes these external sources and is more complex to solve.

What are some real-world applications of the inhomogeneous heat equation?

The inhomogeneous heat equation has many practical applications, such as modeling heat transfer in materials processing, analyzing thermal behavior in buildings, and predicting temperature changes in the Earth's atmosphere. It is also used in fields such as engineering, physics, and meteorology.

What techniques are used to solve the inhomogeneous heat equation?

There are several methods for solving the inhomogeneous heat equation, including separation of variables, Green's functions, and numerical methods such as finite difference and finite element methods. The choice of method depends on the specific problem and the desired level of accuracy.

How does the inhomogeneous heat equation relate to other equations in physics?

The inhomogeneous heat equation is closely related to other equations in physics, such as the diffusion equation and the wave equation. It can also be derived from the laws of thermodynamics and the principles of conservation of energy. Many problems in physics involve solving coupled equations that include the inhomogeneous heat equation.

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