Finding Solution of Inhomogeneous Heat Equation

In summary, the problem involves finding solutions to the Dirichlet problems for the Heat equation. It is shown that if u(x,t) and v(x,t) are solutions to these problems, and certain conditions are met (Φ₂(x) ≤ Φ₁(x) and g₂(t) ≤ g₁(t)), then for all 0 < x < 1 and t > 0, u(x,t) will be greater than or equal to v(x,t). Steps from examples 2 and 3 are followed, but may be difficult to understand.
  • #1
dominic.tsy
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Homework Statement



Show that if u(x,t) and v(x,t) are solutions to the Dirichlet problems for the Heat equation

u_t (x,t) - ku_xx (x,t) = f(x,t), u(x,0) = Φ₁(x), u(0,t) = u(1,t) = g₁(t)

v_t (x,t) - kv_xx (x,t) = f(x,t), v(x,0) = Φ₂(x), v(0,t) = v(1,t) = g₂(t)

and if Φ₂(x) ≤ Φ₁(x) for 0 ≤ x ≤ 1, g₂(t) ≤ g₁(t), t > 0, then for all 0 < x < 1, t >0, we have u(x,t) ≥ v(x,t)

Homework Equations





The Attempt at a Solution



Following steps of example 2 and 3 of the following link, but I don't really understand what they are doing

http://www.math.mcgill.ca/jakobson/courses/ma264/pde-heat.pdf
 
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  • #2
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1. What is an inhomogeneous heat equation?

An inhomogeneous heat equation is a partial differential equation that describes the transfer of heat in a non-uniform medium. It takes into account both the heat source and the heat diffusion in the medium.

2. What is the difference between homogeneous and inhomogeneous heat equations?

A homogeneous heat equation describes the transfer of heat in a uniform medium without any external heat source. In contrast, an inhomogeneous heat equation takes into account both the heat source and the heat diffusion in a non-uniform medium.

3. How do you solve an inhomogeneous heat equation?

To solve an inhomogeneous heat equation, you can use various techniques such as separation of variables, the method of eigenfunctions, or the Green's function method. These methods involve manipulating and solving the partial differential equation to obtain a solution.

4. What are some applications of the inhomogeneous heat equation?

The inhomogeneous heat equation has many applications in physics and engineering, including heat conduction in non-uniform materials, temperature distribution in buildings, and diffusion of pollutants in the atmosphere. It is also used in fields such as thermodynamics, meteorology, and heat transfer.

5. Is the solution to an inhomogeneous heat equation unique?

No, the solution to an inhomogeneous heat equation is not unique. It depends on the initial conditions and boundary conditions of the problem. Different sets of conditions can result in different solutions to the same inhomogeneous heat equation.

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