Preface

This section concerns other heat transfer problems.

Other heat transfer problems

  In this section, we show how the separation of variables method can be extended for inhomogeneous equations and nonuniform boundary conditions.

 

Diffusion equation with nonhomogeneous boundary conditions

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Inhomogeneous heat equation with homogeneous boundary conditions

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Let u(x, t) denote the solution of the heat equation subject to the initial condition:

\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} , \qquad u(x,0) = f(x) , \]

where f(x) is a give function on the interval \( (-\infty , \infty ) . \)

 

Example: Inhomogeneous heat equation

Example: Consider a one-dimensional heat equation

\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \left( t^2 -4t +2 \right) e^{-t} \sin x , \qquad 0 < x < \frac{\pi}{2} , \]

on finite interval x ∈ (0, π/2) subject to boundary conditions

\[ u(0,t) = 0, \qquad \left. \frac{\partial u}{\partial x} \right\vert_{x=\pi /2} , \]

and homogeneous initial condition

\[ u(x,0) = 0 . \]

A unique solution to the above initial boundary value problem is

\[ u(x,t) = \left( \frac{1}{3}\, t^3 - 2\,t^2 + 2t \right) e^{-t} \sin x . \]

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References

2. Gonzalez-Velasco, E., The existence of a steady state solution for a type of parabolic boundary value problem, International Journal of Mathematical Education in Science and Technology, 1988, Vol. 19, No. 3, pp. 413--419. https://doi.org/10.1080/0020739880190307

 

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