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Heat Diffusion Equation and Standard Heat Equation

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A) Let the temperature u inside a solid sphere be a function only of radial distance r from the center and time t. Show that the equation for heat diffusion is now: {see attachment}. This is not an exercise in doing a polar coordinate transformation. First you should derive an integral form for the equation by integrating over an appropriate domain. Then from this obtain the differential equation.
b) Show that a transformation of the form {see attachment} for a suitable choice of m can be used to reduce this equation to the standard heat equation

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Heat Diffusion Equation and Standard Heat Equation are investigated. The solution is detailed and well presented.

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a) Let the temperature u inside a solid sphere be a function only of radial distance r from the center and time t. Show that the equation for heat diffusion is now

.

This is not an exercise in doing a polar coordinate transformation. First you should derive an integral form for the equation
by integrating over an appropriate domain. Then from this obtain the differential equation.

b) Show that a transformation of the form for a suitable choice of m can be used to reduce this equation to the
standard heat equation.

Solution:

a) We will get the heat equation in spherical coordinates by analyzing heat equilibrium on an appropriate domain which, in our case when temperature is depending only on radius (r), will by the volume between 2 spheres, whose radii are (r) and (r + dr):

The thermal equilibrium can be expressed as follows:
( ...

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