# Partial Differential Equation : Diffusion Equation and Explicit Series Solution

Consider the diffusion equation

ut = ku.xx for 0 < < pi and t > 0 with the boundary conditions

ux(0, t) = 0 and u(pi, t) = 0

and the initial condition

u(x,0) = 1.

(a) Find the separated solutions satisfying the differential equation and boundary conditions.

(b) Use these solutions to write an explicit series solution to the differential equation satisfying the boundary conditions and the initial condition.

https://brainmass.com/math/calculus-and-analysis/partial-differential-equation-diffusion-equation-explicit-series-39038

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Solution.

(a) Since this is a diffusion equation, we have k>0. We use separated variable method. Assume that its solution . Then

and

By , we have

=

So,

Letting ,where then we have

................(1)

and

...

#### Solution Summary

A Diffusion Equation and Explicit Series Solution are investigated. The solution is detailed and well presented. The response was given a rating of "5" by the student who originally posted the question.