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Partial Differential Equation : Diffusion Equation and Explicit Series Solution

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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.

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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.

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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 provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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