Please see the attached files and go through it also let us know in case you have any query.
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I am a mechanical engineer and I have worked with heat equation many times. Please allow me to help you. Will do it in 2 hours for you even if I am busy. Thanks and regards, chaks03
I spent 18 years on the Physics faculty at the Florida State University, the University of Cambridge and the University of Edinburgh. So I can deliver the answer to you very quickly as a pdf and/or LaTeX file.