Series or Physics?

A bar of length \(a\) is at zero temperature. At \(t=0\), the end \(x=a\) is raised to a temperature \(u_0\) and the end \(x=0\) is insulated. If the temperature at any point \(x\) of the bar at any time \(t>0\), assuming that the surface of the bar is insulated, can be expressed as:

\[ \displaystyle u(x,t)=u_0+\frac{Au_0}{\pi^B}\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n-1)}e^{-\frac{(2n-1)^C\pi^Dc^2t}{Fa^G}}\cos\left( \frac{(2n-1)\pi x^H}{Ia^J} \right)\]

Evaluate \(A+B+C+D+F+G+H+I+J\)

Details and Assumptions:

Here, \(c\) is arbitrary constant.


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