Let’s consider Uc to be transformed form of h(x,t) by applying Fourier transform
Then solution of Eq 1 by integrating factor is as in Eq 2
And by applying on Eq 2 inverse fourier transform & some simplification gives us final solution as Eq 3
But what if f(t) in Eq 1 is equal to Eq 4
Putting value of f(t) from Eq 4 in Eq 1 gives us Eq 5
Similarly by integrating factor solution of Eq 5 comes out to be as in Eq 6
Applying fourier inverse on Eq 6 gives final solution as Eq 7
The problem is in final equation (Eq 7)I have h(x,tow) on right hand side which is actually same what I want to find out and is on left side of eq 7. How can I modify solution of Eq 5 that h(x,tow) in Eq 6 does not remain on right hand side?
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Thankyou Luan . Yes Eq 2 is obtained by applying the condition Uc(0)=0. I jut did not mention it. Eq 3 is equation number 2.15.18 on page 106, in book 'Integral Transforms and Their Applications' by Debnath and Bhatta. I could not follow the steps exactly how he arrived on Eq 3 or eq 2.15.18 in the book. A similar problem is solved by Dong et al 2012 but by fourier sine transform instead of cosine transform and he arrived at a similar equation. You can safely assume Eq 3 to be correct otherwise I am attaching both the book and paper of Dong. In book, while solving problem they took boundary condition dh/dt= f(t) while in my case I want to take dh/dt=1/a*g(t)-h(x,t).
Note: For shifting from my equations to book consider dh/dt as Ux and D as k (small k)
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