I am currently trying to solve the following EDP, which captures the temperature change (T) as a function of time (t) and displacement in the tank (x):

$ left ( rho C_p right) _m frac {{ partial T} _m} { partial t} + G.C_ {p, f} frac {{ partial T} _m} { partial x} = frac { partial} { xx partial} left (k_m frac {{partial T} _m} { xx partial} right) $

What can be better written as:

$ A frac { partial T} { partial t} + B frac { partial T} { partial x} = C frac { partial ^ 2T} {{ partial x} ^ 2} $

With the following initial condition:

$ T (x, 0) = 20 ° C $

And the following boundary conditions:

$ frac { T partial} { partial x} (0, t) = 0 $

$ frac { T partial} { partial x} (L, t) = 0 $

Since then, I've got literature search results where the loading process of the compacted bed system was as follows:

t = 1 hour

Polynomial solution: $ y = 13728x ^ 6-50447x ^ 5 + 67135x ^ 4-37308x ^ 3 + 6810x ^ 2-445.52x + $ 554.58

t = 1.5 hours

Polynomial solution: $ y = -2710.5x ^ 6 + 4928.9x ^ 5 + 2182.7x ^ 4-7837.1x ^ 3 + 3379.5x ^ 2-480.39x + $ 566.51

t = 2 hours

Polynomial solution: $ y = -5493.2x ^ 6 + 21095x ^ 5-29059x ^ 4 + 17302x ^ 3-4758.7x ^ 2 + 535.14x + 533.75 $

t = 2.5 hours

Polynomial solution: $ y = 1090.7x ^ 6-2259,7x ^ 5 + 742.42x ^ 4 + 496.34x ^ 3-319.33x ^ 2 + 50.762x + 548.15 $

t = 3 hours

Polynomial solution: $ y = 872.72x ^ 6-2893.5x ^ 5 + 3281x ^ 4-1720.8x ^ 3 + 435.16x ^ 2-46.906x + $ 551.42

I hoped that it might be possible to combine these polynomial equations and to approximate them using a sinusoidal sum of waves, maybe by Fourier transform, so that I could build a mathematical model to predict this charging time performance ?? I became completely stuck with this so that any help would be received with gratitude.