How to solve this complicated equation

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AJMIT KUMAR on 19 Jan 2021
Commented: AJMIT KUMAR on 22 Jan 2021
Find the value of D ? (In RHS term, D is also present, see equations below)
(Note - Use N(x,t) instead of N(z,t))
Additional equations are:- (If required then use only) ;(I haven't used these in my calculation)
I have tried this code but getting error that Symbolic parameters not supported in nonpolynomial equations if I use "vpasolve" command; and getting Warning: Unable to find explicit solution, if I use "solve" command : -
( I have denoted "ri+" as y; "ri-" as z ; "gamma" as g ; "beta" as b ; "N_infinity" as u ; "n_infinity" as v ; "Sigma" as O ; "delta" as d ; "i" as s ; )
clear all; clc;
syms x d y z t s D h g b u v
K = (pi*pi*D)/(h^2);
y = 0.5*((K*s*s + g + b)+ sqrt(((K*s*s + g + b)^2) - 4*K*b*s*s));
z = 0.5*((K*s*s + g + b)- sqrt(((K*s*s + g + b)^2) - 4*K*b*s*s));
F = symsum(((-1)^(0.5*(s-1)))*(cos(0.5*pi*s*x/d))*((y*exp(z*t)-z*exp(y*t))/(s*(y-z))),s,1,Inf);
N = (g*v/b)*((1 - 4/pi)*F);
F1 = diff(N,t);
G = symsum(((-1)^(0.5*(s-1)))*(cos(0.5*pi*s*x/d))*(y*z)*((exp(z*t)-exp(y*t))/(s*(y-z))),s,1,Inf);
n = v*(((1 - 4/pi)*F) + ((4/(pi*b))*G));
G1 = diff(n,t);
G2 = diff(n,x,2);
eqn = [D*(G2) == (G1)+ (F1)*(F1)];
eqn = rewrite(eqn,'log'); % Additional step
[R] = vpasolve(eqn,D);
  1 Comment
AJMIT KUMAR on 19 Jan 2021
If you need other information, feel free to ask

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Answers (1)

amin on 20 Jan 2021
Hi Ajmit,
It is not an easy one!
You have made few mistakes:
1) mistake in parantheses in 'n' and 'N'. They should be:
N = (g*v/b)*(1 - 4*F/pi);
n = v*(1 - 4*F/pi + 4*G/(pi*b));
2) You have missed the negative sign in all exp functions.
3) In all series, you have to consider only odd values of i (i=1,3,5,...,inf). So, in the code, you sould replace all 's' by '2*s-1'.
Correcting all these issues, 'solve' function could not solve the eqn and it returns the same warning that you received, which means that Matlab cannot find a closed form explicit solution.
I recommend you to double check for more possible mistakes and then you may try 'ode'.
Hope it helps!
good luck
  1 Comment
AJMIT KUMAR on 22 Jan 2021
@amin Thankyou for pointing out correction and I found the numerical value of many terms here, finally equation solved

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