How to solve this complicated equation

Find the value of D ? (In RHS term, D is also present, see equations below)

where

(Note - Use N(x,t) instead of N(z,t))

where

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);

[AJMIT KUMAR]

If you need other information, feel free to ask

Answer 1

[amin]

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

[AJMIT KUMAR]

@amin Thankyou for pointing out correction and I found the numerical value of many terms here, finally equation solved