Modeling efflux time from a tank. Equation 11 in the attached document

The equation is first-order nonlinear ordinary differential equation. Says to combine Newton-Rhapson and Runge-Kutta methods to solve numerically.

Answer 1

[Sergey Kasyanov]

I think it can be helpful.

You have this function with A, B, C, D are known:

 fun = A * (dHdt)^2 + B * (dHdt)^1.75 + C * (dHdt) + D* H;
 A=number;
 B=number;
 C=number;
 D=number;

Then you define initial condition:

 dHdt0=dHdt0;
 H0=H0;

Then you combine solving of function fun(dHdt) and integrate ODE fun(dHdt,H):

 %create new function
 fun1 = A * (dHdt)^2 + B * (dHdt)^1.75 + C * (dHdt);

 %define interval to integrate
 T=number;

 %define step
 dt = 1e-3;

 dH = [dHdt0,zeros(1,T/dt)];
 H = [H0,zeros(1,T/dt)];
 i=2;

 %integrate
 while i<T/dt+1
     %solving of fun(dHdt)
     dH(i)=vpasolve(fun1 + H(i-1) * D,dHdt);
     %integrate with euler method
     H(i)=H(i-1) + dH(i) * dt;
     i=i+1;
 end

 t=[1:i]*dt;
 plot(t,H);