how to get the numerical solution of Non linear equation for unknown variable

Hello all, I am a beginner in Matlab. I want to solve the above equation numerically to calculate the value of phi0. below is the list of parameters involved. I tried to solve using fsolve function but could not get the solution, because I could not solve the left hand side term with integral. please help me finding the solution.I Will appericiate the help from bottom of my heart. Thank you.

Nd=10^17;

Ld = 1.6923e-05;

T=500;

K=1.3807*10^-23;

r=40*10^-9;

Nt=10^13;

Et-Ef= 1.21*1.6*10^-19

R=40*10^-9;

V= 4*pi*r^3

[madhan ravi]

Upper limit of left hand side?

[Anil Kumar]

Sir, The integration on left hand side is the volume integral of a spherical grain. from V=0 to V=4*pi*r^3 where r= 40*10^-9.

[Torsten]

How is the integral rewritten as an integration over r ? Furthermore, the right-hand side cannot depend on "r" since this is the integration variable and will cancel out during integration.

Answer 1

[John D'Errico]

Confusing. That usually means the person is also confused.

We are shown an integral on the left hand side. It is an integral over the variable V. V appears in the integrand only as dV. I see Nd (known), also R (known) Ld (known.) Only phi0 is missing and unknown in there. But IF phi0 WERE known, then it would be a constant, NOT a function of V, the variable of integration.

So the left integral is trivial to compute, even on paper. The integral of a constant over fixed, known limits is simple. If C is the value of the constant, then the integral of the left hand side is just

4/3*pi*R^3*C

Yes, C is a function of phi0. Why do I care? Once phi0 is known, then it is a constant, so not impacting the value of the integral, and it is clearly not a function of V.

How about the right hand side? Again, all known. The right hand side is a number. Computatable.

In the end, you can do the algebra. You will find an expression of the form

C(phi0) = D

But C(phi0) is trivially solved using a little algebra and the log function. Again, paper and pencil will suffice, were I not too lazy, but this is your problem to solve, not mine, and MATLAB is not even required. You don't need fsolve. You don't even need a computer to solve it. Lets see

4/3*pi*R^3*Nd*(1-exp(-phi0*sinh(r/Ld)/(r/Ld))) = 4*pi*R^2*(Nt/stuff)

I left part of that as just "stuff". But you get the hint. The 4 cancels. The pi cancels. We get

1 - exp(-phi0*sinh(r/Ld)/(r/Ld)) = 3/R*Nt/(Nd*stuff)

Move some more stuff around, take the log. Then solve for phi0. WTP? Using fsolve here is like using a Mack truck to carry a pea to Boston.