Solve this system of equations explicitly for $f_d$ and $\alpha$

Question

In this system of equations $\alpha$ and $f_d$ are unknown. $\alpha$ and $C_{13}$ are complex numbers; the other parameters are all real numbers. I want to solve this system explicitly for $f_d$ (real) and $\alpha$ (complex) and use the results in my programming with MATLAB.

$$ \begin{cases} f_d(1-\alpha)=C_{33}-C_{13}-\frac{2}{5}f_v-\frac{1}{2}f_c\\ f_d(1-|\alpha|^2)=C_{33}-C_{11}-\frac{1}{3}f_v \end{cases} $$

Up to now, I have taken these steps:

$$\frac{1-\alpha}{1-|\alpha|^2}=\frac{C_{33}-C_{13}-\frac{2}{5}f_v-\frac{1}{2}f_c}{C_{33}-C_{11}-\frac{1}{3}f_v}$$

By separating the real and imaginary parts, we have:

$$\frac{(1-\Re(\alpha))+i(-\Im(\alpha))}{1-\Re(\alpha)^2-\Im(\alpha)^2}= \frac{(C_{33}-\Re(C_{13})-\frac{2}{5}f_v-\frac{1}{2}f_c)+i(-\Im(C_{13}))}{C_{33}-C_{11}-\frac{1}{3}f_v}$$

and then we have:

$$\frac{1-\Re(\alpha)}{1-\Re(\alpha)^2-\Im(\alpha)^2}=\frac{C_{33}-\Re(C_{13})-\frac{2}{5}f_v-\frac{1}{2}f_c}{C_{33}-C_{11}-\frac{1}{3}f_v}$$

$$\frac{\Im(\alpha)}{1-\Re(\alpha)^2-\Im(\alpha)^2}=\frac{\Im(C_{13})}{C_{33}-C_{11}-\frac{1}{3}f_v}$$

How do I find explicit expressions for $\Re(\alpha)$ and $\Im(\alpha)$ based on these equations?

Answer 1

Since you're using Matlab, you might want to try solving this via symbolic math:

syms alp C13
syms fd fv fc C11 C33 real
eq1 = fd*(1-alp)==C33-C13-2*fv/5-fc/2;
eq2 = fd*(1-(real(alp)^2+imag(alp)^2))==C33-C11-fv/3;
s = solve(eq1,eq2,fd,alp);
fd_eq = simplify(s.fd)
alp_eq = simplify(s.alp)

which in R2015b returns

fd_eq =

-(120*fc*fv - 240*C33*fv - 300*C33*fc - 600*C33*real(C13) + 300*fc*real(C13) + 240*fv*real(C13) + 300*C33^2 + 75*fc^2 + 48*fv^2 + 300*imag(C13)^2 + 300*real(C13)^2)/(20*(15*fc - 15*C33 - 15*C11 + 7*fv + 30*real(C13)))

and

alp_eq =

(150*C11*fc - 300*C13*fc + 150*C33*fc + 120*C11*fv - 140*C13*fv + 20*C33*fv - 70*fc*fv - 600*C13*real(C13) - 75*fc^2 - 8*fv^2 + 300*C11*C13 - 300*C11*C33 + 300*C13*C33 + 300*imag(C13)^2 + 300*real(C13)^2)/(120*fc*fv - 240*C33*fv - 300*C33*fc - 600*C33*real(C13) + 300*fc*real(C13) + 240*fv*real(C13) + 300*C33^2 + 75*fc^2 + 48*fv^2 + 300*imag(C13)^2 + 300*real(C13)^2)

You should check that is indeed the solution you're looking for (it's possible that there are others that solve didn't find). Also note that I had to explicitly expand abs(alp)^2 to real(alp)^2+imag(alp)^2 in order get a solution.