I want to solve the following problem in Mathematica:
Assume that $\lambda_2=-\frac{1}{2}+ r+\frac{1}{1+r}- r^2\log\frac{1+r}{r}$, $v\in[0,1]$ and $\theta\in[0,1]$. For pre-specified values of $k,r>0$, solve for $\{s,\lambda_1\}$ through the following two equations $$\int_{\left\{\stackrel{2\theta v s+\lambda_2(1-\theta)r\geq\lambda_1}{\theta(v+r)\geq r}\right\}}(1-\theta)r\,dvd\theta+ \int_{\left\{\stackrel{2\theta v s+\lambda_2v\geq\lambda_1}{\theta(v+r)\geq r}\right\}}v\theta\,dvd\theta=s$$ $$\int_{\left\{\stackrel{2\theta v s+\lambda_2(1-\theta)r\geq\lambda_1}{\theta(v+r)\geq r}\right\}}\,dvd\theta+ \int_{\left\{\stackrel{2\theta v s+\lambda_2v\geq\lambda_1}{\theta(v+r)\geq r}\right\}}\,dvd\theta=k$$
For the first step, I tried to calculate the integrals using Integrate
and ImplicitRegion
, but got stuck here because Mathematica refuse to evaluate the expressions. Here are my codes:
λ2 = -(1/2) + r + 1/(1 + r) - r^2 Log[1 + 1/r];
k = 0.5;
r = 0.5;
R1 = ImplicitRegion[θ (v + r) > r && 2 θ v s + (1 - θ) r > λ1, {{θ, 0, 1}, {v, 0, 1}}];
R2 = ImplicitRegion[θ (v + r) < r && 2 θ v s + λ2 v > λ1, {{θ, 0, 1}, {v, 0, 1}}];
Integrate[(1 - θ) r, {θ, v} ∈ R1] + Integrate[θ v, {θ, v} ∈ R2]
Integrate[1, {θ, v} ∈ R1] + Integrate[1, {θ, v} ∈ R2]
Ultimately, I would like to plug back the solved $\{s,\lambda_1\}$, and plot the regions of R1
and R2
. But how can I fix the codes to solve the equations?
s
andλ1
? $\endgroup$Integrate
can be performed, how ares
andλ1
to be determined from the result? $\endgroup$