we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization
and apply this investigated regulaization strategies to Symm's integral equation $$ (K\psi)(t)=-\dfrac{1}{2\pi}\int_{0}^{2\pi}\psi(s)\ln\left(4\sin^2{\dfrac{t-s}{2}}\right)dx+\int_{0}^{2\pi}\psi(s)k(t,s)ds$$ where $\gamma(s)=(\cos{s},2\sin{s}),0\le s\le 2\pi$
for $0\le t\le 2\pi$ with the analytic function $$k(t,s)=-\dfrac{1}{2\pi}\ln{\dfrac{|\gamma(t)-\gamma(s)|^2}{4\sin^2{\dfrac{t-s}{2}}}},t\neq s$$ $$k(t,t)=-\dfrac{1}{\pi}\ln{|\gamma'(t)|},0\le t\le 2\pi$$
we use the trapezoidal rule gor periodic functions,Let $t_{j}=\dfrac{j\pi}{n},j=0,1,\cdots,2n-1$
the smooth part is approximated by $$\int_{0}^{2\pi}k(t,s)\psi(s)ds\approx \dfrac{\pi}{n}\displaystyle\sum_{j=0}^{2n-1}k(t,t_{j})\psi(t_{j}),0\le t\le 2\pi$$
and with simple computation we have
$$-\dfrac{1}{2\pi}\int_{0}^{2\pi}\psi(s)\ln{\left(4\sin^2{\dfrac{t-s}{2}}\right)}ds\approx \displaystyle\sum_{i=0}^{2n-1}\psi(t_{j})R_{j}(t)$$
where $$R_{j}(t)=\dfrac{1}{n}\left(\dfrac{1}{2n}\cos{n(t-t_{j})}+\displaystyle\sum_{m=1}^{n-1}\dfrac{1}{m}\cos{m(t-t_{j})}\right)$$ for $j=0,1,\cdots,2n-1$, therefore, $$(K_{n}\psi)(t):=\displaystyle\sum_{j=0}^{2n-1}\psi(t_{j})\left[R_{j}(t)+\dfrac{\pi}{n}k(t,k_{j})\right]$$
and let $t=t_{k}$, we have $$(K_{n}\psi)(t_{k})=\displaystyle\sum_{j=0}^{2n-1}A_{kj}\psi(t_{j})$$ with the symmetrix matrix $$A_{kj}:=R_{|k-j|}+\dfrac{\pi}{n}k(t_{k},t_{j}),k,j=0,1,\cdots,2n-1$$ where $$R_{l}=\dfrac{1}{n}\left(\dfrac{(-1)^l}{2n}+\displaystyle\sum_{m=1}^{n-1}\dfrac{1}{m}\cos{\dfrac{ml\pi}{n}}\right),l=0,1,\cdots,2n-1$$
for the numerical example,we take $\psi(s)=e^{3\sin{s}},0\le s\le 2\pi$ and take $n=60$
we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization
my qustion :how can Get the following four fig 3.1,or someone can Write the MATLAB source here
?Thank you every much!
