I’m making an attempt to unravel this train and do not know how you can begin or what to do. Can somebody assist me or clarify me how you can resolve this?

Take into account the next one-dimensional boundary worth drawback

$ start{align}-(omega u_x)_x = f , textual content{for} , -1 < x < 1 u(-1) = u(1) = Zero finish{align} ,(1)$

with $omega(x) := sqrt{1-x^2}$ and $f(x):= x$ for $-1<x<1$.

Moreover, the useful area $H_{0}^{1}(-1,1,omega):= { vvert v(-1)=v(1)=0, vertvert v vert vert_{L^2(-1,1,omega)} < infty, vertvert v_x vert vert_{L^2(-1,1,omega)} < infty }$ is given with $vertvert v vert vert_{L^2(-1,1,omega)}^{2} := int_{-1}^{1} vert v(x) vert^2 omega(x) dx$

i) Decide a linear and restricted useful $l : H_{0}^{1}(-1,1,omega) rightarrow mathbb{R}$ and

present that the issue (1) could be transferred to the next normal variation equation $a(u,v) = l(v)$ for all $v in H_{0}^{1}(-1,1,omega)$

ii) Use the Lemma from Lax-Milgram to point out that the issue (1) has precisely one resolution in room $H_{0}^{1}(-1,1,omega)$.

iii) For $j = 1,dots,N$ the next strategy capabilities are given $ phi^{(j)}(x) := T_{j+1}(x) – T_{j-1}(x), xin[-1,1]$

${T}_{jin mathbb{N}_0}$ are the Chebyshev polynomials of the primary sort, that are outlined recursively for $x in [-1,1]$ as

$ T_{j+1} (x) = 2xTj_ (x) – T_{j-1} (x) $ for $j>1$ with $T_0(x) = 1$ and $T_1(x) = x$.

Present that $phi^{(j)} (-1) = phi^{(j)} (1) = 0$ for $ j = 1, dots ,N$.

Notice: The primary kind of Chebyshev polynomials fulfill the equation $ T_j (cos (x)) = cos (jx)$ for $ xin mathbb{R}$ and $ j in mathbb{N}_0 $

iv)Calculate for the above drawback (1) and $N = 2$ the coefficient matrix $A in mathbb{R}^{2times2}$ in addition to the suitable facet $b in mathbb{R}^2$, which end result from the equation of variation from job i) and the strategy capabilities from job iii). Additionally resolve the equation system $Ac = b$ and specify the discrete resolution $u= sum_{j=1}^{2} c_j phi^{(j)} $

Notice: For the derivation of the Chebyshev polynomials of the primary sort the next applies $(T_j)x = jU_{j-1}$ for $ j geq 1$. ${U_j}_{j in mathbb{N}_0}$ are the second kind of Chebyshev polynomials, that are outlined recursively for $x in [-1,1]$ as $U_{j+1} (x) = 2xU_j (x) – U_{j-1} (x)$ for $ j > 1$ with $U_0(x)=1$ and $U_1(x) =2x$.The Chebyshev polynomials of the primary kind and second kind have the orthogonality properties

$int_{-1}^{1} T_i(x) T_j(x) frac{1}{omega(x)} dx =left{%

start{array}{ll}

pi, & hbox{i=j=0,}

frac{pi}{2}, & hbox{i=j $neq$ 0,}

0, & hbox{i $neq$j}

finish{array}%

proper.$

and

$int_{-1}^{1} U_i(x) U_j(x) omega(x) dx =left{%

start{array}{ll}

frac{pi}{2}, & hbox{i=j}

0, & hbox{i $neq$j}

finish{array}%

proper.$