Write a single python programme to solve the given poisson equation:$-$

d$^2$T$/$dx$^2+$ d$^2$t$/$dx$^2=2$ Sin$(10($x$-0\mathrm{.}5))+40($x$-0\mathrm{.}5)$cosh$(10($x$-0\mathrm{.}5))+100(($x$-0\mathrm{.}5)^2)$sinh$(10($x$-0\mathrm{.}5))+2$ Sin$(10($y$-0\mathrm{.}5))+$sinh$(10($y$-0\mathrm{.}5))+4($x$^2+$y$^2).$e$^2$xy

with the following methods

$1.$ Gauss Jacobi

$2.$ Gauss Seidel

$3.$ Successive over relaxation

$4.$ Red Black Gauss Seidel method

$5.$ Alternating Direction Implicit

in a square domain of unit length using Gauss Jacobi method on a $41*41$ and $81*81$ grids for convergence set to tolerance $10^-6$ Compare the residual plot and comment on the rate of convergence and compare your converged solution and exact solution. Boundary conditions are

T$(0,$y$)=0\mathrm{.}25*$sinh$(-5)+($y$-0\mathrm{.}5)^2.$sinh$(10($y$-0\mathrm{.}5))+1$

T$(1,$y$)=0\mathrm{.}25*$sinh$(5)+($y$-0\mathrm{.}5)^2.$sinh$(10($y$-0\mathrm{.}5))+$ e$^2$y$.$

T$($x$,0)=0\mathrm{.}25*$sinh$(-5)+($x$-0\mathrm{.}5)^2.$sinh$(10($x$-0\mathrm{.}5))+1.$

T$($x$,1)=0\mathrm{.}25*$sinh$(5)+($x$-0\mathrm{.}5)^2.$sinh$(10($x$-0\mathrm{.}5))+$ e$^2$x$(\}$