Consider the following Dirichlet problem: ${}_{}+{}_{yy}=0$ subject to $(x,y)=\frac{1}{sinh}sinx*sinh(1-y)x=y=0\mathrm{.}2{}_{i,j}^0=0ij|{}_{i,j}^{n+1}-{}_{i,j}^n|0\mathrm{.}001ij=sinx$;$0.$

Using the method $of$ separation $of$ variables, $we$ had $(x,y)=\frac{1}{sinh}sinx*sinh(1-y)$

Let $x=y=0\mathrm{.}2$ and the initial estimate ${}_{i,j}^0=0$ for all interior i and $j.$ Solve this problem $by$ hand for $2$ iterations using Gauss$-$Seidel method. $(Roundedto$ four digits$)$

Solve the above problem $by$ computer using Jacobi and Gauss$-$Seidel methods until $|{}_{i,j}^{n+1}-{}_{i,j}^n|0\mathrm{.}001$ for all interior i and $j.$ Compute the $absolute$ and relative errors $of$ the last iteration and discuss the results.