Question

please answer all parts

Please answer all parts, thank you Problem 3: Linear system for linear BVPs& Consider the linear BVP y(0) = -1 y(1)1 You will define a set of linear equations for yi, i-o, (y.* y(m), i = 0, ,n) and the Net of n(xk, is , n, where yi İs the approximate solution on node i with x-ih,i-0,n and h n is a fixed positive integer. (a) Write the forward difference approximation for y' on the nodes. Use the approximations yi's Include the relevant set of i values for which the approximation holds. (bWrite the centered difference approximation for y" on the nodes. Use the approximations y's Include the relevant set of i values for which the approximation holds. (c)Write the linear equations you will need to solve for the yi's. Inelude the relevant set of i values. (d Based on the previous part, write a linear system in matrix form for the approximations y's, considering the boundary conditions as well.

Answer 1

h orward difference approximation or y' is h-i n iven as Xi Cleorly, he appoximation holds For the inner nodes ercluding the end nodes, i.e., For i = 1, 2, 3, , n-1 S0, Hi8i, where iei, 2,... , n-1) The centerd difference oppauximation for y" is given (b) as 2 Again, the appnximotion holds for all the inner hodes ie nodes With index,-1, 2, 3, Also, since the rodes are uniformly spaced, therefore, Thus, the approximah oms fo y' and y" can be writfen as Ђе linear BVP on any inner node can be written as (c) Substitule " and i From parts to) and (b), to obtain,

4- 4 h#0, therefore since, where, i-1, 2, ,(n-1), and y, -1, yn1 (d) Use ean.) writlen For the i th node to urite systmor linear equahions for nodes with index i-i, 2, .. (n-1) For i-i therefore, y,--1, Since , For i- 3 For 1n-2 For n-1 since, yn -1, therefore, Rewrite the syskm of equa homs From (2)-(6> in fhe matrix fom

A , where 4 where h -13h44) and, the coeFFicient matrix, A is given as 4- 4 4 -3h-8 3R4 4- Note this is a triadiagonal matrix of order (n-1) th other entries being