Problem Statement: MULTIMEDIA UNIVERSITY An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation 1. = a²y_a² y 812 Ox² (1) Equation 1 can be written in the form of Equation 2 using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations, while superscripts represent time steps. y=By+Cy+ Dy + Ey 1+1 (2) a) Express B, C, D and E in terms of A t and A x. Note that the coefficient of y, is equal to one. b) The string is discretised into five nodes equally distanced, with A t = 0.2. Write down equations as a result of applying Equation 2 to the domain. Subscripts must be replaced with correct node numbers, while superscripts can be retained. Students are not required to incorporate the boundary and initial conditions yet. c) Assemble the three equations above in the form of matrix {y}+1 = [A] {y}'+ {y}+{BC}, where [A] is a square matrix, {BC} is a vector containing the boundary conditions, {y}+1, {y} and {y} are vectors containing nodal variables at times t+1, t, and t-1 respectively. d) Initially, the string is at rest at the profile {y2 y3 y4} = {0.4 0.7 0.3}. Use the answer in part c) to find {y2 y3 y4}¹. Show all the necessary calculations. e) Thus, further calculations to show one cycle (periodical process) are obtained by tabulating a table. f) Based on the problem statement given in part e), students must use MS Excel to plot a graph of displacement y against x. Please clearly show the theoretical derivation, relevant plots, and any numerical calculations. Students must show all the working solutions, including the tabulated table and plotted graph.