Question

# 2. Consider the equation for the domain (a) Obtain an analytical solution to the above problem, i.e. Yexact (2). Box your answer. (b) Goal is to also solve the differential equation using a Forward Euler scheme (i.e. obtain YFE (2₂) at some discrete , locations). i. Using uniform grid spacing Ar between discrete points, first obtain a general finite- difference approximation (i.e. the equation is written for a general point z;, where j can be 0,1,2,3...etc.) using the Forward Euler scheme. Box your answer. ii. Rewriting the above equation as Conduct linear stability analysis to identify the maximum possible step size for a stable solution over the domain 0≤x≤ 10. Note that you will have to compare the allowable step sizes at different a locations, and choose the most restrictive one over - the entire domain i.e. use different values of z between 0 and 10, and see which one gives the most restrictive step size. iii. Write a Matlab program to solve the differential equation over the region 0 < r ≤ 10. Use three different step sizes Az - 0.1, 0.5, 1.0 and compare your numerical solution YFE(T;) to the exact solution by plotting them on the same graph [i.e. For each time-step, plot yexact versus z and ypg versus z on one plot. You will thus have three different figures for each Ar. Clearly label your plots, clearly identify exact and numerical solution on each plot. Use axis range of [0, 8] for the y axis and [0, 10] for the x-axis. Note that the exact solution should be plotted at finer intervals to obtain a smooth curve. Provide your matlab code and graphs.  Fig: 1  Fig: 2