-The exam is open adopted textbook, open class notes (posted notes and solutions on the class' Canvas site only) and you may use Matlab's build-in help system, but only to look up Matlab syntax questions; - no collaboration is allowed; no help, including the tutoring center, may be sought to solve the problems; - exam questions may only be asked to the instructor via private Ed Discussion posts or during the instructor's office hours; - for non Matlab Grader problems, document all steps you took to solve the problem. This can be handwritten, but must be legible for credit. If the problem states 'By hand', do not use any script/function to actually solve the problem, however, you may use a non-programmable calculator or script/functions coded in this class to help in verifying the numerical results of individual steps; - on Gradescope associate/select your answer pages with the corresponding problem numbers. Failure to do so may result in no points given initially and will require a re-grade request to fix; - if the problem header contains MatlabGrader, you must submit your code to Matlab Grader using the Exam - Problem link available in the Canvas Module; - point scores received for MatlabGrader problems during the exam are provisional and may change even after the deadline in case the MatlabGrader assessment codes contain bugs; - point scores you may see on Canvas after submission of Canvas multiple choice answers are arbitrary and may not reflect your graded final score for such problems; if a problem requires submission of a printout, submit a pdf of your source code and results. Do not submit a picture/screen capture of your screen. No credit will be given in that case. See Module 0 on Canvas for a how-to. - only use functions developed in this class to solve the exam problems. If an equivalent build-in Matlab function is used for a function developed in this class, no credit will be given. - brute force searches are neither acceptable nor a numerical method in this class and therefore will give no credit. - to earn credit, your solution must contain all steps used to determine the solution. This includes all code (functions) not previously submitted in homework or exam problems to determine the solution. - answers given in code will not receive credit. - exam bonus problems give bonus points for the indicated exam and additional opportunities to demonstrate core outcome mastery. However, a module's exam score is capped at 100%; Problem 1 (5 points, CCOs #5, 10)/Gradescope The temperature of a fluid in a pipe of length L can be described by the following 1D non-dimensional equation ᎧᎢ Ət ƏT Əx a²T ==V + a(T). + Q(x,t) მx2 (1) with time t, temperature T, constant velocity u, temperature dependent thermal diffusivity a(T), and a source term describing additional heating/cooling along the pipe, Q(x, t). On the left end, the fluid entering the pipe has a prescribed temperature of T(x=0,t) = 5 - 4 sin 20πT 11 On the right end of the pipe, the fluid leaving the pipe has a prescribed temperature gradient, эт 4π (x = L,t) = -2-3 sin -t მე 5 At t = 0 the temperature of the fluid in the pipe is T(x, t = 0) = 5 The thermal diffusivity of the rod can be modeled by a third-degree polynomial a(T) = ao a₁T+α₂T² +αзT³ (2) (3) (4) (5) For a grid of M + 1 equally spaced points along the rod and Ax the spacing between points, do the following tasks: a) write the equation for the temperature at the first point Tr² = ... using Eq. (2); b) write the equation for the temperature at the last point TM+1 = ... as a function of the neighboring points using a third order accurate finite difference approximation to Eq. (3). Potential third-order finite difference formulas using the notation of Table 8.1 are: -2f(x-3)+9f(xi-2) - 18f(xi-1) +11f(xi)) f'(xi) f'(xi) = 6Ax +0(Ax³) (6) f(x-2)-6f(xi-1)+3f(xi) +2f(xi+1) + O(Ax³) (7) 6Ax f'(xi) = -2f(x-1)-3f(xi) + 6f(xi+1) − f(xi+2) + O(Ax³) (8) f'(xi) 6Ax -11f(xi)+18f(xi+1) - 9f(xi+2)+2f(xi+3) = + O(Ax³) (9) 6Ax 1 c) at all non-boundary mesh points i, using second-order central finite differences write the equation for the right-hand-side fr of Eq. (1), incorporating Eq. (5), leaving Q(x, t) as Q(xi, t") or Q. Use only the above variable names and not any specific parameter values given in later problems. d) write the equation for the initial temperature at every interior point T =... In all answers do not forget the superscript where appropriate, e.g., n, to indicate the discrete time level and the sub-script to indicate the appropriate location, e.g., i. No points will be awarded if these are missing or wrong. Required submission: handwritten or typeset answers using the correct variable names and all sub- and superscripts submitted to Gradescope Problem 2 (3 points)/MatlabGrader Develop a Matlab function myBCE 9 that applies the boundary conditions of problem 1 tasks a)-b) to the dependent variable T. The function input shall be t: scalar value of time t • T: column vector of temperature values T at all M + 1 mesh points at time t The function output shall be • T: column vector of temperature T at all M + 1 mesh points at time t with the boundary conditions applied The function shall use the following global variable • dx: grid spacing Ax Do not set values for global variables inside the function. Required submission: ☐ well commented function source code submitted to Matlab Grader using the Canvas link for Exam 9 - Problem 2