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 эт Ət эт = =U + a(T). მე a²T მ2 +Q(x,t) (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, ᎧᎢ 4π მე: (x = L,t) = -2 - 3 sin 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 +α3T³. (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 T = ... using Eq. (2); b) write the equation for the temperature at the last point T+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) + O(Ax³) 6Ax f(xi-2) - 6f(xi-1)+3f(xi) +2f(xi+1) + O(Ax³) 6Ax -2f(x-1)-3f(xi) +6f(xi+1) − f(xi+2) f'(xi) = + O(Ax³) 6Ax f'(xi) = −11f(xi) + 18f(xi+1) − 9f (xi+2) + 2f (xi+3) + O(Ax³) 6Ax 1 (6) (7) (8) (9) 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 Problem 3 (4 points)/MatlabGrader Develop a Matlab function myRHSE9 that calculates the right hand side of Eq. (1) at every interior mesh point according to problem 1 task c). 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 ⚫ f: column vector for all M + 1 mesh points containing the right-hand side values of Eq. (1) in the interior of the mesh and zero on the boundaries. The function shall use the following global variables • dx: grid spacing Ax • x: column vector of grid point coordinates ⚫u: scalar value of velocity u • a: column vector containing ao to a3 (see Eq. (5)) ⚫ Q: anonymous function for Q(x, t) Do not set values for global variables inside the function. Apply boundary conditions by calling my BCE 9 before calculating f. Required submission: ☐ well commented function source code submitted to Matlab Grader using the Canvas link for Exam 9 - Problem 3 Problem 4 (4 points, CCOs #2 & #234)/MatlabGrader To determine the model constants ao to aз in Eq. (5), a series of measurements of a as a function of T is taken. The measured data is available on Canvas in the MAT file data.mat that must be copied into your working directory and loaded using the Matlab command load("data.mat"); This loads the Matlab column vector variables Texp and alphaExp containing 800 measured data pairs (T, a). Calculate ao to a3 in Eq. (5) that best fit the measured data and store the results in the global column vector variable a. If you are unable to complete this task, use a = [0.2; 0.2; -0.01; -0.0002]; in the following problems. Note: the MatlabGrader assessments for this problem use random data contained in the .mat file. Functions developed in this class required to solve this problem are provided on Matlab Grader. These can be called in your script, but their source code should not be included in your problem submission. Required submission: ☐ well commented script source code submitted to Matlab Grader using the Canvas link for Exam 9 - Problem 4 values of ao to a4 stored in the global column vector a 2 Problem 5 (20 points, CCOs #5, 8, 10, 11, 12)/ MatlabGrader For 3 5π Q(x,t) = 2+ cos(3πt) cos( " (10) L = 4, M = 80, u = 4.8, and At = 16Ax², calculate and graph the solution of Eq. (1), T(x, t), at t = 0,1, 2, 3, 4, and 5 in one figure, and at t = 6, 7, 8, 9, and 10 in a second figure. Do not use subfigures. In a third figure, plot T(x = L, t) as a function of t. Recall that all global variables used in functions, e.g., see Problem 3, must be declared global and set to their respective values in the script. You must use the third-order Runge-Kutta method from Exam 8 to solve this problem. Notes for the MatlabGrader script submission: • Comment out clear or clear all in your script when submitting to MatlabGrader. • Generate and store the figure handles examFigl, examFig2, and examFig3 for the requested 3 figures before doing any plot commands, using examFigl = figure (1); etc. • Properly label all axis using the appropriate variable names, e.g., (x, T), respective (t,T(x = L, t)). • Inlcude legends in the first 2 figures listing the times t of each line. • Do not include the source code for any functions in your script submission. Necessary functions are provided on Matlab- Grader. • Start your script code from scratch and do not try to modify the recitation codes. ⚫ The script submitted to this problem must contain the script portion from problem 4 calculating the global variable a or setting the alternate values given there. Required submission: Problem 5: PDE solution see pdf. Important ■ Make sure you define ALL global variables in your script, i.e., declare them as global and set their value, including the anonymous function Q! ■ If you time out, make sure you are calculating the solution only once! ■ Make sure to use the correct initial condition from the updated assignment pdf: T(x, t = 0) = 5 ■ The notes for problem 5 listed the y-axis label as T(x=0,t), but the problem requests plotting T(x=L,t) in figure 3. Thus please use T(x=L,t) as the y-axis label. ■ Recall: boundary conditions take precedent over initial conditions. Thus you have to apply boundary conditions after setting the initial conditions. My Solutions > ■ The Matlab Grader assessments for this problem are very "absolutist": to pass an assessment, the entire result must be correct, i.e., even if a single result point/value is incorrect, you likely will be unable to pass an assessment. The final grading for this problem likely will include "partial credit" for some assessments to account for "minor" mistakes. It's impossible to account for these beforehand, since there are way too many ways to make such mistakes. I will analyze failing codes when doing the final grading for this problem to account for such "minor" mistakes. (That's the long explanation for the statement in bold under Required submission for this problem). Script 1 Assessment: clear is not used no function definitions in script variables examFig1, examFig2, examFig3 exist solution figures 1 & 2 axis labels solution figure 3 axis labels legend used solutions in figure 1 correct solutions in figure 2 correct solution in figure 3 correct figure 3 from t=0 to t=10 with initial boundary conditions? 眉 Save CReset MATLAB Documentation Open Problem in MATLAB Online Run Script Submit ?