1. A. Start with the Matlab listing for the 6 differential equation CV model. Run the model for normal Emaxl (3) and plot left ventricular volume versus time in the steady state. Estimate ejection fraction (EF) and submit plot. B. Change Emaxl to half normal (1.5) and estimate EF and submit plot. 2) Add Matlab code to include baroreceptor control of systemic resistance R. G AR R 100= P0? ? CV model Pa Ro Pa G=-.01 R0 = 1.3 ? = 6 sec 100-Pa =100-P(5)=input to differential equation $v_p(7) = ((100-P(5))*G-v(7))/ au$ R=1.3+v(7) Initial condition for v(7)=0; Repeat plots and ejection fraction measurements for normal Emaxl, Emaxl/2, and add plot of v(:,7) versus time. 3) Add one more differential equation similar to v(7) above but with input Plv (P(4)), G=1, and ?=.01 sec. This allows plotting Plv versus time after slight low pass filtering. Plot this Plv (v(:,8))versus time for the same conditions as question 2 above. Include a listing of your final Matlab code.
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This project introduces approximations to Ordinary Differential Equations using numerical methods. You will program three numerical solvers: Euler's method, Improved Euler's method, and 4th order Runge-Kutta (RK4). You are required to write your own numerical methods in either MATLAB or MS Excel. You are not allowed to use numerical solvers written by anyone else. Problem 1: Consider the following Initial Value Problem (IVP) where y is the dependent variable and t is the independent variable: y' = sin(t) * (1 - y) with y(0) = y0 and t ≥ 0 Note: the analytic solution for this IVP is: y(t) = 1 + (y0 - 1)e^cos(t) - 1 Part 1A: Approximate the solution to the IVP using Euler's method with the following conditions: Initial condition y0 = -1/2; time step h = 1/16; and time interval t ∈ [0,20] + Derive the recursive formula for Euler's method applied to this IVP + Plot the Euler's method approximation + Plot the absolute error between the approximation and the exact solution using a semilog plot Part 1B: Approximate the solution to the IVP using the Improved Euler's method with the following conditions: Initial condition y0 = -1/2; time step h = 1/16; and time interval t ∈ [0,20] + Derive the recursive formula for the Improved Euler's method applied to this IVP + Plot the Improved Euler's method approximation + Plot the absolute error between the approximation and the exact solution using a semilog plot Part 1C: Approximate the solution to the IVP using the RK4 method with the following conditions: Initial condition y0 = -1/2; time step h = 1/16; and time interval t ∈ [0,20] + Plot the RK4 method approximation + Plot the absolute error between the approximation and the exact solution using a semilog plot Problem 2: Consider the following Initial Value Problem (IVP) where y(t) is the dependent function: y' = y - y^2 + 1.14 cos(et/2) with y(0) = y0 and t ≥ 0 Part 2A: Approximate the solution to the IVP using the Improved Euler's method with the following conditions: Initial condition y0 = 1; time steps h = 1/8, 1/16, 1/32, 1/64; and time interval t ∈ [0,20] + Plot the Improved Euler's method approximation for all 4 time steps + Discuss the results of these approximations Part 2B: Approximate the solution to the IVP using the RK4 method with the following conditions: Initial condition y0 = 1; time steps h = 1/8, 1/16, 1/32, 1/64; and time interval t ∈ [0,20] + Plot the RK4 approximation for all 4 time steps + Discuss the results of these approximations
Sri K.
3.2 Spectrogram of Periodic Full-Wave Rectified Sinusoid A periodic signal is known to have a Fourier Series, which is usually described as a harmonic line spectrum because the only frequencies present in the spectrum are integer multiples of the fundamental frequency. With the spectrogram, it is easy to exhibit this harmonic line characteristic. More detail about the materials below can be found in Lectures 7 & 8, and the Fourier Series text in Section 3-5 and Appendix C. (a) Write a simple MATLAB script that will generate a periodic full-wave rectified sine wave once the period is given. The peak amplitude should be equal to 1. Here is a MATLAB one-liner that can form the basis of this script: tt=0:(1/fs):tStop;xx=Amp*abs(sin(2*pi*tt/T)); The values of fs, tStop, T, Amp will have to be determined. (b) Generate a full-wave rectified sine wave with T=1 sec, using a sampling rate of fs = 1000 Hz. The duration should be 5 secs, and Amp = 1. (c) Make a spectrogram with a long section duration. It is important to pick a section duration that is equal to an integer number of periods of the periodic full-wave rectified sine waveform created in the previous part. Define TSECT to get exactly 5 periods, and then determine the section length LSECT (an integer) to be used in plotspec. (d) You should expect to see a "harmonic line spectrum" in the spectrogram. Since frequency is along the vertical axis, the harmonic lines will appear as horizontal lines in the spectrogram. Make a list of all the harmonic frequencies that you can see in the spectrogram. Note the Fourier series coefficient can be found as (see Section 3-5.2): ak = 2 / (pi * (1-4k^2)) = 2 / (pi * (4k^2-1)) * e^(j*pi). (e) Determine the fundamental frequency for the harmonic lines. Note here the fundamental frequency doubles the original frequency of the sine wave after the full-wave rectification operation. (f) Measure the magnitudes of the first and third harmonic lines by using the fseriesdemo GUI first (see Slide 21 of Lecture 07 for an illustration. Another example in in Figure 2 in the Pre-Lab part on Page 6 of Lab 05 here) and then confirm your values with the Fourier Series coefficient formula: ak = 2 / (pi * (1-4k^2)) = 2 / (pi * (4k^2-1)) * e^(j*pi).
Akash M.
Problem Statement: The following simultaneous ordinary differential equations present a mathematical model for a system: Mz̈ + Cż + Kz = u(t) where; M = ⌈4 0 0⌉, K = ⌈100 -50 -40⌉, C = 0.2 M + 0.01*K and u = { 10, x0(t), 4 }. ⌊0 3 0⌋ ⌊-50 200 -10⌋ ⌊0 0 4⌋ ⌊-40 -10 150⌋ Considering matrix D = M⁻¹ K; and the initial conditions are all zeros, and x₀ (t) is given as shown in figure 1, determine: 1. Determine and plot x₀ (t) using the first five terms of its Fourier series for the time duration of [0 to 40 sec.] with equal time increments of 0.02 sec. 2. Compute eigen values and eigen vectors of matrix D. 3. Using the result obtained in part 1, plot the vector z(t) for the time duration of [0 to 40 sec.] with equal time increments of 0.02 sec. x₀(t) = -t² + 2 t Figure 1. Variation of x₀ (t) versus time
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