Differential Equations using Matlab, lab3
%% Lab 3 – Your Name – MAT 275 Lab
% Introduction to Numerical Methods for Solving ODEs
%% Exercise 1
% Part (a)
% NOTE: We often define the right-hand side of an ODE as some function, say
% f, in terms of the input and output variables in the ODE. For example,
% dy/dt = f(t,y) = 2y. Although the ODE does not explicitly depend on t, it does
% so implicitly since y is a function of t. Therefore, t is an input
% variable of f.
% Define ODE function f for ODE dy/dt = f(t,y) = 2y.
% Define vector t of time-values over the interval [0,0.5] to compute
% analytical solution vector.
% Create vector of analytical solution values at corresponding t values.
% NOTE: For the following steps, please use notation consistent with the
% protocol–e.g., use t5 to represent the time vector produced from Euler’s
% method using 5 timesteps and use y500 to represent the numerical solution
% vector produced from Euler’s method using 500 timesteps. In the following
% comments, I refer to “Euler’s method” as “forward Euler’s method,” which
% is more precise since there is also a backward Euler’s method. I use IVP
% as an abbreviation for initial-value problem, which for our purposes here
% means an ODE with associated initial condition.
% Solve IVP numerically using forward Euler’s method with 5 timesteps.
% Solve IVP numerically using forward Euler’s method with 50 timesteps.
% Solve IVP numerically using forward Euler’s method with 500 timesteps.
% Solve IVP numerically using forward Euler’s method with 5000 timesteps.
% Compute numerical solution error at t=0.5 for forward Euler with 5
% timesteps.
% Here, we define error as |analytical solution value – numerical solution
% value|.
% Compute numerical solution error at t=0.5 for forward Euler with 50
% timesteps.
% Compute numerical solution error at t=0.5 for forward Euler with 500
% timesteps.
% Compute numerical solution error at t=0.5 for forward Euler with 5000
% timesteps.
% Compute ratio of errors between N=5 and N=50.
% Compute ratio of errors between N=50 and N=500.
% Compute ratio of errors between N=500 and N=5000.
% NOTE: To complete the following table, simply run this section after
% you’ve finished the steps above and copy and paste your errors and ratios
% into the follwing display commands. Make sure you re-align each column
% after entering in values so that it looks pretty.
% Display table of errors (and ratios of consecutive errors) for the
% numerical solution at t=0.5 (the last element in the solution vector).
disp(‘———————————————-‘)
disp(‘| N | approximation | error | ratio |’)
disp(‘|——|—————–|———-|——–|’)
disp(‘| 5 | 7.4650 | 0.6899 | N/A |’)
disp(‘| 50 | | 0.0801 | 8.6148 |’)
disp(‘| 500 | | | |’)
disp(‘| 5000 | | | |’)
disp(‘———————————————-‘)
%%
% Part (b). Your answer here. Notice that by not adding a title to this
% section and skipping to the next line to start our comment, this comment
% will display in black (not green) when we publish it.
%%
% Here, we can add another paragraph to our essay answer for part (b) by
% once again partitioning off a new section with no title and beginning our
% comment on the second line of the section.
%%
% Part (c). Your answer here.
%% Exercise 2
% Part (a)
% Plot slopefield for dy/dt = -2y.
%%
% Part (b)
% Define vector t of time-values over the interval [0,10] to define
% analytical soution vector.
% Create vector of analytical solution values at corresponding t values.
% NOTE: The “hold on” command in the segment of code to plot the slope
% field indicates to MATLAB to add future plots to the same window unless
% otherwise specified using the “hold off” command.
% Plot analytical solution vector with slopefield from (a).
%%
% Part (c)
% NOTE: Recall we can define the right-hand side of an ODE as some function
% f. In the case of a one-dimensional autonomous ODE, we may define f such
% that dy/dt = f(t,y).
% Define ODE function.
% Compute numerical solution to IVP with 8 timesteps using forward Euler.
% Plot numerical solution with analytical solution from (b) and slopefield
% from (a) using circles to distinguish between the approximated data
% (i.e., the numerical solution values) and actual (analytical) solution.
hold off; % end plotting in this figure window
%%
% Your response to the essay question for part (c) goes here. Please do not
% answer the question with “the numerical solution is inaccurate because
% the stepsize is too big” or “because there are not enough points in the
% numerical solution.” What is it about this particular solution that makes
% forward Euler so inaccurate for this stepsize? Think about
%%
% Part (d)
% Define new grid of t and y values at which to plot vectors for slope
% field.
% Plot slope field for dy/dt = -2y corresponding to the new grid.
% Define vector t of time-values over the interval [??what??] to define
% analytical soution vector.
% Create vector of analytical solution values at corresponding t values.
% Define ODE function.
% Compute numerical solution to IVP with 16 timesteps using forward Euler.
%%
% Your essay answer goes here.
%% Exercise 3
% Display contents of impeuler M-file.
type ‘impeuler.m’
% NOTE: Make sure you document your code in impeuler.m appropriately for
% full credit.
% Define ODE function for dy/dt = f(t,y) = 2y.
% Compute numerical solution to IVP with 5 timesteps using “improved
% Euler.”
%%
% How does your output compare to that in the protocol?
%% Exercise 4
% NOTE: Here, we repeat all the steps from exercise 1, except using
% “improved Euler” instead of forward Euler (or regular Euler). Make sure
% to document your code appropriately.
%% Exercise 5
% NOTE: Here, we repeat all the steps from exercise 2, except using
% improved Euler rather than forward Euler. Remember code doc.
%%
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