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Numerical Solutions to Differential Equations. Beginning Programming for Engineers . Learning goals for class 5. Understand why numeric solutions of ODEs is necessary.

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## Beginning Programming for Engineers

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**Numerical Solutions to Differential Equations**Beginning Programming for Engineers**Learning goals for class 5**• Understand why numeric solutions of ODEs is necessary. • Understand, by referring to Euler's method, how it could be possible to numerically solve an ODE, even when their is no algebraic solution. • Know how to use Matlab's ODE solvers to solve ODE IVP problems, with one or more equations, and with higher level derivatives; in particular, how to use ode45. • Understand why more than one ODE solver is needed, and have some advice how to choose.**Initial Value Problems**We wish to "solve" the ordinary differential equation (ODE): where we have the initial conditions: By "solve", we want to compute for**Matlab's ODE Solvers**Matlab provides a selection of solvers to integrate an ODE function over a range. General form: [t,y] = solver(odefun, tspan, y0) The odefun is the function to be integrated over the interval tspan with initial conditions y0. The result is the vector t for values of the independent variable in the range tspan, and estimated values y for each value t. The solver is typically ode45.**Simple usage of ode45**We would like to solve this system numerically: We need a function to calculate function [dy] = simple_ode(t, y) dy = -3.7*y;end Call the solver: [t,y] = ode45(@simple_ode, [0 1], 5);**Using ode45, continued**Given the example: [t,y] = ode45(@simple_ode, [0 1], 5); • The interval [0 1] is the range over which the independent variable (often x or t) varies. The solver chooses its own values to use. Actual values chosen are sorted in t, a column vector. • Corresponding values of the integrated function are stored in the column vector y. • Instead of [0 1], additional points can be given, e.g, 0:0.1:1 -- which will be used as the values of the output t (and to compute y).**Function Handles**The notation @simple_ode is a function handle, to allow the solver to call our function simple_ode. Small functions can be defined anonymously: odefun = @(T,Y) -3.7*Y; [t,y] = ode45(odefun, [0 1], 5); or even: [t,y] = ode45(@(T,Y)-3.7*Y, [0 1], 5);**Multiple ODEs**The Matlab ODE solvers can solve systems of related equations over the same interval of the independent variable. Example: The Lotka-Volterra predator-prey model can be used to model the population on rabbits (r) and foxes (f): Model this with**Rabbits and Foxes (2) ...**Need the ODE function: function [dy] = foxrabbitode(t,y) % FOXRABBITODE calculates population changes % y(1) = rabbits % y(2) = foxes alpha = 0.01; r = y(1); f = y(2); dy = [ 2*r - alpha*r*f -f + alpha*r*f ]; end • Calculate on y, a column vector, and produce a column vector. Each row corresponds to one of the equations.**Rabbits and Foxes (3) ...**Create column vector of initial conditions (r0 and f0) y0 = [ 300; 150 ]; [t,y] = ode45(@foxrabbitode, [0 20], y0); figure; hold on; plot(t, y(:,1), 'b'); % rabbits plot(t, y(:,2), 'r'); % foxes Each column of dy corresponds to the points computed for one of the equations.**Function handles and arguments**We might want to pass alpha as an argument to the ODE function: function [dy] = foxrabbitode2(t,y,alpha) r = y(1); f = y(2); dy = [ 2*r - alpha*r*f -f + alpha*r*f ]; end Need anonymous function to pass this to ODE function: odefun = @(T,Y) foxrabbitode2(T,Y,alpha);[t,y] = ode45(odefun, interval, y0);**Higher-order ODEs**Treat higher-order ODEs as systems of related ODEs. The ODE function receives both y and y', and computes y' and y''. (Simply copy the received y' value!) We need initial conditions for y and y'. Consider solving: Let: y(1) = 0, y'(1) = 1 Solve for y(t), for t in [1 30].**Function for simple second-order ODE**function [dy] = second_ode(t,y) % SECOND_ODE is a simple second order ODE. % % y(1) = f(t) % 1st equation % y(2) = f'(t) % 2nd equation % % dy(1) = f'(t) % 1st equation % dy(2) = f''(t) % 2nd equation % Note how we can just copy y(2) to dy(1). dy = [ y(2) (-1/t)*sin(y(1)) ];**Solving the second-order ODE**% Compute the solutions. [t,y] = ode45(@second_ode, [1 30], [0;1]); % y(:,1) is y(t), and y(:,2) is y'(t). plot(t, y(:,1))**Trajectory as ODE**function [dpv] = update_path(t, pv) % pv = position and velocity state vector % pv(1) = x position, xpos, at time t % pv(2) = y position, ypos, at time t % pv(3) = x velocity, vx, at time t % pv(4) = y velocity, vy, at time t x = pv(1); y = pv(2); xv = pv(3); yv = pv(4); dpv = [xv % dx = x velocity yv % dy = y velocity 0 % d2x = d(x velocity) = x acceleration -9.81 ]; % d2y = d(y velocity) = y acceleration**Event functions**Sometimes we need to stop the integration due to some event, especially if we don't know how long to run! You can create an "event function" to stop integration: function [value,isterminal,direction] = event(t,y) This accepts t,y like the ODE function. It computes a value at t,y. The output isterminal, if 1, means to stop the integration if value is 0. The direction indicates when we care about value being 0; if it is 0, then we always care.**Event functions: Trajectory**Sample for testing if above ground: function [value, isterminal, direction] = ... trajectory_event(t, pv) if pv(2) > 0 value = 1; % Above the ground else value = 0; % Hit the ground end isterminal = 1; % Stop when value=0 direction = 0; % Always care about zerosend**Using an event function**Use odeset to configure options to the ODE solvers, including events functions: options = odeset('Event', @trajectory_event);[t,y] = ode45(@update_path, 0:dt:ft, ... [x;y;v0x;v0y], options); • You can use [0 Inf] as an interval when you don't know when to finish, but have an Event function to stop. • You can use more complicated function handles to pass information to the Event function, e.g., the terrain.**Other solvers**• Some problems are stiff, needing slower specialized solvers. Other solvers are faster than ode45, but not as accurate. • Usually try ode45 first. If it works but is too slow, consider ode23. • If ode45 is so slow it is unusable, you might have a stiff problem. Try using ode15s or ode23s. • See doc ode45 for much more information, including guidance about choosing a solver, options you can pass to solvers, etc.

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