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MATLAB Initial Value Problems In this livescript, you will learn how To solve first order initial value problems in MATLAB We’ll consider the solution of the initial value problem \frac{dx}{dt}=\arctan{\{\sin^{2}{(x)}}\} with the initial condition x(0)=\pi/4 . As always, it’s always worh a shot to use \texttt{dsolve} to see whether an analytical solution exists syms x(t) a
eqn = diff(x,t) == atan(sin(x).^2);
init_cond = x(0)==pi/4;
dsolve(eqn,init_cond) Unfortunately, no analytical solution exists, but that shouldn’t worry us as MATLAB has numerous numerical solvers for initial value problems including \texttt{ode45}
Uses Runge-Kutta method of orders 4 and 5 \texttt{ode23}
Uses Runge-Kutta method of orders 2 and 3 \texttt{ode23t } Uses the the Crank-Nicolson method Many more solvers using more complicated algorithms that won’t be covered in this course are \texttt{ode113} \texttt{ode15s} \texttt{ode23s} \texttt{ode23tb} \texttt{ode15i} There are situations where each solver has better performance, but typically it’s recommended to try \texttt{ode45}
first. Since each solver has the same syntax \texttt{[t,x] = ode\_\_(odefun,tspan,x0)} we’ll just cover the use of \texttt{ode45}
. The first input \texttt{odefun} is a function handle that specifies our differential equation. odefun = @(t,x) atan(sin(x).^2); \texttt{tspan}
defines the size of the time domain that we want to solve the IVP in, \texttt{tspan = [ti tf]} . To solve the IVP within the interval 0\leq t \leq 100 , we have tspan = [0 100] \texttt{x0}
is a vector of the initial conditions, which in our case is given by x0 = pi/4; With these pieces, we can then call \texttt{ode45}
using the following command [t,x] = ode45(odefun,tspan,x0); Plotting the position and velocity, we find plot(t,x)
xlabel(‘t’)
ylabel(‘x’)
xlim([0,100]) (b) Change the code to use \texttt{ode23}
and \texttt{ode23t} to solve the initial value problem.
manual code ready 0.4 warning Warning: Unable to find symbolic solution. 4 text
ans =
[ empty sym ]
false false 4 matrix tspan 1×2 1 2 double 0 100
double double [[{“value”:”{\”value\”:\”0\”}”},{“value”:”{\”value\”:\”100\”}”}]] 6 figure 6810095d-34e6-4df3-967a-b157ac2f115c 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×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 577 433 8 9 10 11 8 9 10 11 12 true false 0 9 9 false false 1 10 10 false false 2 11 11 false true 3 12 12 0 1 true false 4 24 24 false false 5 26 26 2 false false 6 28 28 false false 7 30 30 false false 8 32 32 3 false false 9 33 33 3 false false 10 34 34 3 false true 11 35 35 3
2020-05-07T07:31:41Z 2020-10-09T06:11:03Z
application/vnd.mathworks.matlab.code MATLAB Code R2020b
9.9.0.1444674 fb5e7875-1a73-4d5e-9382-8d99b8af09a2
9.9.0.1467703
R2020b
Aug 26 2020
2314982500