200
18.11 A MATLAB M-file for the midpoint method can be developed as
function [t,y] = midpointdydt,tspan,y0,h [t,y] = midpointdydt,tspan,y0,h:
uses the midpoint method to integrate an ODE input:
dydt = name of the M-file that evaluates the ODE tspan = [ti, tf] where ti and tf = initial and
final values of independent variable y0 = initial value of dependent variable
h = step size output:
t = vector of independent variable y = vector of solution for dependent variable
ti = tspan1; tf = tspan2;
t = ti:h:tf; n = lengtht;
if necessary, add an additional value of t so that range goes from t = ti to tf
if tntf tn+1 = tf;
n = n+1; end
y = y0onesn,1; preallocate y to improve efficiency for i = 1:n-1
hh = ti+1 - ti; k1 = fevaldydt,ti,yi;
ymid = yi + k1hh2; k2 = fevaldydt,ti+hh2,ymid;
yi+1 = yi + k2hh; end
plott,y
Here is the test of the solution of Prob. 18.5. First, an M-file holding the differential equation is written as
function dp = dpdtt, p dp = 0.0261-p12000p;
201 Then the M-file can be invoked as in
[t,p]=midpointdpdt,[1950 2000],2555,5; disp[t,p]
1.0e+003 1.9500 2.5550
1.9550 2.8260 1.9600 3.1163
1.9650 3.4253 1.9700 3.7521
1.9750 4.0953 1.9800 4.4529
1.9850 4.8227 1.9900 5.2021
1.9950 5.5881 2.0000 5.9776
The following plot is generated
18.12 A MATLAB M-file for the fourth-order RK method can be developed as
function [t,y] = rk4dydt,tspan,y0,h [t,y] = rk4dydt,tspan,y0,h:
uses the fourth-order Runge-Kutta method to integrate an ODE input:
dydt = name of the M-file that evaluates the ODE tspan = [ti, tf] where ti and tf = initial and
final values of independent variable y0 = initial value of dependent variable
h = step size output:
t = vector of independent variable y = vector of solution for dependent variable
ti = tspan1; tf = tspan2;
t = ti:h:tf; n = lengtht;
if necessary, add an additional value of t
202
so that range goes from t = ti to tf if tntf
tn+1 = tf; n = n+1;
end y = y0onesn,1; preallocate y to improve efficiency
for i = 1:n-1 hh = ti+1 - ti;
k1 = fevaldydt,ti,yi; ymid = yi + k1hh2;
k2 = fevaldydt,ti+hh2,ymid; ymid = yi + k2hh2;
k3 = fevaldydt,ti+hh2,ymid; yend = yi + k3hh;
k4 = fevaldydt,ti+hh,yend; phi = k1+2k2+k3+k46;
yi+1 = yi + phihh; end
plott,y
Here is the test of the solution of Prob. 18.2. First, an M-file holding the differential equation is written as
function dy = dydxx, y dy = 1+2xsqrty;
Then the M-file can be invoked as in [x,y] = rk4dydx,[0 1],1,0.1;
disp[x,y] 0 1.0000
0.1000 1.1130 0.2000 1.2544
0.3000 1.4280 0.4000 1.6384
0.5000 1.8906 0.6000 2.1904
0.7000 2.5440 0.8000 2.9584
0.9000 3.4410 1.0000 4.0000
The following plot is generated
203
18.13 Note that students can take two approaches to developing this M-file. The first program
Kategori