Ecuaciones diferenciales My Solutions

Runge-Kutta 2 del punto medio

Sea la ecuación diferencial

y′ = f (t, y), y(t0) = y0

Dado un tamaño de paso h , se tiene el método queda definido por las fórmulas

k1 = f (ti, yi)

1 1
k2 = f ti + h, yi + hk1
( )
2 2

yi + 1 = yi + hk2

Se quiere simular los valores de la ecuación diferencial

−t
2 e
y′ − y = , y(2) = 0
2
ln (t) t

en el intervalo t ∈ [2, 3] , considerando un tamaño de paso h por determinar.

Realice lo siguiente

1. Defina la función f (t, y) para la ecuación diferencial a estudiar. Guarde el valor de f (2, 0) en la variable fvalue.
2. Use la función de MATLAB ode45 para aproximar y(3) , sea N la cantidad de iteraciones empleadas. Calcule h = 1/ N , almacene el resultado en la variable h.
3. Implemente la función rk2_midlle para el método de Runge-Kutta 2 del punto medio empleando la plantilla dada.
4. Aproxime el valor de y(3) con el método implementado, considere el valor de h previamente calculado. Almacene el resultado en la variable y_rk2.
5. Se sabe que el valor exacto de y(3) es 0.069899. Guarde el error relativo de la aproximación en la variable xi_rk2.
6. Corrobore que la implementación del método de Runge-Kutta 2 del punto medio trabaja de manera vectorial; es decir, puede resolver el problema dado en la plantilla

Script   Save  Reset  MATLAB Documentation (https://www.mathworks.com/help/) Open Item in MATLAB Online 
A

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6
% f(t,y)
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f=@ (t,y) ( exp(-t)/t^2 )+( 2*y/log(t) )
8
y0=0 % y(2)=0
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fvalue=f(2,0)
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% h
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[tsol,sol]=ode45(f,[2 3],y0);
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N=length(sol)
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h=1/N
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% RK2
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[t_rk2,y_rk2]=rk2_midle(f,[2 3],y0,h);
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y_rk2=y_rk2(:,end)
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% Error de la aproximación
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exacto=0.069899
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xi_rk2=abs(exacto-y_rk2)/abs(exacto)
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% Funcionamiento vectorial
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tspan = [0 10], h = 0.1; y0 = [1; 0]; % Initial condition and solver parameters
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[t_vec, y_vec] = rk2_midle(@ftest, tspan, y0, h);
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z_vec = [t_vec; y_vec]
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function dydt = ftest(t,y)
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dydt = [y(2); -y(1)];
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end
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function [t,y] = rk2_midle(f, tspan, y0, h)
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% Integrate a system of ordinary differential equations in the form
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% dy/dt = f(t,y) as an initial value problem using Middle Point method.
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% Initialize the output solution arrays.
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nSteps = floor( (tspan(2)-tspan(1))/h );
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m = length(y0);
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t = zeros(1,nSteps+1);
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y = zeros(m,nSteps+1);
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t(1) = tspan(1);
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y(:,1) = y0;
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43 A
% Integrate the ODE's using Middle Point method.
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%
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% Complete here
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%
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for i = 1:nSteps
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k1 = f(t(i), y(:,i));
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k2 = f(t(i) + h/2, y(:,i) + (h/2)*k1);
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y(: i+1) = y(: i) + h * k2;

 Run Script 

Tests: 3 of 5 Tests Passed (60%) Run Pretest  Submit (max. reached) 

  Define correctamente f(t,y) (Pretest) 20% (20%)

 Determina correctamente el tamaño de paso h 20% (20%)

 Aproxima y(3) correctamente con el método de Runge-Kutta 2 del punto medio 0% (20%)
Variable y_rk2 has an incorrect value.

 Calcula el error relativo de la aproximación 0% (20%)

Variable xi_rk2 has an incorrect value.

 El método de Runge-Kutta 2 del punto medio implementado trabaja de manera vectorial 20% (20%)

Total: 60%

Output
A
f =

function_handle with value:

@(t,y)(exp(-t)/t^2)+(2*y/log(t))
y0 =

fvalue =

0.033833820809153

N =

49

h =

0.020408163265306

y_rk2 =

0.069862957926970

exacto =

0.069899000000000
A

xi_rk2 =

5.156307390703199e-04
tspan =

0 10

z_vec =

Columns 1 through 9

0 0.100000000000000 0.200000000000000 0.300000000000000 0.400000000000000 0.500000000000000 0.6000

1.000000000000000 0.995000000000000 0.980025000000000 0.955224875000000 0.920848000625000 0.877238765621875 0.8248
0 -0.100000000000000 -0.199000000000000 -0.296007500000000 -0.390049950000000 -0.480184500312500 -0.5655

Columns 10 through 18

0.900000000000000 1.000000000000000 1.100000000000000 1.200000000000000 1.300000000000000 1.400000000000000 1.5000

0.620507623055981 0.538970697569426 0.452028552416600 0.360552647472168 0.265459915339205 0.167703570236663 0.0682
-0.784343873712754 -0.842472916649789 -0.892157621823482 -0.932899688956025 -0.964290455258462 -0.986014994516090 -0.9978

Columns 19 through 27

1.800000000000000 1.900000000000001 2.000000000000000 2.100000000000001 2.200000000000001 2.300000000000001 2.4000

-0.230165601960146 -0.326353044497550 -0.419271202449929 -0.507988489551696 -0.591615409977885 -0.669313476592057 -0.7403
-0.973382705472044 -0.945499231748669 -0.908136431140171 -0.861668628739477 -0.806561436640610 -0.743367088459618 -0.6727

Columns 28 through 36

2.700000000000001 2.800000000000001 2.900000000000001 3.000000000000001 3.100000000000001 3.200000000000002 3.3000

-0.906286272396483 -0.944101081940442 -0.972452223508186 -0.991055240313800 -0.999723393410688 -0.998369536192461 -0.9870
-0.423462409059409 -0.330716469774464 -0.234652779231547 -0.136234292984571 -0.036447597488268 0.063706979840242 0.1632 A

Columns 37 through 45

3.600000000000002 3.700000000000002 3.800000000000002 3.900000000000002 4.000000000000002 4.100000000000001 4.2000

-0.894497686986399 -0.845217355282953 -0.787462487584503 -0.721811864576326 -0.648923436360197 -0.569527728483850 -0.4844
 0.448078432685137 0.535287809220351 0.617133105702544 0.692793688932482 0.761510906945452 0.822595696046745 0.8754

Columns 46 through 54

4.500000000000000 4.600000000000000 4.699999999999999 4.799999999999999 4.899999999999999 4.999999999999998 5.0999

-0.203595059961603 -0.104614098257184 -0.004581905693694 0.095498721279190 0.194624475586924 0.291801597670757 0.3860
0.979629864046115 0.995091220722044 1.000577174444153 0.996032479141301 0.981502444617676 0.957132484835895 0.9231

Columns 55 through 63

5.399999999999997 5.499999999999996 5.599999999999996 5.699999999999996 5.799999999999995 5.899999999999995 5.9999

0.642034395647914 0.715579857310972 0.781953469541029 0.840489657579242 0.890601400204925 0.931786116587119 0.9636
0.767556336412973 0.699515115166117 0.624459553859189 0.543141909135790 0.456377233832187 0.365035207642533 0.2700

Columns 64 through 72

6.299999999999994 6.399999999999993 6.499999999999993 6.599999999999993 6.699999999999992 6.799999999999992 6.8999

1.000415346958592 0.992683151516785 0.974999114176136 0.947540268614938 0.910581645389736 0.864493517203921 0.8097
-0.027301187070141 -0.127206215830650 -0.225838499903175 -0.322209218821273 -0.415352199588660 -0.504333603129690 -0.5882

Columns 73 through 81

7.199999999999990 7.299999999999990 7.399999999999990 7.499999999999989 7.599999999999989 7.699999999999989 7.7999

0.599351830186449 0.516194046710448 0.427859338971587 0.335233132991843 0.239243899198706 0.140853845585257 0.0410
-0.801610243250691 -0.857537375053083 -0.904869092848862 -0.943130681281776 -0.971938341174552 -0.991003039388550 -1.0001

Columns 82 through 90

8.099999999999987 8.199999999999987 8.299999999999986 8.399999999999986 8.499999999999986 8.599999999999985 8.6999

-0.256835703875825 -0.352301845642654 -0.444238548060460 -0.531724057451520 -0.613881220304363 -0.689886277852726 -0.7589
-0.967503202862081 -0.936982116460189 -0.897067021313622 -0.848157831401008 -0.790744636498851 -0.725402791285921 -0.6527 A

Columns 91 through 99

8.999999999999984 9.099999999999984 9.199999999999983 9.299999999999983 9.399999999999983 9.499999999999982 9.5999

-0.918223722090240 -0.953522163661918 -0.979262428003925 -0.995186230011801 -1.001133688158858 -0.997044929768577 -0.9829
-0.398895601821292 -0.305078751603162 -0.208201141478954 -0.109233892971167 -0.009169100505131 0.090990113813280 0.1902

Columns 100 through 101

9.899999999999981 9.999999999999980
-0.882636140767323 -0.830954421124928
0.472685389385588 0.558585576515392

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