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# ¿Qué es un método numérico? - page 77 / 82

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k2 = f(xi + p1h, yi + q11k1h) = f(xi,yi) + p1hf/x + q11k1hf/y

yi+1 = yi + {a1f(xi,yi) + a2[f(xi,yi) + p1hf/x + q11k1hf/y]}h

= yi + [a1f(xi,yi) + a2f(xi,yi)]h + [a2p1f/x + a2q11k1f/y]}h2__ (5.11)

Comparando (5.11) con (5.10) se obtienen tres ecuaciones, con cuatro incógnitas:

a1 + a2 = 1 ;a2p1 = 1/2 ;a2q11  = 1/2__ (5.12)

que generan una familia infinita de soluciones.

Si se fija a2 = 1/2 , entonces:a1 = 1/2 ,p1 = 1  yq11  = 1

se obtiene el método de Heun:

yi+1 = yi + (k1/2 + k2/2)h = yi + (k1 + k2)h/2

k1 = f(xi,yi) ;k2 = f(xi + h, yi + k1h)

yi+1 = yi + [f(xi,yi) + f(xi + h, yi + k1h)]h/2

= yi + [f(xi,yi) + f(xi+1, yi+1)]h/2

donde:yi+1 = yi + k1h = yi + f(xi,yi)  es un predictor.

Si se fija a2 = 1 , entonces:a1 = 0 ,p1 = 1/2 ,q11  = 1/2

yi+1 = yi + k2h

k1 = f(xi,yi) ;k2 = f(xi + h/2, yi + k1h/2)

yi+1 = yi + f(xi + h/2, yi + k1h/2)h

= yi + f(xi+0.5, yi+0.5)]h

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