Question

the general form of modified eluer method is​

Answer 1

Answer:

The scheme so obtained is called modified Euler's method. It works first by approximating a value to yi+1 and then improving it by making use of average slope. Truncation error: yi+1 = yi + h y'i + h2yi'' /2 + h3yi''' /3!

Step-by-step explanation:

Modified Euler's Method :

The Euler forward scheme may be very easy to implement but it can't give accurate solutions. A very small step size is required for any meaningful result. In this scheme, since, the starting point of each sub-interval is used to find the slope of the solution curve, the solution would be correct only if the function is linear. So an improvement over this is to take the arithmetic average of the slopes at xi and xi+1(that is, at the end points of each sub-interval). The scheme so obtained is called modified Euler's method. It works first by approximating a value to yi+1 and then improving it by making use of average slope.

yi+1 = yi+ h/2 (y'i + y'i+1)

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

If Euler's method is used to find the first approximation of yi+1 then

yi+1 = yi + 0.5h(fi + f(xi+1, yi + hfi))

Truncation error:

yi+1 = yi + h y'i + h2yi'' /2 + h3yi''' /3! + h4yiiv /4! + . . .

fi+1 = y'i+1 = y'i + h y''i + h2yi'''' /2 + h3yiiv /3! + h4yiv /4! + . . .

By substituting these expansions in the Modified Euler formula gives

yi + h y'i + h2yi'' /2 + h3yi''' /3! + h4yiiv /4! + . . . = yi+ h/2 (y'i + y'i + h y''i +

h2yi'''' /2 + h3yiiv /3! + h4yiv /4! + . . . )

So the truncation error is: - h3yi''' /12 - h4yiiv /24 + . . . that is, Modified Euler's method is of order two.

Hope it will help you....