Page 162 - Advanced engineering mathematics

142 CHAPTER 5 Approximation of Solutions

1. y = y sin(x); y(0) = 1 4. y = 2 − x; y(0) = 1

2. y = x + y; y(1) =−3 5. y = y − cos(x); y(1) =−2
2

3. y = 3xy; y(0) = 5 6. y = x − y ; y(0) = 4

5.3 Taylor and Modified Euler Methods
We will develop two other numerical approximation schemes, both of which are (in general)
more accurate than Euler’s method.
Under certain conditions on f and h, we can use Taylor’s theorem with remainder to write
1 2 1 3
(3)

y(x k+1 ) = y(x k ) + hy (x k ) + h y (x k ) + h y (ξ k )
2 6
for some ξ k in [x k , x k+1 ]. If the third derivative of y(x) is bounded, we can make the last term in
this sum as small as we like by choosing h to be small enough, leading to the approximation
1 2
y k+1 ≈ y(x k ) + hy (x k ) + h y (x k ). (5.1)

2
Now, y(x)= f (x, y(x)). This suggests that in equation (5.1) we consider f (x k , y k ) as an approx-
imation of y (x k ) if y k is an approximation of y(x k ). This leaves the term y (x k ) in equation (5.1)

to approximate. To do this, differentiate the equation y (x) = f (x, y(x)) with respect to x to get
∂ f ∂ f

y (x) = (x, y) + (x, y)y (x).
∂x ∂y
We are therefore led to approximate
∂ f ∂ f

y (x k ) ≈ (x k , y k ) + (x k , y k )y (x k ).
∂x ∂y
Insert these approximations of y (x k ) and y (x k ) into equation (5.1) to get

1 2 ∂ f ∂ f

y k+1 ≈ y k + hf (x k , y k ) + h (x k , y k ) + (x k , y k )y (x k ) .
2 ∂x ∂y
The second-order Taylor method consists of using this expression to approximate y(x k+1 )
by y k+1 We can simplify this expression for the approximate value of y k+1 by using the
notation
f k = f (x k , y k ),
∂ f ∂ f
= f x , = f y ,
∂x ∂y
∂ f ∂ f
(x k , y k ) = f xk , (x k , y k ) = f yk .
∂x ∂y
With this notation, the second-order Taylor approximation is
1 2
y k+1 ≈ y k + hf k + h ( f xk + f k f yk ).
2

The second-order Taylor method is a one-step method because it approximates the solution
value at x k using the approximation made at x k−1 , which is just one step back. Euler’s method is
also one-step.

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October 14, 2010 14:19 THM/NEIL Page-142 27410_05_ch05_p137-144