Initial Value Problems - Numerical Methods in Engineering with MATLAB - page 253

Using the finite difference approximation

y ( m ) ( x y ( m ) ( x ) + h ) −

y ( m + 1) ( ξ ) ≈

h

weobtain the more usable form

h m
( m 1)! y ( m ) ( x

y ( m ) ( x ) E ≈ + h ) − (7.7)

+

which couldbe incorporatedinthe algorithm to monitor the errorineach integration
step.

taylor

The function taylor implements the Taylor seriesmethod of integration of order four.
It can handle any number of first-orderdifferentialequations y i = f i ( x , y 1 , y 2 ,..., y n ),

i = 1 , 2 ,..., n . The useris required to supply the function deriv that returns the 4 × n

array

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

( y ) T
( y ) T
( y ) T
( y (4) ) T

y 1 y 2 y n ···

y 1 y 2

···

y n

d =

y
1

y
2

y
n

···

y (4)
1

y (4)
2

··· y (4)

n

The functionreturns the arrays xSol and ySol thatcontain the values of x and y
at intervals h .

function [xSol,ySol] = taylor(deriv,x,y,xStop,h)
% 4th-order Taylor series method of integration.
% USAGE: [xSol,ySol] = taylor(deriv,x,y,xStop,h)
% INPUT:
% deriv = handle of function that returns the matrix
% d = [dy/dx dˆ2y/dxˆ2 dˆ3y/dxˆ3 dˆ4y/dxˆ4].
% x,y = initial values; y must be a row vector.
% xStop = terminal value of x
% h = increment of x used in integration (h > 0).
% OUTPUT:
% xSol = x-values at which solution is computed.
% ySol = values of y corresponding to the x-values.

ifsize(y,1)>1;y=y';end %ymustberowvector
xSol = zeros(2,1); ySol = zeros(2,length(y));
xSol(1) = x; ySol(1,:) = y;