RKF

Type:

Command

Description:

Solve for Initial Values (Runge–Kutta–Fehlberg) Command: Computes the solution to an initial

 

value problem for a differential equation, using the Runge-Kutta-Fehlberg (4,5) method.

 

RKF solves y'(t) = f(t,y), where y(t0) = y0. The arguments and results are as follows:

 

•

{ list } contains three items in this order: the independent (t) and solution (y) variables, and the

 

 

right-hand side of the differential equation (or a variable where the expression is stored).

 

•

xtol sets the absolute error tolerance. If a list is used, the first value is the absolute error

 

 

tolerance and the second value is the initial candidate step size.

 

•

xTfinal specifies the final value of the independent variable.

 

RKF repeatedly calls RKFSTEP as it steps from the initial value to xTfinal.

Access:

…µRKF

Input/Output:

L3/A1L2/A2L1/A3L2/I1L1/I2

{ list }

xtol

xT final

→

{ list }

xtol

{ list }

{ xtol xhstep }

xT final

→

{ list }

xtol

L = Level; A = Argument; I = item

Example: Solve the following initial value problem for y(8), given that y(0) = 0:

 

′

 

 

1

 

2

 

y

= 1

+ t2

− 2y

 

= f (t, y)

 

 

1.Store the independent variable’s initial value, 0, in T.

2.Store the dependent variable’s initial value, 0, in Y.

3. Store the expression,

 

1

− 2y2 , in F.

 

+ t2

1

 

4. Enter a list containing these three items: { T Y F }.

5.Enter the tolerance. Use estimated decimal place accuracy as a guideline for choosing a tolerance: 0.00001.

6.Enter the final value for the independent variable: 8.

The stack should look like this:

{ T Y F }

.00001

8

7.Press RKF. The variable T now contains 8, and Y now contains the value .123077277659. The actual answer is .123076923077, so the calculated answer has an error of approximately

.00000035, well within the specified tolerance.

See also: RKFERR, RKFSTEP, RRK, RRKSTEP, RSBERR

RKFERR

Type: Command

Description: Error Estimate for Runge–Kutta–Fehlberg Method Command: Returns the absolute error estimate for a given step h when solving an initial value problem for a differential equation.

The arguments and results are as follows:

•{ list } contains three items in this order: the independent (t) and solution (y) variables, and the right-hand side of the differential equation (or a variable where the expression is stored).

•h is a real number that specifies the step.

•ydelta displays the change in solution for the specified step.

Full Command and Function Reference 3-203