TI-84 Plus and HP Prime: Differential Equations and Half-Increment
Solution, Numerical Methods
Introduction
The program HALFSTEP
solves the numerical differential equation
d^2y/dt^2 =
f(dy/dt, y, t) given the initial
conditions y(t0) = y0 and dy/dt (t0) = dy0
In this
notation, y is the independent variable and t is the dependent variable.
The Method
Let C =
f(dy/dt, y, t). Give the change of t as Δt.
First Step:
With t = t0:
h_1/2 = dy0 + C
* Δt/2
y1 = y0 + dy0 * Δt
Loop:
t = t0 + Δt
h_I+1/2
= h_I-1/2 + C * Δt
y_I+1 = y_I +h_I+1/2 * Δt
Repeat as many
steps as desired.
This method was
presented by Robert M. Eisberg in his 1976 calculator programming book (see
source below).
Variables
The program
uses the following variables:
C: d^2y/dt^2.
Represent dy/dt as the variable A, y as the variable Y, and t as the
variable T.
The program
will always designate Y as the
independent variable and T as the dependent variable.
Examples:
Application
|
C
|
C
for HALFSTEP
|
Free-Fall
|
d^2y/dt^2 = g
|
“9.80665”
(SI) or “32.1740468” (US)
|
Free-Fall with
Friction
|
d^2y/dt^2 = g
- α (dy/dt)^2
(α = F/m)
|
“g - α * A^2”
(sub numeric
values for g, α)
|
Spring
|
d^2x/dt = -k/m
* x
|
“-k/m * T”
(sub numeric
values for k, m)
|
Pendulum
|
d^2θ/dt = -α*sin(θ)
(α = -g/l)
|
“-α * sin(Y)”
(sub numeric
values for α)
|
Damped,
Driven Oscillations
|
d^2x/dt = -α*x
– β*dx/dt + γ * sin(ω*t)
|
“-α*Y-β*A+γ*sin(ω*T)”
(sub numeric
values for α, β, γ)
|
HP Prime Program HALFSTEP
Input: C. Use
single quotes to enclose d^2y/dt^2.
Represent dy/dt as A, y as Y, and t as T.
Output: A matrix of two columns, t and y.
EXPORT HALFSTEP(c,A,Y,D,tmax)
BEGIN
// d^2y/dt^2=C,dy0,y0,Δt,tmax
// EWS 2016-11-17
// C use single quotes
// 'dy=A, y=Y, t=T'
// Radian mode
HAngle:=0;
LOCAL mat:=[[0,Y]],T,H;
LOCAL K:=3,I;
T:=D;
H:=A+EVAL(c)*D/2;
Y:=Y+H*D;
mat:=ADDROW(mat,[D,Y],2);
FOR I FROM 2*D TO tmax STEP D DO
T:=I; A:=H;
H:=H+EVAL(c)*D;
Y:=Y+H*D;
mat:=ADDROW(mat,[I,Y],K);
K:=K+1;
END;
RETURN mat;
END;
TI-84 Plus Program
HALFSTEP
Input: For C, use enclose d^2y/dt^2 in quotes. Represent dy/dt as A, y as Y, and t as
T.
Output: A matrix of two columns, t and y.
"EWS
2016-11-27"
Func
Radian
Disp
"D²Y/DT²=C"
Disp "USE
A=DY/DT,Y,T"
Input "C, USE A
STRING:",Y1
Input
"DY0:",A
Input
"Y0:",Y
Input "DELTA
TIME:",D
Input "TIME
MAX:",N
[[0][Y]]→[A]
D→T
A+Y1*D/2→H
Y+H*D→Y
augment([A],[[D][Y]])→[A]
For(I,2D,N,D)
I→T:H→A
H+Y1*D→H
Y+H*D→Y
augment([A],[[I][Y]])→[A]
End
[A]^T→[A]
Examples:
Please see the
screen shots below. Both are screen
shots from the TI-84 Plus.
Source: Eiseberg, Robert M. Applied Mathematical Physics with
Programmable Pocket Calculators
McGraw-Hill, Inc: New York. 1976.
ISBN 0-07-019109-3
This blog is
property of Edward Shore, 2016.