Sunday, November 27, 2016

TI-84 Plus and HP Prime: Differential Equations and Half-Increment Solution, Numerical Methods

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.

1 comment:

  1. Hi Eddie, I tried to use the Halfstep program on my TI-84 Plus CE, unfortunately the error occurs here
    augment([A],[[D][Y]])→[A].
    I copied your program to TI-Connect CE software and send it to the calc with the sam program.
    Are there some differences between TI-84 Plus and TI-84 Plus CE?

    ReplyDelete

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