RungeKutta 4th Order Method  Differential Equations
The program RK4 estimates solutions to the differential equation y dy/dx = f(x,y) given the initial condition (x_n, y_n). With step size h, the point (x_n+1, y_n+1) is calculated by:
x_n+1 = x_n + h
y_n+1 = y_n + (k1 + 2*k2 + 2*k3 + k4)/6
Where:
k1 = h * f(x_n, y_n)
k2 = h * f(x_n + h/2, y_n + k1/2)
k3 = h * f(x_n + h/2, y_n + k2/2)
k4 = h * f(x_n + h, y_n + k3)
The program RK4 uses a subroutine FXY. The program FXY is where the y'(x) is stored in terms of X and Y. The result is stored in the variable Z.
Variables used in RK4 and FXY:
A = x_n
B = y_n
C = x_n+1
D = y_n+1
H = step
K = k1
L = k2
M = k3
N = k4
E, X, Y, and Z are also used
Remember for the fx5800p, all variables (AZ, and Z[#]s) are global, meaning they will carry over from programs and subroutines.
Anything following the double slash (//) is a comment.
Once the first point (x_n+1, y_n+1) is calculated, RK4 will ask if you want the next point (x_n+2, y_n+2) calculated. Enter 1 at the prompt for "Yes".
Program FXY
type f(X,Y) here → Z
// A RETURN command is implicit.
Program RX4
"INITIAL COND."
"X0"?→ A
"Y0"?→ B
"STEP"?→ H
Lbl 0 // main routine
A → X // k1
B → Y
Prog "FXY"
H * Z → K
A + H ÷ 2 → X // k2
B + K ÷ 2 → Y
Prog "FXY"
H * Z → L
B + L ÷ 2 → Y // k3
Prog "FXY"
H * Z → M
A + H → X // k4
B + M → Y
Prog "FXY"
H * Z → N
X → C // x_n+1
B + (K + 2*L + 2*M + N) ÷ 6 → D // y_n+1
"RESULT"
"X1 = "
C ◢
"Y1 = "
D ◢
"NEXT POINT?"
"1 = YES 0 = NO"
? → E
E ≠ 1 ⇒ Goto E
C → A
D → B
Goto 0
Lbl E
"DONE"
Example: y'(x) = y  x with the initial condition (0,2).
Find y when x = 0.1 and x = 0.2, respectively.
In this case, our step is h = 0.1.
Program FXY will have this:
Y  X → Z
Running RK4 gives these results:
x = 0.1, y = 2.205170833
x = 0.2, y = 2.421402571
Eddie
This blog is property of Edward Shore. 2015
A blog is that is all about mathematics and calculators, two of my passions in life.
Monday, January 5, 2015
fx5800p: RungeKutta 4th Order Method  Differential Equations
Subscribe to:
Post Comments (Atom)
HP Prime and TI84 Plus: Basic Wheatstone Full Bridge Circuit
HP Prime and TI84 Plus: Basic Wheatstone Full Bridge Circuit The program WHEATSTONE (HP Prime) and WHTSTONE (TI84 Plus CE) deals wi...

TI36X Pro Review This is a review of the TI36X Pro Calculator by Texas Instruments. History Originally, this was the TI30X Pro that w...

Casio fx991EX Classwiz Review Casio FX991EX The next incarnation of the fx991 line of Casio calculators is the fx991 EX. ...

One of the missing features of the TI82/83/84 family is the ability to convert between bases. Here are two programs in TIBasic to help...
No comments:
Post a Comment