**Runge-Kutta 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 fx-5800p, all variables (A-Z, 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

### fx-5800p: Runge-Kutta 4th Order Method - Differential Equations

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