restart:

# RUNGE-KUTTA FOR SYSTEMS OF DIFFERENTIAL EQUATIONS ALGORITHM 5.7

# TO APPROXIMATE THE SOLUTION OF THE MTH-ORDER SYSTEM OF FIRST-

# ORDER INITIAL-VALUE PROBLEMS

# UJ` = FJ( T, U1, U2, ..., UM ), J = 1, 2, ..., M

# A <= T <= B, UJ(A) = ALPHAJ, J = 1, 2, ..., M

# AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].

# INPUT: ENDPOINTS A,B: NUMBER OF EQUATIONS M: INITIAL

# CONDITIONS ALPHA1, ..., ALPHAM: INTEGER N.

# OUTPUT: APPROXIMATION WJ TO UJ(T) AT THE (N+1) VALUES OF T.

print(`This is the Runge-Kutta Method for Systems of m equations`):

OK := FALSE:

while OK = FALSE do

print(`Input the number of equations`):

M := scanf(`%d`)[1]:print(`Number of equations is `):print(M):

if M <= 0 then

print(`Number must be a positive integer\134n`):

else

OK := TRUE:

fi:

od:

for I1 from 1 to M do

print(`Input the function f[i](t,y1 ... yM) in terms of t, y1, ... for i = `):print(I1):

print(`For example: y1-t^2+1`):

F[I1] := scanf(`%a`)[1]:print(`function is `):print(F[I1]):

F[I1] := unapply(F[I1],t,evaln(y || (1..M))):

od:

OK := FALSE:

while OK = FALSE do

print(`Input left and right endpoints separated by blank`):

A := scanf(`%f`)[1]:

B := scanf(`%f`)[1]:print(`a = `):print(A): print(`b = `):print(B):

if A >= B then

print(`Left endpoint must be less than right endpoint`):

else

OK := TRUE:

fi:

od:

for I1 from 1 to M do

print(`Input the initial condition for i = `):print(I1):

ALPHA[I1] := scanf(`%f`)[1]:print(`initial condition is `):print(ALPHA[I1]):

od:

OK := FALSE:

while OK = FALSE do

print(`Input a positive integer for the number of subintervals`):

N := scanf(`%d`)[1]:print(`N = `):print(N):

if N <= 0 then

print(`Number must be a positive integer`):

else

OK := TRUE:

fi:

od:

if OK = TRUE then

print(`Choice of output method:`):

print(`1. Output to screen`):

print(`2. Output to text file`):

print(`Please enter 1 or 2`):

FLAG := scanf(`%d`)[1]:print(`Entry = `):print(FLAG):

if FLAG = 2 then

print(`Input the file name in the form - drive:\134\134name.ext`):

print(`For example A:\134\134OUTPUT.DTA`):

NAME := scanf(`%s`)[1]:print(`File name is `):print(NAME):

OUP := fopen(NAME,WRITE,TEXT):

else

OUP := default:

fi:

fprintf(OUP, `RUNGE-KUTTA METHOD FOR SYSTEMS OF DIFFERENTIAL EQUATIONS\134n\134n`):

fprintf(OUP, ` T`):

for I1 from 1 to M do

fprintf(OUP, ` W%d`, I1):

od:

# Step 1

H := (B-A)/N:

T := A:

# Step 2

for J from 1 to M do

W[J] := ALPHA[J]:

od:

# Step 3

fprintf(OUP, `\134n%5.3f`, T):

for I1 from 1 to M do

fprintf(OUP, ` %11.8f`, W[I1]):

od:

fprintf(OUP, `\134n`):

# Step 4

for L from 1 to N do

# Step 5

for J from 1 to M do

K[1,J] := H*F[J](T,seq(W[i1],i1=1..M)):

od:

# Step 6

for J from 1 to M do

K[2,J] := H*F[J](T+H/2.0,seq(W[i1]+K[1,i1]/2.0,i1=1..M)):

od:

# Step 7

for J from 1 to M do

K[3,J] := H*F[J](T+H/2.0,seq(W[i1]+K[2,i1]/2.0,i1=1..M)):

od:

# Step 8

for J from 1 to M do

K[4,J] := H*F[J](T+H,seq(W[i1]+K[3,i1],i1=1..M)):

od:

# Step 9

for J from 1 to M do

W[J] := W[J]+(K[1,J]+2.0*K[2,J]+2.0*K[3,J]+K[4,J])/6.0:

od:

# Step 10

T := A+L*H:

# Step 11

fprintf(OUP, `%5.3f`, T):

for I1 from 1 to M do

fprintf(OUP, ` %11.8f`, W[I1]):

od:

fprintf(OUP, `\134n`):

od:

# Step 12

if OUP <> default then

fclose(OUP):

print(`Output file`,NAME,` created successfully`):

fi:

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RUNGE-KUTTA METHOD FOR SYSTEMS OF DIFFERENTIAL EQUATIONS T W1 W2 0.000 0.00000000 0.00000000 0.100 0.53825520 0.31962624 0.200 0.96849874 0.56878217 0.300 1.31071904 0.76073313 0.400 1.58126524 0.90632062 0.500 1.79350749 1.01440242

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