restart:

# WAVE EQUATION FINITE-DIFFERENCE ALGORITHM 12.4

# To approximate the solution to the wave equation:

# subject to the boundary conditions

# u(0,t) = u(l,t) = 0, 0 < t < T = max t

# and the initial conditions

# u(x,0) = F(x) and Du(x,0)/Dt = G(x), 0 <= x <= l:

# INPUT: endpoint l: maximum time T: constant ALPHA: integers m, N.

# OUTPUT: approximations W(I,J) to u(x(I),t(J)) for each I = 0, ..., m

# and J=0,...,N.

print(`This is the Finite-Difference Method for the Wave Equation.`):

print(`Input the functions F(X) and G(X) in terms of x, separated by a

space.`):

print(`For example: sin(3.141592654*x) 0`):

F := scanf(`%a`)[1]:

G := scanf(`%a`)[1]:

print(`F(x) = `):print(F):print(`G(x) = `):print(G):

F := unapply(F,x):

G := unapply(G,x):

print(`The lefthand endpoint on the X-axis is 0.`):

OK := FALSE:

while OK = FALSE do

print(`Input the righthand endpoint on the X-axis.`):

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

print(`Righthand endpoint = `):print(FX):

if FX <= 0 then

print(`Must be a positive number.`):

else

OK := TRUE:

fi:

od:

OK := FALSE:

while OK = FALSE do

print(`Input the maximum value of the time variable T.`):

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

print(`Maximum time value = `):print(FT):

if FT <= 0 then

print(`Must be a positive number.\134n`):

else

OK := TRUE:

fi:

od:

print(`Input the constant alpha.`):

ALPHA := scanf(`%f`)[1]:print(`alpha = `):print(ALPHA):

OK := FALSE:

while OK = FALSE do

print(`Input integer m := number of intervals on X-axis`):

print(`and N := number of time intervals - separated by a blank.`):

print(`Note that m must be 3 or larger.`):

M := scanf(`%d`)[1]:

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

print(`Number of intervals on x-axis m = `):print(M):

print(`Number of time intervals n = `):print(N):

if M <= 2 or N <= 0 then

print(`Numbers are not within correct range.`):

else

OK := TRUE:

fi:

od:

if OK = TRUE then

M1 := M+1:

M2 := M-1:

N1 := N+1:

N2 := N-1:

# Step 1

# V is used in place of lambda

H := FX/M:

K := FT/N:

V := ALPHA*K/H:

# Step 2

for J from 2 to N1 do

W[0,J-1] := 0:

W[M1-1,J-1] := 0:

od:

# Step 3

W[0,0] := evalf(F(0)):

W[M1-1,0] := evalf(F(FX)):

# Step 4

for I2 from 2 to M do

W[I2-1,0] := F(H*(I2-1)):

W[I2-1,1] := (1-V^2)*F(H*(I2-1))+V^2*(F(I2*H)+F(H*(I2-2)))/2+K*G(H*(I2-1)):

od:

# Step 5

for J from 2 to N do

for I2 from 2 to M do

W[I2-1,J] :=

evalf(2*(1-V^2)*W[I2-1,J-1]+V^2*(W[I2,J-1]+W[I2-2,J-1])-W[I2-1,J-2]):

od:

# Step 6

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(`Input is `):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(`Output file is `):print(NAME):

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

else

OUP := default:

fi:

fprintf(OUP, `FINITE DIFFERENCE METHOD FOR THE WAVE EQUATION\134n\134n`):

fprintf(OUP, ` I X(I) W(X(I),%12.6e)\134n`, FT):

for I2 from 1 to M1 do

X := (I2-1)*H:

fprintf(OUP, `%3d %11.8f %13.8f\134n`, I2, X, W[I2-1,N1-1]):

od:

if OUP <> default then

fclose(OUP):

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

fi:

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IiIi

FINITE DIFFERENCE METHOD FOR THE WAVE EQUATION I X(I) W(X(I),1.000000e+00) 1 0.00000000 0.00000000 2 0.10000000 0.30901699 3 0.20000000 0.58778525 4 0.30000000 0.80901699 5 0.40000000 0.95105652 6 0.50000000 1.00000000 7 0.60000000 0.95105652 8 0.70000000 0.80901699 9 0.80000000 0.58778525 10 0.90000000 0.30901699 11 1.00000000 0.00000000

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