alg124.txt

Numerical Anaysis 8th Edition Burden and Faires (Maple Source)

> 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.
> alg124 := proc() local F, G, OK, FX, FT, ALPHA, M, N, M1, M2, N1, N2, H, K, V, J, W, I, flag, NAME, OUP, X;
> printf(`This is the Finite-Difference Method for the Wave Equation.\n`);
> printf(`Input the functions F(X) and G(X) in terms of x, separated by a 
> space.\n`);
> printf(`For example:  sin(3.141592654*x)  0\n`);
> F := scanf(`%a`)[1];
> G := scanf(`%a`)[1];
> F := unapply(F,x);
> G := unapply(G,x);
> printf(`The lefthand endpoint on the X-axis is 0.\n`);
> OK := FALSE;
> while OK = FALSE do
> printf(`Input the righthand endpoint on the X-axis.\n`);
> FX := scanf(`%f`)[1];
> if FX <= 0 then
> printf(`Must be a positive number.\n`);
> else
> OK := TRUE;
> fi;
> od;
> OK := FALSE;
> while OK = FALSE do
> printf(`Input the maximum value of the time variable T.\n`);
> FT := scanf(`%f`)[1];
> if FT <= 0 then
> printf(`Must be a positive number.\n`);
> else
> OK := TRUE;
> fi;
> od;
> printf(`Input the constant alpha.\n`);
> ALPHA := scanf(`%f`)[1];
> OK := FALSE;
> while OK = FALSE do
> printf(`Input integer m := number of intervals on X-axis\n`);
> printf(`and N := number of time intervals - separated by a blank.\n`);
> printf(`Note that m must be 3 or larger.\n`);
> M := scanf(`%d`)[1];
> N := scanf(`%d`)[1];
> if M <= 2 or N <= 0 then
> printf(`Numbers are not within correct range.\n`);
> 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 I from 2 to M do
> W[I-1,0] := F(H*(I-1));
> W[I-1,1] := (1-V^2)*F(H*(I-1))+V^2*(F(I*H)+F(H*(I-2)))/2+K*G(H*(I-1));
> od;
> # Step 5
> for J from 2 to N do
> for I from 2 to M do
> W[I-1,J] := 
> evalf(2*(1-V^2)*W[I-1,J-1]+V^2*(W[I,J-1]+W[I-2,J-1])-W[I-1,J-2]);
> od;
> od;
> # Step 6
> printf(`Choice of output method:\n`);
> printf(`1. Output to screen\n`);
> printf(`2. Output to text file\n`);
> printf(`Please enter 1 or 2.\n`);
> FLAG := scanf(`%d`)[1];
> if FLAG = 2 then
> printf(`Input the file name in the form - drive:\\name.ext\n`);
> printf(`for example:  A:\\OUTPUT.DTA\n`);
> NAME := scanf(`%s`)[1];
> OUP := fopen(NAME,WRITE,TEXT);
> else
> OUP := default;
> fi;
> fprintf(OUP, `FINITE DIFFERENCE METHOD FOR THE WAVE EQUATION\n\n`);
> fprintf(OUP, `  I    X(I)     W(X(I),%12.6e)\n`, FT);
> for I from 1 to M1 do
> X := (I-1)*H;
> fprintf(OUP, `%3d %11.8f %13.8f\n`, I, X, W[I-1,N1-1]);
> od;
> if OUP <> default then
> fclose(OUP):
> printf(`Output file %s created successfully`,NAME);
> fi;
> fi;
> # Step 7
> RETURN(0);
> end;
> alg124();