# Forward difference method for heat equation

1. Apr 3, 2009

### ahmedo047

I don't be able to convert the following code(HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM in the Burden-faires numerical analysis book).I need HEAT EQUATION FORWARD-DIFFERENCE ALGORITHM C like following code.I dont be able to convert FORWARD-DIFFERENCE the following code .Please help me.

Code (Text):
/*
*   HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM 12.2
*
*   To approximate the solution to the parabolic partial-differential
*   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), 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 = 1, ..., m-1 and J = 1, ..., N.
*/

#include<stdio.h>
#include<math.h>
#define pi 4*atan(1)
#define true 1
#define false 0

double F(double X);
void INPUT(int *, double *, double *, double *, int *, int *);
void OUTPUT(double, double, int, double *, double);

main()
{
double W[25], L[25], U[25], Z[25];
double FT,FX,ALPHA,H,K,VV,T,X;
int N,M,M1,M2,N1,FLAG,I1,I,J,OK;

INPUT(&OK, &FX, &FT, &ALPHA, &N, &M);
if (OK) {
M1 = M - 1;
M2 = M - 2;
N1 = N - 1;
/* STEP 1 */
H = FX / M;
K = FT / N;
VV = ALPHA * ALPHA * K / ( H * H );
/* STEP 2 */
for (I=1; I<=M1; I++) W[I-1] = F( I * H );
/* STEP 3 */
/* STEPS 3 through 11 solve a tridiagonal linear system
using Algorithm 6.7 */
L[0] = 1.0 + 2.0 * VV;
U[0] = -VV / L[0];
/* STEP 4 */
for (I=2; I<=M2; I++) {
L[I-1] = 1.0 + 2.0 * VV + VV * U[I-2];
U[I-1] = -VV / L[I-1];
}
/* STEP 5 */
L[M1-1] = 1.0 + 2.0 * VV + VV * U[M2-1];
/* STEP 6 */
for (J=1; J<=N; J++) {
/* STEP 7 */
/* current t(j) */
T = J * K;
Z[0] = W[0] / L[0];
/* STEP 8 */
for (I=2; I<=M1; I++)
Z[I-1] = ( W[I-1] + VV * Z[I-2] ) / L[I-1];
/* STEP 9 */
W[M1-1] = Z[M1-1];
/* STEP 10 */
for (I1=1; I1<=M2; I1++) {
I = M2 - I1 + 1;
W[I-1] = Z[I-1] - U[I-1] * W[I];
}
}
/* STEP 11 */
OUTPUT(FT, X, M1, W, H);
}
/* STEP 12 */
return 0;
}

/* Change F for a new problem */
double F(double X)
{
double f;

f =  sin(pi * X);
return f;
}

void INPUT(int *OK, double *FX, double *FT, double *ALPHA, int *N, int *M)
{
int FLAG;
char AA;

printf("This is the Backward-Difference Method for Heat Equation.\n");
printf("Has the function F been created immediately\n");
printf("preceding the INPUT procedure? Answer Y or N.\n");
scanf("\n%c", &AA);
if ((AA == 'Y') || (AA == 'y')) {
printf("The lefthand endpoint on the X-axis is 0.\n");
*OK =false;
while (!(*OK)) {
printf("Input the righthand endpoint on the X-axis.\n");
scanf("%lf", FX);
if (*FX <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
*OK = false;
while (!(*OK)) {
printf("Input the maximum value of the time variable T.\n");
scanf("%lf", FT);
if (*FT <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
printf("Input the constant alpha.\n");
scanf("%lf", ALPHA);
*OK = false;
while (!(*OK)) {
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");
scanf("%d %d", M, N);
if ((*M <= 2) || (*N <= 0))
printf("Numbers are not within correct range.\n");
else *OK = true;
}
}
else {
printf("The program will end so that the function F can be created.\n");
*OK = false;
}
}

void OUTPUT(double FT, double X, int M1, double *W, double H)
{
int I, J, FLAG;
char NAME[30];
FILE *OUP;

printf("Choice of output method:\n");
printf("1. Output to screen\n");
printf("2. Output to text file\n");
scanf("%d", &FLAG);
if (FLAG == 2) {
printf("Input the file name in the form - drive:name.ext\n");
printf("for example:   A:OUTPUT.DTA\n");
scanf("%s", NAME);
OUP = fopen(NAME, "w");
}
else OUP = stdout;
fprintf(OUP, "THIS IS THE BACKWARD-DIFFERENCE METHOD\n\n");
fprintf(OUP, "  I        X(I)    W(X(I),%12.6e)\n", FT);
for (I=1; I<=M1; I++) {
X = I * H;
fprintf(OUP, "%3d %11.8f    %14.8f\n", I, X, W[I-1]);
}
fclose(OUP);
}

2. Apr 6, 2009

### ahmedo047

Isn't there anyone to help me.

if I write VV = - [ALPHA * ALPHA * K / ( H * H )]; instead of VV = ALPHA * ALPHA * K / ( H * H );in the HEAT EQUATION BACKWARD-DIFFERENCE ALGORITHM then have I obtained HEAT EQUATION FORWARD-DIFFERENCE ALGORITHM?

3. May 11, 2009

### ahmedo047

problem is solved:
finally code
Code (Text):
#include<stdio.h>
#include<math.h>
#define pi 4*atan(1)
#define true 1
#define false 0

double F(double X);
void INPUT(int *, double *, double *, double *, int *, int *);
void OUTPUT(double, double, int, double *, double);

main()
{
double W[25], U[25], Z[25];
double FT,FX,ALPHA,H,K,VV,T,X;
int N,M,M1,M2,N1,FLAG,I1,I,J,JJ,OK;

INPUT(&OK, &FX, &FT, &ALPHA, &N, &M);
if (OK) {
M1 = M - 1;
M2 = M - 2;
N1 = N - 1;
/* STEP 1 */
H = FX / M;
K = FT / N;
VV = ALPHA * ALPHA * K / ( H * H );
/* STEP 2 */
W[M]=0;
W[0]=0;
for (I=1; I<=M1; I++) W[I] = F( I * H );
/* STEP 3 */
for(J=1;J<=N;J++)
{ T=J*K;
for (I=1;I<=M1;I++)
U[I]=VV*W[I+1]+(1-2*VV)*W[I]+VV*W[I-1];
for(JJ=1;JJ<=M1;JJ++)
W[JJ]=U[JJ];
}
/* STEP 11 */
OUTPUT(FT, X, M1, U, H);
}
/* STEP 12 */
return 0;
}

/* Change F for a new problem */
double F(double X)
{
double f;

f =  sin(pi * X);
return f;
}

void INPUT(int *OK, double *FX, double *FT, double *ALPHA, int *N, int *M)
{
int FLAG;
char AA;

printf("This is the Backward-Difference Method for Heat Equation.\n");
printf("Has the function F been created immediately\n");
printf("preceding the INPUT procedure? Answer Y or N.\n");
scanf("\n%c", &AA);
if ((AA == 'Y') || (AA == 'y')) {
printf("The lefthand endpoint on the X-axis is 0.\n");
*OK =false;
while (!(*OK)) {
printf("Input the righthand endpoint on the X-axis.\n");
scanf("%lf", FX);
if (*FX <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
*OK = false;
while (!(*OK)) {
printf("Input the maximum value of the time variable T.\n");
scanf("%lf", FT);
if (*FT <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
printf("Input the constant alpha.\n");
scanf("%lf", ALPHA);
*OK = false;
while (!(*OK)) {
printf("Input integer m = number of intervals on X-axis\n");
printf("and N = number of time intervals - separated by a.\n");
printf("Note that m must be 3 or larger.\n");
scanf("%d %d", M, N);
if ((*M <= 2) || (*N <= 0))
printf("Numbers are not within correct range.\n");
else *OK = true;
}
}
else {
printf("The program will end so that the function F can be\n");
*OK = false;
}
}

void OUTPUT(double FT, double X, int M1, double *W, double H)
{
int I, J, FLAG;
char NAME[30];
FILE *OUP;

printf("Choice of output method:\n");
printf("1. Output to screen\n");
printf("2. Output to text file\n");
scanf("%d", &FLAG);
if (FLAG == 2) {
printf("Input the file name in the form - drive:name.ext\n");
printf("for example:   A:OUTPUT.DTA\n");
scanf("%s", NAME);
OUP = fopen(NAME, "w");
}
else OUP = stdout;
fprintf(OUP, "THIS IS THE BACKWARD-DIFFERENCE METHOD\n\n");
fprintf(OUP, "  I        X(I)    W(X(I),%12.6e)\n", FT);
for (I=1; I<=M1; I++) {
X = I * H;
fprintf(OUP, "%3d %11.8f    %14.8f\n", I, X, W[I]);
}
fclose(OUP);
}