/*
*   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");
   printf("Please enter 1 or 2.\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);
}