C     EEE2022-Ex4

! U(1) = u(x,t)
! U(2) = w(x,t) [Forward Integral from 0 to x]

C-----------------------------------------------------------------------
      SUBROUTINE F(T, X, U, UX, UXX, FVAL, NPDE)
C-----------------------------------------------------------------------
C PURPOSE:
C       THIS SUBROUTINE DEFINES THE RIGHT HAND SIDE VECTOR OF THE
C       NPDE DIMENSIONAL PARABOLIC PARTIAL DIFFERENTIAL EQUATION
C                        UT = F(T, X, U, UX, UXX).
C
C-----------------------------------------------------------------------
        IMPLICIT NONE
C SUBROUTINE PARAMETERS:
C INPUT:
        INTEGER                 NPDE
C                               THE NUMBER OF PDES IN THE SYSTEM.
C
        DOUBLE PRECISION        T
C                               THE CURRENT TIME COORDINATE.
C
        DOUBLE PRECISION        X
C                               THE CURRENT SPATIAL COORDINATE.
C
        DOUBLE PRECISION        U(NPDE)
C                               U(1:NPDE) IS THE APPROXIMATION OF THE
C                               SOLUTION AT THE POINT (T,X).
C
        DOUBLE PRECISION        UX(NPDE)
C                               UX(1:NPDE) IS THE APPROXIMATION OF THE
C                               SPATIAL DERIVATIVE OF THE SOLUTION AT
C                               THE POINT (T,X).
C
        DOUBLE PRECISION        UXX(NPDE)
C                               UXX(1:NPDE) IS THE APPROXIMATION OF THE
C                               SECOND SPATIAL DERIVATIVE OF THE
C                               SOLUTION AT THE POINT (T,X).
C
C OUTPUT:
        DOUBLE PRECISION        FVAL(NPDE)
C                               FVAL(1:NPDE) IS THE RIGHT HAND SIDE
C                               VECTOR F(T, X, U, UX, UXX) OF THE PDE.
C-----------------------------------------------------------------------
C
C     ASSIGN FVAL(1:NPDE) ACCORDING TO THE RIGHT HAND SIDE OF THE PDE
C     IN TERMS OF U(1:NPDE), UX(1:NPDE), UXX(1:NPDE).
C
C     Identify solution components: 
C     u = U(1), w = U(2)

c     Heat Equation
      FVAL(1) = UXX(1) + EXP(T) * (X**2 - 2.0D0)

C     ODE for w: w_x = u, in residual form
      FVAL(2) = UX(2)-U(1)

      RETURN
      END

      SUBROUTINE BNDXA(T, U, UX, BVAL, NPDE)
C-----------------------------------------------------------------------
C PURPOSE:
C       THE SUBROUTINE IS USED TO DEFINE THE BOUNDARY CONDITIONS AT THE
C       LEFT SPATIAL END POINT X = XA.
C                           B(T, U, UX) = 0
C
C-----------------------------------------------------------------------
        IMPLICIT NONE

C SUBROUTINE PARAMETERS:
C INPUT:
        INTEGER                 NPDE
C                               THE NUMBER OF PDES IN THE SYSTEM.
C
        DOUBLE PRECISION        T
C                               THE CURRENT TIME COORDINATE.
C
        DOUBLE PRECISION        U(NPDE)
C                               U(1:NPDE) IS THE APPROXIMATION OF THE
C                               SOLUTION AT THE POINT (T,XA).
C
        DOUBLE PRECISION        UX(NPDE)
C                               UX(1:NPDE) IS THE APPROXIMATION OF THE
C                               SPATIAL DERIVATIVE OF THE SOLUTION AT
C                               THE POINT (T,XA).
C
C OUTPUT:
        DOUBLE PRECISION        BVAL(NPDE)
C                               BVAL(1:NPDE) IS THE BOUNDARY CONTIDITION
C                               AT THE LEFT BOUNDARY POINT.
c-----------------------------------------------------------------------

C       Identify solution components: 
C       u = U(1), w = U(2)

C       Left BC of PDE: u = 0
        BVAL(1) = U(1) 

C       Left BC for w: w = 0
        BVAL(2) = U(2)

      RETURN
      END

      SUBROUTINE BNDXB(T, U, UX, BVAL, NPDE)
C-----------------------------------------------------------------------
C PURPOSE:
C       THE SUBROUTINE IS USED TO DEFINE THE BOUNDARY CONDITIONS AT THE
C       RIGHT SPATIAL END POINT X = XB.
C                           B(T, U, UX) = 0
C
C-----------------------------------------------------------------------
        IMPLICIT NONE

C SUBROUTINE PARAMETERS:
C INPUT:
        INTEGER                 NPDE
C                               THE NUMBER OF PDES IN THE SYSTEM.
C
        DOUBLE PRECISION        T
C                               THE CURRENT TIME COORDINATE.
C
        DOUBLE PRECISION        U(NPDE)
C                               U(1:NPDE) IS THE APPROXIMATION OF THE
C                               SOLUTION AT THE POINT (T,XB).
C
        DOUBLE PRECISION        UX(NPDE)
C                               UX(1:NPDE) IS THE APPROXIMATION OF THE
C                               SPATIAL DERIVATIVE OF THE SOLUTION AT
C                               THE POINT (T,XB).
C
C OUTPUT:
        DOUBLE PRECISION        BVAL(NPDE)
C                               BVAL(1:NPDE) IS THE BOUNDARY CONTIDITION
C                               AT THE RIGHT BOUNDARY POINT.
C-----------------------------------------------------------------------
 

C       Identify solution components: 
C       u = U(1), w = U(2)

C       Residual form of right BC of PDE:  w = target
        BVAL(1) = U(2) - EXP(T)/3.0D0 

C       There is no right BC for w so use the ODE itself:
C       Residual form of w_x = u, evaluated at x=1
        BVAL(2) = UX(2)-U(1)         

      RETURN
      END

      SUBROUTINE UINIT(X, U, NPDE)
C-----------------------------------------------------------------------
C PURPOSE:
C       THIS SUBROUTINE IS USED TO RETURN THE NPDE-VECTOR OF INITIAL
C       CONDITIONS OF THE UNKNOWN FUNCTION AT THE INITIAL TIME T = T0
C       AT THE SPATIAL COORDINATE X.
C
C-----------------------------------------------------------------------
        IMPLICIT NONE

C SUBROUTINE PARAMETERS:
C INPUT:
        DOUBLE PRECISION        X
C                               THE SPATIAL COORDINATE.
C
        INTEGER                 NPDE
C                               THE NUMBER OF PDES IN THE SYSTEM.
C
C OUTPUT:
        DOUBLE PRECISION        U(NPDE)
C                               U(1:NPDE) IS VECTOR OF INITIAL VALUES OF
C                               THE UNKNOWN FUNCTION AT T = T0 AND THE
C                               GIVEN VALUE OF X.
        REAL(8), PARAMETER :: PI = 4.d0*datan(1.d0)

c-----------------------------------------------------------------------

C
C     ASSIGN U(1:NPDE) THE INITIAL VALUES OF U(T0,X).
C
C     Initial Condition for u
      U(1) = X**2

C     Initial Condition for w
      U(2) = (X**3) / 3.0D0
C
      RETURN
      END

      SUBROUTINE TRUU(T, X, U, NPDE)
C-----------------------------------------------------------------------
C PURPOSE:
C     THIS FUNCTION PROVIDES THE EXACT SOLUTIONS
C-----------------------------------------------------------------------
        IMPLICIT NONE

C SUBROUTINE PARAMETERS:
C INPUT:
        INTEGER                 NPDE
C                               THE NUMBER OF PDES IN THE SYSTEM.
C
        DOUBLE PRECISION        T
C                               THE CURRENT TIME COORDINATE.
C
        DOUBLE PRECISION        X
C                               THE CURRENT SPATIAL COORDINATE.
C
C OUTPUT:
        DOUBLE PRECISION        U(NPDE)
C                               U(1:NPDE) IS THE EXACT SOLUTION AT THE
C                               POINT (T,X).
        REAL(8), PARAMETER :: PI = 4.d0*datan(1.d0)

C-----------------------------------------------------------------------

C       Exact solution for u
        U(1) = EXP(T) * (X**2)

C       Exact solution for w
        U(2) = EXP(T) * (X**3 / 3.0D0)

      RETURN
      END