3 December 2023 2:40:33.057 PM fem1d_nonlinear(): Fortran77 version. Solve a nonlinear boundary value problem: -d/dx (p(x) du/dx) + q(x)*u + u*u' = f(x) on an interval [xl,xr], with the values of u or u' specified at xl and xr. Solving problem # 1 The equation is to be solved for X greater than XL = 0.0000000000000000 and less than XR = 1.0000000000000000 The boundary conditions are: At X=XL, U= 0.0000000000000000 At X=XR, U'= 1.0000000000000000 This is test problem #1: P(X) = 1, Q(X) = 0, F(X) = X. Boundary conditions: U(0) = 0, U'(1) = 1. The exact solution is U(X) = X Number of quadrature points per element is 1 Number of iterations is 10 Node Location 0 0.00000 1 0.100000 2 0.200000 3 0.300000 4 0.400000 5 0.500000 6 0.600000 7 0.700000 8 0.800000 9 0.900000 10 1.00000 Subint Length 1 0.100000 2 0.100000 3 0.100000 4 0.100000 5 0.100000 6 0.100000 7 0.100000 8 0.100000 9 0.100000 10 0.100000 Subint Quadrature point 1 0.500000E-01 2 0.150000 3 0.250000 4 0.350000 5 0.450000 6 0.550000 7 0.650000 8 0.750000 9 0.850000 10 0.950000 Subint Left Node Right Node 1 0 1 2 1 2 3 2 3 4 3 4 5 4 5 6 5 6 7 6 7 8 7 8 9 8 9 10 9 10 Node Unknown 0 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 Printout of tridiagonal linear system: Equation ALEFT ADIAG ARITE RHS 1 20.0000 -10.0000 0.100000E-01 2 -10.0000 20.0000 -10.0000 0.200000E-01 3 -10.0000 20.0000 -10.0000 0.300000E-01 4 -10.0000 20.0000 -10.0000 0.400000E-01 5 -10.0000 20.0000 -10.0000 0.500000E-01 6 -10.0000 20.0000 -10.0000 0.600000E-01 7 -10.0000 20.0000 -10.0000 0.700000E-01 8 -10.0000 20.0000 -10.0000 0.800000E-01 9 -10.0000 20.0000 -10.0000 0.900000E-01 10 -10.0000 10.0000 1.04750 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.149750 2 0.200000 0.298500 3 0.300000 0.445250 4 0.400000 0.589000 5 0.500000 0.728750 6 0.600000 0.863500 7 0.700000 0.992250 8 0.800000 1.11400 9 0.900000 1.22775 10 1.00000 1.33250 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.849454E-01 2 0.200000 0.170164 3 0.300000 0.255932 4 0.400000 0.342534 5 0.500000 0.430266 6 0.600000 0.519437 7 0.700000 0.610377 8 0.800000 0.703436 9 0.900000 0.798989 10 1.00000 0.897438 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.105202 2 0.200000 0.210298 3 0.300000 0.315181 4 0.400000 0.419746 5 0.500000 0.523887 6 0.600000 0.627501 7 0.700000 0.730483 8 0.800000 0.832732 9 0.900000 0.934149 10 1.00000 1.03464 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100039 2 0.200000 0.200077 3 0.300000 0.300111 4 0.400000 0.400140 5 0.500000 0.500163 6 0.600000 0.600179 7 0.700000 0.700189 8 0.800000 0.800194 9 0.900000 0.900195 10 1.00000 1.00020 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.100000 2 0.200000 0.200000 3 0.300000 0.300000 4 0.400000 0.400000 5 0.500000 0.500000 6 0.600000 0.600000 7 0.700000 0.700000 8 0.800000 0.800000 9 0.900000 0.900000 10 1.00000 1.00000 Compare computed and exact solutions: X Computed U Exact U 0.00000 0.00000 0.00000 0.125000 0.125000 0.125000 0.250000 0.250000 0.250000 0.375000 0.375000 0.375000 0.500000 0.500000 0.500000 0.625000 0.625000 0.625000 0.750000 0.750000 0.750000 0.875000 0.875000 0.875000 1.00000 1.00000 1.00000 Solving problem # 2 The equation is to be solved for X greater than XL = 0.0000000000000000 and less than XR = 1.0000000000000000 The boundary conditions are: At X=XL, U= 0.0000000000000000 At X=XR, U'= 1.0000000000000000 This is test problem #2: P(X) = 1, Q(X) = 0, F(X) = -0.5*pi*cos(0.5*pi*X) + 2*sin(0.5*pi*X)*(1-cos(0.5*pi*X)/pi. Boundary conditions: U(0) = 0, U'(1) = 1. The exact solution is U(X) = 2*(1-cos(pi*x/2))/pi Number of quadrature points per element is 1 Number of iterations is 10 Node Location 0 0.00000 1 0.100000 2 0.200000 3 0.300000 4 0.400000 5 0.500000 6 0.600000 7 0.700000 8 0.800000 9 0.900000 10 1.00000 Subint Length 1 0.100000 2 0.100000 3 0.100000 4 0.100000 5 0.100000 6 0.100000 7 0.100000 8 0.100000 9 0.100000 10 0.100000 Subint Quadrature point 1 0.500000E-01 2 0.150000 3 0.250000 4 0.350000 5 0.450000 6 0.550000 7 0.650000 8 0.750000 9 0.850000 10 0.950000 Subint Left Node Right Node 1 0 1 2 1 2 3 2 3 4 3 4 5 4 5 6 5 6 7 6 7 8 7 8 9 8 9 10 9 10 Node Unknown 0 -1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 Printout of tridiagonal linear system: Equation ALEFT ADIAG ARITE RHS 1 20.0000 -10.0000 -0.154454 2 -10.0000 20.0000 -10.0000 -0.147799 3 -10.0000 20.0000 -10.0000 -0.136149 4 -10.0000 20.0000 -10.0000 -0.119284 5 -10.0000 20.0000 -10.0000 -0.972917E-01 6 -10.0000 20.0000 -10.0000 -0.706000E-01 7 -10.0000 20.0000 -10.0000 -0.399792E-01 8 -10.0000 20.0000 -10.0000 -0.651058E-02 9 -10.0000 20.0000 -10.0000 0.284722E-01 10 -10.0000 10.0000 1.02308 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.279485E-01 2 0.200000 0.713424E-01 3 0.300000 0.129516 4 0.400000 0.201305 5 0.500000 0.285022 6 0.600000 0.378468 7 0.700000 0.478975 8 0.800000 0.583479 9 0.900000 0.688634 10 1.00000 0.790942 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.148389E-02 2 0.200000 0.184564E-01 3 0.300000 0.504114E-01 4 0.400000 0.965231E-01 5 0.500000 0.155664 6 0.600000 0.226426 7 0.700000 0.307153 8 0.800000 0.395967 9 0.900000 0.490808 10 1.00000 0.589466 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.978230E-02 2 0.200000 0.350230E-01 3 0.300000 0.751252E-01 4 0.400000 0.129110 5 0.500000 0.195640 6 0.600000 0.273059 7 0.700000 0.359430 8 0.800000 0.452588 9 0.900000 0.550198 10 1.00000 0.649816 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.777244E-02 2 0.200000 0.310140E-01 3 0.300000 0.691526E-01 4 0.400000 0.121249 5 0.500000 0.186017 6 0.600000 0.261860 7 0.700000 0.346904 8 0.800000 0.439050 9 0.900000 0.536021 10 1.00000 0.635418 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Computed solution: Node X(I) U(X(I)) 0 0.00000 0.00000 1 0.100000 0.776462E-02 2 0.200000 0.309987E-01 3 0.300000 0.691306E-01 4 0.400000 0.121221 5 0.500000 0.185985 6 0.600000 0.261824 7 0.700000 0.346867 8 0.800000 0.439012 9 0.900000 0.535982 10 1.00000 0.635380 Compare computed and exact solutions: X Computed U Exact U 0.00000 0.00000 0.00000 0.125000 0.135731E-01 0.122325E-01 0.250000 0.500647E-01 0.484598E-01 0.375000 0.108198 0.107290 0.500000 0.185985 0.186462 0.625000 0.283085 0.282933 0.750000 0.392940 0.392996 0.875000 0.511740 0.512421 1.00000 0.635380 0.636620 fem1d_nonlinear(): Normal end of execution. 3 December 2023 2:40:33.057 PM