Numerical Analysis Problems and Solutions

Boundary Value Problem

Solve boundary value problem of the form

1. Collocation method

subroutine collocation_method(x_min, x_max, n, solution, solution_ext, f, a, p, q, basic, &
            x_points, is_print, is_draw, info)

2. Integral Least Squares method

subroutine int_least_squares_method(x_min, x_max, n, solution, solution_ext, f, a, p, q, basic, &
            x_points, is_print, is_draw, info)

3. Discrete Least Squares method

subroutine disc_least_squares_method(x_min, x_max, n, solution, solution_ext, f, a, p, q, basic, &
            x_points, is_print, is_draw, info)

4. Galerkin method

subroutine galerkin_method(x_min, x_max, n, solution, solution_ext, f, a, p, q, basic, &
            x_points, is_print, is_draw, info)

5. Finite Difference method

subroutine finite_difference_method(AA, BB, x_min, x_max, n, f, a, p, q, alpha1, beta1, &
            alpha2, beta2, y_coef, y_coef_0, y_coef_n, x_grid, sol, is_print, is_draw, info)

Examples

• example1

• example2

• example3

• example4

Heat Transfer

Estimate the solution of the time-dependent (time-independent) heat equation over a one dimensional region.

Time-independent form:

Time-dependent form:

1. Finite Difference methods (1D Steady State Heat Equation)

subroutine fd_centered_heat_transfer_steady(n, a, b, Ua, Ub, k, f, x_grid, sol, &
            is_print, is_draw, info)

subroutine fd_balance_heat_transfer_steady(n, a, b, Ua, Ub, k, f, x_points, sol, &
            is_print, is_draw, info, psources_)

2. Finite Difference method (Time Dependent 1D Heat Equation using Explicit Time Stepping)

subroutine fd_heat_transfer_explicit(n, m, a, b, t, g1, g2, phi, k, f, x_list, t_list, &
            matrix, is_check_cfl, bc, is_print, is_draw, info)

3. Finite Difference method (Time Dependent 1D Heat Equation using Implicit Time Stepping)

subroutine fd_heat_transfer_implicit(n, m, a, b, t, g1, g2, phi, k, f, x_list, t_list, &
            matrix, bc, is_print, is_draw, info)

Examples

1. Modeling of steady-state thermal conductivity processes.

• example1
• example2

2. Modeling of unsteady-state thermal conductivity processes.

• example3
• example4
• example5