Numerical Solution of Boundary Value Problems for ODE (requires further development)

This program solves a second‑order boundary value problem (BVP) of the form

y'' = f(x, y, y'),    x ∈ [a, b],
y(a) = q_a,   y(b) = q_b,

on the interval [0, 1] with arbitrary boundary conditions q_a, q_b ∈ [-7, 7]. The right‑hand side is chosen as:

• Linear case: f(x, y, y') = y'·cos(x) + y·x³ + tan(x)
• Nonlinear case: f(x, y, y') = (y')²·cos(x) + y²·x³ + tan(x)

Four numerical methods are implemented and compared:

• Linear problem:
  • Finite difference method with tridiagonal solver (Thomas algorithm)
  • Shooting method (using 4th‑order Runge–Kutta for the auxiliary Cauchy problems)
• Nonlinear problem:
  • Newton's method (for solving the nonlinear shooting equation)
  • Secant method (for solving the nonlinear shooting equation)

The accuracy is estimated by comparing solutions on two grids (step h and h/2). Absolute and relative errors are computed in three vector norms: L₁, L₂ and L∞.

Requirements

• C++ compiler with C++17 support (e.g., g++, clang++)
• GNU Make
• Python 3 with matplotlib and numpy (for visualisation)

Quick Start

git clone <your-repo-url>
cd <project-folder>
make          # compile the program
make run      # compile and run

Alternatively, you can run manually:

make
./cauchy_boundary

The program will:

1. Solve the linear problem using the finite difference method (with tridiagonal solver) and the shooting method.
2. Solve the nonlinear problem using Newton's method and the secant method.
3. Print a table of errors (comparing h and h/2) for each method.
4. Generate parameters.txt with all numerical solutions.
5. Launch create_plot.py to produce two plots:
   • plot_linear.png – comparison of finite difference and shooting methods for the linear problem.
   • plot_nonlinear.png – comparison of Newton and secant methods for the nonlinear problem.

To clean up build artifacts:

make clean

Example Output

After running ./cauchy_boundary, the console shows a table similar to the following (values are illustrative):

         paragraph 5 - printing errors for y for h / 2 and h
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|          |   abs 1   |   abs 2   |  abs inf  |   rel 1   |   rel 2   |  rel inf  |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
| fin diff | 1.234e-05 | 1.567e-05 | 1.890e-05 | 2.345e-05 | 2.678e-05 | 2.901e-05 |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|   shoot  | 3.456e-06 | 4.567e-06 | 5.678e-06 | 6.789e-06 | 7.890e-06 | 8.901e-06 |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|  newtons | 1.234e-06 | 1.567e-06 | 1.890e-06 | 2.345e-06 | 2.678e-06 | 2.901e-06 |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|  secants | 1.234e-06 | 1.567e-06 | 1.890e-06 | 2.345e-06 | 2.678e-06 | 2.901e-06 |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

(The exact values depend on the chosen boundary conditions and step sizes. The finite difference method typically shows higher error due to its lower order, while shooting and Newton/secant methods provide more accurate results.)

Results

The program automatically generates two plots:

• Linear problem – comparison of finite difference (green) and shooting (purple) methods.
• Nonlinear problem – comparison of Newton (blue) and secant (brown) methods.

Figure 1: Numerical solutions for the linear BVP.

Figure 2: Numerical solutions for the nonlinear BVP.

(Replace the placeholder images with actual screenshots after running the program.)

Project Structure

.
├── Makefile
├── README.md
├── .gitignore
├── LICENSE
├── create_plot.py          # Python script for visualisation
├── parameters.txt          # generated data (ignored by Git)
├── src/
│   ├── main.cpp
│   ├── functions.cpp
│   └── functions.h
├── docs/
│   └── Kraevye_zadachi.pdf
└── build/                  # object files (ignored by Git)

Author

Name: @g30613740 Philip K.

Year: 2025

License

This project is distributed under the MIT License. See the LICENSE file for details.