zjj0930 / numerical-methods-fortran

Solving linear, nonlinear equations, ordinary differential equations, ... using numerical methods in fortran

Numerical methods in fortran

Solving linear, nonlinear equations, integral, ordinary differential equations, ... using numerical methods in fortran

1. Linear equations:

LU
PLU (TODO)
QR (TODO)

2. Non-uniform random number generator

Normal distribution
    - Box–Muller transform
    - Ratio-of-uniforms method

3. Nonlinear equations

Newton-Raphson
Fix point

4. Integration Methods

One-dimensional
    - Rectangle rule.
    - Trapezoidal rule.
    - Simpson's rule.   
    - Gauss-Hermite     (TODO)
    - Gauss-Laguerre    (TODO)

N-dimensional
    - Monte Carlo       (TODO)
    - Sparse grids      (TODO)
    - Bayesian Quadrature (TODO)

5. Ordinary differential equations (ODE)

Monostep
    - Euler explicit
    - Euler implicit (TODO)
    - Runge-Kutta fourth order method (classical)
Multistep
    - Adams-Bashforth
    - Adams-Moulton   (TODO)
    - Nyström         (TODO)
    - Mile-Simpson    (TODO)
    - Backward differentiation formula (TODO)

6. Stochastic Ordinary Differential Equations (SDE)

- Euler-Maruyama method
- Milstein Method (TODO)
- Strong Order 1.0 Runge-Kutta Method 
- Strong Order 1.5 Taylor Method (TODO)
- Weak Order 2 Taylor Method (TODO)
- Weak Order 2 Runge-Kutta Method

Examples

pgplot and dislin libraries are necessary to plot the examples: pgplot, dislin

Example 1: ordinary differential equation

Example 2: Lotka–Volterra 1

Example 3: Lotka–Volterra 2

Example 4: isotherm transesterification reaction at 50 C (link)

Where:

• [TG] Triglycerides.
• [DG] Diglycerides.
• [MG] Monoglycerides.
• [GL] Glycerin.
• [A] Alcohol.
• [E] Ethylester.

Example 5: Bogdanov-Takens bifurcation

Example 6: pendulum

Example 7: Lorenz system

Example 8: Stochastic Ordinary Differential Equation (SDE)

Force Regulation by Nascent Adhesion Sites (Robijn Bruinsma)

Where:

• G(0,1) standard normal distribution.