numericalSolutions

This repo is my implementation of the Euler and Runga-Kutta methods for numerical solution to first-order differential equations. In output.txt I have provided my results for problems 11-15 on page 101 of the book "Differential Equations and Linear Algebra" by Goode.

I solved problem 11 and problem 12 in Latex but I provided the PDFs for reference. This allowed me to graph and compare the actual solution against the numerical methods.

Summary of Results and Graphs

The results are as expected. The Runga-Kutta approximaton is considerably more accurate in both cases.

Problem 11:

Problem 12:

Description

I wanted to avoid hard coding individual solutions, so I came up with a more general approach. Using parse_expr, sympy is able to take a string and convert it into an expression. For example:

>>> from sympy import parse_expr
>>> str_expr = "x**2"
>>> parse_expr(str_expr, {"x": 8})
64

The rest is trivial. I just substitute for x and y, define the needed variables and evaluate the formula:

  exact_h = Float(h)

  k_1 = exact_h * parse_expr(function, values)
  k_2 = exact_h * parse_expr(function, {
      "x": values["x"] + (0.5 * exact_h),
      "y": values["y"] + (0.5 * k_1)
  })
  k_3 = exact_h * parse_expr(function, {
      "x": values["x"] + (0.5 * exact_h),
      "y": values["y"] + (0.5 * k_2)
  })
  k_4 = exact_h * parse_expr(function, {
      "x": values["x"] + exact_h,
      "y": values["y"] + k_3
  })

  .... values["y"] + (1 / 6)*(k_1 + (2 * k_2) + (2 * k_3) + k_4)

The graphs were made using matplotlib.

Bugs

I struggled with rounding errors. This was my first time using sympy and it does not play well with the decimal module out of the box. You will notice that some solutions are more accurate than others.

How to use

This program is written in Python3 using the sympy and tabulate libraries. Once you have all dependencies installed, run:

python3 ./solutions.py