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An Order Six Numerical Method for Direct Solution of General Second Order ODEs

Industrial Mathematics | Federal University of Technology, Akure (FUTA)

Student Name: OGUNDARE OLAMIDE EMMANUEL | MTS/19/2584

📂 Documentation

📌 Project Overview

This repository contains the full mathematical derivation and computational implementation of a Sixth-Order Numerical Method designed for the direct solution of general second-order Initial Value Problems (IVPs): $$y'' = f(x, y, y'), \quad y(x_0) = y_0, \quad y'(x_0) = y'_0$$

Developed as a final year project in the Department of Mathematical Sciences, FUTA, this method utilizes a Power Series (Taylor Series) basis function to maintain the original second-order structure, avoiding the computational doubling required by reduction to first-order systems.

🛠️ Technical Stack

  • Computational Engine: Maple (Matrix inversion, Taylor expansions, Numerical Experiments)
  • Typesetting: LaTeX (High-precision mathematical documentation)
  • Mathematical Theory: Collocation & Interpolation, Linear Multistep Methods (LMM).

Find the full project here, Maple scripts here and LateX source here.

Mathematical Methodology

1. The Basis Function

We assume an approximate solution of the form: $$y(x) = \sum_{j=0}^{7} a_j x^j$$ By collocating at $p=5$ points and interpolating at $q=3$ points, we construct the following matrix system $AX = B$:

$$ \begin{bmatrix} 1 & x_n & x_n^2 & x_n^3 & x_n^4 & x_n^5 & x_n^6 & x_n^7 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 2 & 6x_{n+4} & 12x_{n+4}^2 & 20x_{n+4}^3 & 30x_{n+4}^4 & 42x_{n+4}^5 \end{bmatrix} \begin{bmatrix} a_0 \ \vdots \ a_7 \end{bmatrix} = \begin{bmatrix} y_n \ \vdots \ f_{n+4} \end{bmatrix} $$

2. The Derived Discrete Scheme

Evaluating the continuous form at $t=4$ yields the main discrete scheme: $$y_{n+4} - 2y_{n+2} + y_n = \frac{h^2}{15} (f_{n} + 16f_{n+1} + 26f_{n+2} + 16f_{n+3} + f_{n+4})$$

Differentiating the continuous scheme and evaluating at $t=4$ yields: $$y'{n+4} = - \frac{149}{42}y{n+2} + \frac{128}{21}y_{n+1} - \frac{107}{42}y_{n} + \frac{h^{2}}{1260} \left( 325f_{n} + 4048f_{n+1} + 1106f_{n+2} + 1744f_{n+3} + 397f_{n+4} \right)$$

Analysis of Basic Properties

  • Order ($p$): 6
  • Error Constant ($C_8$): $-\frac{2}{945} \approx -0.002116$
  • Consistency: The method is proven consistent as $p \geq 1$ and it satisfies $\rho(1)=\rho'(1)=0$ and $\rho''(1) = 2\sigma(1)$.
  • Zero-Stability: The first characteristic polynomial $\rho(z) = z^4 - 2z^2 + 1$ has roots $z=1$ (double) and $z=-1$ (double), satisfying the root condition for second-order ODEs.
  • Stability Interval: Determined via boundary locus method to be $(0, -0.2742)$.

Numerical Results & Benchmarks

Problem 1: Linear IVP ($y''=y$)

  • Exact Solution: $y(x) = e^x$
  • Absolute Error: 2.5827e-10
  • Benchmark: Outperforms Olayemi (2015)

Problem 1 Graph

Problem 2: Non-Linear IVP ($y''=x(y')^2$)

  • Exact Solution: $y(x) = 1 + \frac{1}{2}\log\left(\frac{2+x}{2-x}\right)$
  • Absolute Error: 4.4697e-02
  • Benchmark: Validated against Omole (2023)

Problem 2 Graph

Problem 3: Stiff IVP ($y''=y'$)

  • Exact Solution: $y(x) = 1-e^x$
  • Absolute Error: 4.0000e-10
  • Benchmark: Validated against Kuboye (2021)

Problem 3 Graph

Problem 4: Oscillatory IVP ($y''+y=2\cos x$)

  • Exact Solution: $y(x) = \cos x + x\sin x$
  • Absolute Error: 3.6358e-04
  • Benchmark: Validated against Adesanya (2011)

Problem 4 Graph

📁 Repository Structure

├── Documentation/
│   └── MTS192584_OGUNDARE_THESIS.pdf  
│   └── MTS192584_OGUNDARE_THESIS.tex   # Full Thesis Document (LaTeX)
├── Scripts/
│   └── Maple_Calculations_1.mw         # Maple Worksheet for Derivations
│   └── Maple_Calculations_2.mw  
│   └── Maple_Calculations_3.mw  
│   └── Maple_Calculations_4.mw  
├── Images/
│   └── problem1_graph.png              # Graphs
│   └── problem2_graph.png  
│   └── problem3_graph.png  
│   └── problem4_graph.png  
└── README.md

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