# Preface

This section demnstrates how some nonlinear differential equations can be converted into dynamic system with polynomial inputs.

Return to computing page for the first course APMA0330

Return to computing page for the second course APMA0340

Return to Mathematica tutorial for the first course APMA0330

Return to Mathematica tutorial for the second course APMA0340

Return to the main page for the first course APMA0330

Return to the main page for the second course APMA0340

Return to Part IV of the course APMA0340

Introduction to Linear Algebra with *Mathematica*

## Glossary

# System Conversion

Practical applications of popular methods to solve differential equations become unsuitable when slope functions are complicated. For instance, Picard's, Adomian', power series methods etc. are efficient mostly when input is not a polynomial. This section illustrates how differential equations can be converted into polynomial dynamic systems. Since the best way to learn this approach is to work with example, we demonstrate convertion of differential equations into more suitable forms by providing several typical examples.

**Example: **
Consider the autonomous differential equation

**Example: **
The famous pendulum equation is

*t*) is the angle the bob makes with the doownward vertical position. We reduce thisnonlinear second order differential equation to the system of first order differential equations with four unknowns:

**Example: **
Consider the differential equation

*J*

_{1}(

*y*) is the Bessel function of first order. This equation could stand for aharmonic oscillator with position-dependent elastic constant. Upon introducing new variables

**Example: **
The **Morse potential**, named after physicist Philip M. Morse (1903--1985), is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the QHO (quantum harmonic oscillator) because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface.

The Morse potential energy function is of the form

*q*stands for the bond length,

*D*for the dissociation energy, and

*r*for the anharmonic parameter. The Hamiltonian becomes

*E*is the total energy, ω is the anharmonic frequency of the oscillator given by \( \omega = \left( 2D - 2E \right)^{1/2} , \) and φ

_{0}the initial phase.

In order to reduce the above system to polynomial form, we introduce the new variable \( z = e^{-rq} . \) This yields the following polynomial system of differential equations:

- Fairen, V., López, V., Conde, L., Power series approximation to solutions of nonlinear systems of differential equations, American Journal of Physics, 1988, Vol. 56,pp. 57 ; doi: 10.1119/1.15432

Return to Mathematica page

Return to the main page (APMA0340)

Return to the Part 1 Matrix Algebra

Return to the Part 2 Linear Systems of Ordinary Differential Equations

Return to the Part 3 Non-linear Systems of Ordinary Differential Equations

Return to the Part 4 Numerical Methods

Return to the Part 5 Fourier Series

Return to the Part 6 Partial Differential Equations

Return to the Part 7 Special Functions