Preface

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

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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

\[ y' = \left( \beta + \alpha\,y^2 \right)^{1/2} , \]
for which Picard's iteration procedure fails. Upon introducing a new variable \( z = \left( \beta + \alpha\,y^2 \right)^{1/2} , \) we obtain an equivalent linear system of differential equations:
\[ \begin{split} y' &= z , \\ z' &= \alpha \,y . \end{split} \]
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Example: The famous pendulum equation is

\[ \ddot{\theta} + \sin\theta =0 , \]
where θ(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:
\[ \begin{split} \frac{{\text d}y_1}{{\text d}t} &= y_2 , \\ \frac{{\text d}y_2}{{\text d}t} &= - y_3 , \\ \frac{{\text d}y_3}{{\text d}t} &= y_2 y_4 , \\ \frac{{\text d}y_4}{{\text d}t} &= - y_2 y_3 , \end{split} \]
where
\[ y_1 = \theta , \qquad y_2 = \dot{y}_1 = \dot{\theta} , \qquad y_3 = \sin y_1 = \sin \theta , \qquad y_4 = \cos y_1 = \cos \theta . \]
The initial conditions for new variables follow from the initial conditions for the angle:
\[ y_1 (0) = \theta_0 , \quad \dot{y}_1 (0) = y_2 (0) = v_0 , \qquad \Longrightarrow \qquad y_3 (0) = \sin \theta_0 , \quad y_4 (0) = \cos \theta_0 . \]
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Example: Consider the differential equation

\[ \frac{{\text d}y}{{\text d}t} = - \left[ 1 + J_1 (y) \right] y , \]
where J1(y) is the Bessel function of first order. This equation could stand for aharmonic oscillator with position-dependent elastic constant. Upon introducing new variables
\[ y_1 = y, \qquad y_2 = \dot{y}_1 , \qquad y_3 = J_1 (y_1 ) , \qquad y_4 = J_0 (y_1 ) , \]
we reduce the given single differential equation to the dynamic system of polynomial equations:
\[ \begin{split} \frac{{\text d}y_1}{{\text d}t} &= y_2 , \\ \frac{{\text d}y_2}{{\text d}t} &= - \left( 1 + y_3 \right) y_1 , \\ \frac{{\text d}y_3}{{\text d}t} &= \frac{1}{2}\,y_2 \left( y_4 - y_3 \right) , \\ \frac{{\text d}y_4}{{\text d}t} &= - y_2 y_3 . \end{split} \]
The initial conditions become
\[ y_1 (0) = y_0 = y(0), \quad y_2 (0) = v_0 = \dot{y} (0) , \quad y_3 (0) = J_1 (y_0 ) , \quad y_4 (0) = J_0 (y_0 ) . \]
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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

\[ V(x) = D\left( 1 - e^{-rq} \right)^2 , \]
where q stands for the bond length, D for the dissociation energy, and r for the anharmonic parameter. The Hamiltonian becomes
\[ H(p,q) = \frac{p^2}{2} + D\left( 1 - e^{-rq} \right)^2 , \]
and the Hamiltonian equations of motion are
\[ \begin{split} \frac{{\text d}q}{{\text d}t} &= p , \\ \frac{{\text d}p}{{\text d}t} &= -2D \left( 1 - e^{-rq} \right) e^{-rq} , \end{split} \]
of which the exact solution is
\[ q(t) = - \frac{1}{r} \,\ln \frac{1 - (E/D)^{1/2} \cos (\omega t + \varphi_0 )}{1 - E/D} , \]
where E is the total energy, ω is the anharmonic frequency of the oscillator given by \( \omega = \left( 2D - 2E \right)^{1/2} , \) and φ0 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:

\[ \begin{split} \frac{{\text d}q}{{\text d}t} &= p , \\ \frac{{\text d}p}{{\text d}t} &= -2D\,z \left( 1-z \right) , \\ \frac{{\text d}z}{{\text d}t} &= -z\,p . \end{split} \]
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  1. 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

 

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