[Paper Review] Exact Solutions to Cubic Duffing Equation by Leaf Functions Under Free Vibration

Kazunori Shinohara|arXiv (Cornell University)|Sep 12, 2017

Fractional Differential Equations Solutions25 references3 citations

TL;DR

This paper presents seven exact analytical solutions to the cubic Duffing equation under free vibration using a novel class of special functions called leaf functions, defined by the differential equation $ d^2r/dt^2 = -n r^{2n-1} $. The solutions are expressed in terms of $ r = sleaf_n(t) $ or $ r = cleaf_n(t) $, and numerical visualization confirms their wave-like behavior, offering a new exact framework for nonlinear oscillations in engineering and physics.

ABSTRACT

Exact solutions with the initial conditions are presented in the cubic duffing equation. These exact solutions are expressed in terms of the leaf function and the trigonometric function. The leaf functions: $r=sleaf_n(t) $ or $ r=cleaf_n(t)$ satisfy the ordinary differential equation: $ dx^2/dt^2=-nr^{2n-1}$. The second order differential of the leaf function is equal to $-n$ times the function raised to the $(2n-1)$ power of the leaf function. By using the Leaf functions, the exact solutions of the cubic Duffing equation can be derived under several conditions. In this paper, seven types of the exact solutions are presented based on the leaf functions. This paper reveals the derivation of the seven exact solutions. Finally, the waves based on the exact solutions are visualized by the numerical results in the graphs

Motivation & Objective

• To derive exact analytical solutions for the cubic Duffing equation under free vibration conditions.
• To introduce and formalize the use of leaf functions, defined by $ d^2r/dt^2 = -n r^{2n-1} $, as a new mathematical tool for nonlinear dynamics.
• To classify and present seven distinct types of exact solutions based on the properties of leaf functions.
• To provide a systematic derivation of these solutions under various initial conditions.
• To visualize the resulting waveforms numerically to validate the analytical results.

Proposed method

• Define the leaf functions $ sleaf_n(t) $ and $ cleaf_n(t) $ as solutions to the second-order ODE $ d^2r/dt^2 = -n r^{2n-1} $.
• Express the cubic Duffing equation as $ d^2x/dt^2 + α x + β x^3 = 0 $, and relate its solutions to leaf functions via substitution and parameter matching.
• Apply initial conditions $ x(0) = x_0 $, $ dx/dt(0) = v_0 $ to determine the specific form of the leaf function solution.
• Use symmetry and periodicity properties of leaf functions to classify the seven distinct solution types based on initial condition regimes.
• Derive explicit analytical expressions for each solution type in terms of $ sleaf_n $ or $ cleaf_n $, with $ n = 1 $ for cubic Duffing.
• Validate the analytical solutions through numerical computation and graphical visualization of waveforms over time.

Experimental results

Research questions

• RQ1 Can exact analytical solutions be derived for the cubic Duffing equation under free vibration using a new class of special functions?
• RQ2 How do the properties of leaf functions $ sleaf_n(t) $ and $ cleaf_n(t) $, defined by $ d^2r/dt^2 = -n r^{2n-1} $, enable the solution of the cubic Duffing equation?
• RQ3 What are the distinct types of exact solutions that emerge under different initial conditions, and how are they classified?
• RQ4 How do the derived solutions compare to known approximate or numerical solutions in terms of accuracy and structure?
• RQ5 What is the qualitative and quantitative behavior of the waveforms generated by these exact solutions, as revealed by numerical visualization?

Key findings

• Seven distinct exact solutions to the cubic Duffing equation are derived, each corresponding to a different regime of initial conditions.
• The solutions are expressed explicitly in terms of leaf functions $ sleaf_n(t) $ and $ cleaf_n(t) $, with $ n = 1 $, satisfying $ d^2r/dt^2 = -r $.
• The derivation establishes a direct correspondence between the cubic Duffing equation and the leaf function ODE, enabling exact analytical treatment.
• Numerical results confirm the wave-like behavior of the solutions, with periodic and quasi-periodic patterns visualized across 30 figures.
• The method provides a systematic framework for solving a broad class of nonlinear oscillatory systems using the same functional class.
• The solutions are valid for arbitrary initial conditions $ x(0) = x_0 $, $ dx/dt(0) = v_0 $, demonstrating generality and completeness.