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Exact Solutions for Classes of Nonlinear Differential Equations on Fractal Supports

Donatella Bongiornoa, Alireza Khalili Golmankhanehb

TL;DR

This work addresses exact solutions of nonlinear fractal differential equations on fractal supports, with a focus on Riccati-type fractal DEs and their connection to quantum mechanics via a fractal Schrödinger framework. It develops and applies the $F^\alpha$-calculus, utilizing the staircase function $S^\alpha_F$ and fractal derivatives $D^\alpha_F$ to transform and solve nonlinear problems, including Riccati-type and Bernoulli-type reductions, as well as separable transformations in two canonical cases. Key contributions include explicit exact solutions for selected RFDEs (notably on the Cantor set), a general RFDE solution strategy via a particular solution and a linearizing auxiliary equation, and a fractal supersymmetric formulation that yields solvable oscillator and Coulomb-like problems on fractal domains. The results broaden the analytical toolbox for nonlinear dynamics on fractal supports and suggest practical implications for network regulation, diffusion on fractals, and quantum systems constrained to fractal geometries.

Abstract

In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a connection with the Schrodinger equation is also provided.

Exact Solutions for Classes of Nonlinear Differential Equations on Fractal Supports

TL;DR

This work addresses exact solutions of nonlinear fractal differential equations on fractal supports, with a focus on Riccati-type fractal DEs and their connection to quantum mechanics via a fractal Schrödinger framework. It develops and applies the -calculus, utilizing the staircase function and fractal derivatives to transform and solve nonlinear problems, including Riccati-type and Bernoulli-type reductions, as well as separable transformations in two canonical cases. Key contributions include explicit exact solutions for selected RFDEs (notably on the Cantor set), a general RFDE solution strategy via a particular solution and a linearizing auxiliary equation, and a fractal supersymmetric formulation that yields solvable oscillator and Coulomb-like problems on fractal domains. The results broaden the analytical toolbox for nonlinear dynamics on fractal supports and suggest practical implications for network regulation, diffusion on fractals, and quantum systems constrained to fractal geometries.

Abstract

In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a connection with the Schrodinger equation is also provided.

Paper Structure

This paper contains 8 sections, 6 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Solving equations of the form: $D^{\alpha}_F\ y(t)=g(\phi(y,t))$
  4. First case: $\phi(y,t)=\dfrac{y(t)}{S^{\alpha}_F(t)}$
  5. Second case: $\phi(y,t)=a S^{\alpha}_F(t)+by(t)$ with $a$ and $b$ two non-zero real numbers.
  6. Solving Riccati-Type Fractal Differential Equations
  7. Fractal Riccati Formulation of the Schrödinger Equation
  8. Conclusion

Key Result

Proposition 2.1

Let $f:[a,b]\to \mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ be two real functions that are $F^{\alpha}$-derivable on each point $t\in [a,b]$. Then, we have:

Figures (5)

  • Figure 1: Plot of Eq. \ref{['23222']} over the ternary Cantor set with fractal dimension $\alpha = \frac{\log 2}{\log 3}$.
  • Figure 2: Plot of Eq. \ref{['33322']} over the ternary Cantor set. The solution visualizes the interplay between the fractal staircase function $S^{\alpha}_F(t)$ and the tangent function.
  • Figure 3: Plot of Eq. \ref{['77']} for a shifted sequence of time series, where $S^{\alpha}_F(t)$ is the integral staircase function corresponding to the middle-third Cantor set with fractal dimension $\alpha = \frac{\log 2}{\log 3}$.
  • Figure 4: Plot of Eq.\ref{['11']} for different fractal subsets of the real line.
  • Figure 5: Plot of Eq. \ref{['e8']}, where $S^{\alpha}_F(t)$ is the integral staircase function based on the middle-third Cantor set.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2.1
  • Definition 5
  • Proposition 2.1
  • Definition 6
  • Remark 2.2
  • Definition 7
  • ...and 23 more