On stability of a class of second alpha-order fractal differential equations

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-μ Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.

• Publication year

  2020

• Venue

  Unknown venue

• Publication date

  2020-02-11

• Fields of study

  Mathematics

• Identifiers
  DOI 10.3934/math.2020141
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation

  2019cited by this paper
• Existence And Controllability For Nonlinear Fractional Control Systems With Damping in Hilbert Spaces

  2019cited by this paper
• Sumudu transform in fractal calculus

  2019cited by this paper
• Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions

  2018cited by this paper
• Sub- and super-diffusion on Cantor sets: Beyond the paradox

  2018cited by this paper
• Waves Along Fractal Coastlines: From Fractal Arithmetic to Wave Equations

  2017cited by this paper
• New methodologies in fractional and fractal derivatives modeling

  2017cited by this paper
• Lyapunov functions for Riemann-Liouville-like fractional difference equations

  2017cited by this paper
• On the Lipschitz condition in the fractal calculus

  2017cited by this paper
• A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

  2016cited by this paper
• A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging

  2016cited by this paper
• CALCULUS ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH ORDINARY CALCULUS

  2011cited by this paper
• Approximation of infinite dimensional fractals generated by integral equations

  2010cited by this paper
• CALCULUS ON FRACTAL CURVES IN Rn

  2009cited by this paper
• Finite‐ and infinite‐dimensional attractors for porous media equations

  2008cited by this paper
• On a new dimension estimate of the global attractor for chemotaxis-growth systems

  2008cited by this paper
• On the asymptotic behavior of solutions of certain second-order differential equations

  2007cited by this paper
• An optimization algorithm based on chaotic behavior and fractal nature

  2007cited by this paper
• Quantum–classical transition in scale relativity

  2004cited by this paper
• A new approximation procedure for fractals

  2003cited by this paper
• Calculus on fractal subsets of real line - I: formulation

  2003cited by this paper
• Fractal analysis of chaotic classical scattering in a cut-circle billiard with two openings.

  2002cited by this paper
• Local Fractional Fokker-Planck Equation

  1998cited by this paper
• Fractional differentiability of nowhere differentiable functions and dimensions.

  1996cited by this paper
• Fractals and nonstandard analysis

  1984cited by this paper
• Axiomatic basis for spaces with noninteger dimension

  1977cited by this paper

• Fuzzy logic-based reinforcement learning control approach to fractal-order nonlinear chaotic systems

  2025cites this paper
• Boundedness of Vector Linéard Equation with Multiple Variable Delays

  2024cites this paper
• Analyzing the stability of fractal delay differential equations

  2024cites this paper
• Power series solution for fractal differential equations

  2024cites this paper
• Stability and complex dynamical analysis of COVID-19 epidemic model with non-singular kernel of Mittag-Leffler law

  2024cites this paper
• ON HOMOGENEOUS SYSTEM OF FRACTAL DIFFERENTIAL EQUATIONS

  2024cites this paper
• Higher Order Fractal Differential Equations

  2024cites this paper
• Controllability of Second Order Semilinear Random Differential Equations in Fréchet Spaces

  2023cites this paper
• Stability tests and solution estimates for non-linear differential equations

  2023cites this paper
• On Existence and Continuity Results of Solution for Multi-time Scale Fractional Stochastic Differential Equation

  2023cites this paper
• Discrete Bazykin’s Prey–Predator Model with Stability, Control and Bifurcation

  2023cites this paper
• A Mechanical Picture of Fractal Darcy’s Law

  2023cites this paper
• Results in Control and Optimization

  2023cites this paper
• Results in Control and Optimization

  2023cites this paper
• Fractal Mellin transform and non-local derivatives

  2023cites this paper
• A fractal–fractional COVID-19 model with a negative impact of quarantine on the diabetic patients

  2023cites this paper
• Results in Control and Optimization

  2023cites this paper
• Stability analysis for nonlocal evolution equations involving infinite delays

  2022cites this paper
• Fractional order COVID-19 model with transmission rout infected through environment

  2022cites this paper
• U-Model-Based Dynamic Inversion Control for a Class of Nonlinear Dynamical Systems

  2022cites this paper
• On fractal-fractional Covid-19 mathematical model

  2022cites this paper
• Results on controllability for Sobolev type fractional differential equations of order $ 1 < r < 2 $ with finite delay

  2022cites this paper
• Analysis and numerical simulation of tuberculosis model using different fractional derivatives

  2022cites this paper
• XAI hybrid multi-staged algorithm for routine & quantum boosted oncological medical imaging

  2022cites this paper
• A Semi-Analytical Solution of Inverse Laplace Transform

  2022cites this paper
• Fractal Continuum Calculus of Functions on Euler-Bernoulli Beam

  2022cites this paper
• Uniform-ultimate boundedness of solutions to vector lienard equation with delay

  2022cites this paper
• Uniform Convergence of Successive Approximations for Hybrid Caputo Fractional Differential Equations

  2022cites this paper
• New Results on the Qualitative Analysis of Solutions of Vides by the Lyapunov–Razumikhin Technique

  2022cites this paper
• Chaos, Solitons and Fractals

  2022cites this paper
• Chaos, Solitons and Fractals

  2022cites this paper
• Dynamics of an arbitrary order model of toxoplasmosis ailment in human and cat inhabitants

  2021cites this paper
• Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems

  2021cites this paper
• Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

  2021cites this paper
• Fractional aspects of coupled mass-spring system

  2021cites this paper
• A Fractional Bihari Inequality and Some Applications to Fractional Differential Equations and Stochastic Equations

  2021cites this paper
• Computational solutions of the HIV-1 infection of CD4+T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy

  2020cites this paper
• Input-to-State Stability for Impulsive Gilpin-Ayala Competition Model With Reaction Diffusion and Delayed Feedback

  2020cites this paper
• On a new structure of the pantograph inclusion problem in the Caputo conformable setting

  2020cites this paper
• Green–Haar wavelets method for generalized fractional differential equations

  2020cites this paper
• On existence and stability results to a class of boundary value problems under Mittag-Leffler power law

  2020cites this paper
• On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications

  2020cites this paper