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Nonlinear Fractional Differential Equations: Theory and Applications

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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 March 2027 | Viewed by 5134

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Editors

Dr. Jiawei He

E-Mail Website1 Website2
Guest Editor

School of Mathematics, Guangxi University, Nanning 530004, China
Interests: fractional differential equations; evolution equations; differential equations and dynamic systems

Dr. Junfang Zhao

E-Mail Website
Guest Editor

School of Science, China University of Geosciences, Beijing 100083, China
Interests: boundary value problem; partial differential equations; critical points theory; physical information neural network; machine learning

Special Issue Information

Dear Colleagues,

Fractional calculus is a sub-branch of mathematics and applied mathematics, and the fractional problem has become an absorbing field for scientists and mathematicians due to its widespread applications in modeling, engineering, mathematical biology, financial modeling, fluid flows, and so on.

The Special Issue aims to bring together the latest research and developments in the field of nonlinear fractional calculus and nonlinear partial differential equations. This area of study has become increasingly important due to its ability to model complex systems that exhibit memory and hereditary properties, which are often encountered in real-world applications. The issue will feature articles that cover a wide range of topics, including the theoretical foundations of nonlinear fractional differential equations, and their applications in various fields such as physics, engineering, biology, and finance. The goal is to provide an overview of the current state of research in this area and to highlight the potential for future developments and applications. This Special Issue will be of interest to researchers, academics, and practitioners who are involved in the study of nonlinear dynamics, fractional calculus, partial differential equations, and their applications in various scientific and engineering disciplines.

Dr. Jiawei He
Dr. Junfang Zhao
Guest Editors

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Keywords

fractional calculus and applications
fractional order nonlinear systems
partial differential equations
integrodifferential equations
critical point theory
control theory
continuous and discrete dynamical systems
initial and boundary value problems
stability theory

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Published Papers (6 papers)

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Research

25 pages, 1683 KB

Open AccessArticle

Analytical Study of Impulsive Hilfer-Type Fractional p-Laplacian Problems Using Neural Networks and Finite-Difference Methods

by Rahman Ullah Khan, Ioannis K. Argyros, Taha Radwan and Yousif Altayeb

Axioms 2026, 15(8), 591; https://doi.org/10.3390/axioms15080591 - 5 Aug 2026

Viewed by 276

Abstract

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter

ϑ \in [0, 1]

is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting [...] Read more.

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter

ϑ \in [0, 1]

is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting cases of the formulation, not as separate cases. The variational functional is then built by adding the point-impulse contribution to the distributed potential and the use of an appropriate space of the Hilfer fractional derivative. Using variants of the fountain theorem, we prove the existence of two infinite sequences of weak solutions, one of which is of unbounded energy and another of which is of small energy and tends to zero from below. The weak residual based stability analysis is further developed, and local generalized Hyers–Ulam and Hyers–Ulam–Rassias stability estimates are obtained. Because of multiplicity of solutions, a uniqueness-based argument for stability, Ulam’s approach, is not possible and stability is instead achieved by providing residual-based arguments.The assumptions are verified through illustrative examples. Lastly, we examine the convergence behavior, residual decay, and effect of the Hilfer type parameter in conjunction with a Hilfer-type parameter neural surrogate with boundary constraints based on a discrete Hilfer scheme. The study, in general, proves a link between the solution multiplicity, residual stability, and the numerical realization in one impulsive fractional p-Laplacian framework. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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33 pages, 5619 KB

Open AccessArticle

Nonlinear Wave Structures in a Truncated M-Fractional Complex mKdV System: Soliton Dynamics and Numerical Simulations

by Reem Abdullah Aljethi and Ejaz Hussain

Axioms 2026, 15(6), 454; https://doi.org/10.3390/axioms15060454 - 17 Jun 2026

Viewed by 290

Abstract

In this study, a detailed analytical-numerical study of the complex modified Korteweg–De Vries (mKdV) model with truncated M-fractional derivative is carried out to investigate the effects of the fractional order on nonlinear wave propagation. The fractional partial differential equation is solved by an appropriate fractional traveling wave transformation, which transforms it into a nonlinear ordinary differential equation. Two very powerful analytical methods are then used: the modified sub-equation method and the Kumar–Malik method, which give the exact closed-form solutions. The obtained semi-analytical numerical approximations are then obtained from the Differential Transformation Method (DTM). Bright and dark solitons, kink-type waves, periodic and rational solutions, exponential solutions, and Jacobi elliptic functions are found for a variety of parametric regimes. Explicit compatibility conditions and parametric constraints, which control the amplitude, width, and propagation, are derived. The DTM approximations are found to converge to the exact solutions with good accuracy, and the absolute errors are almost negligible, which validates the accuracy of the approximations and reliability of the solution. The three-dimensional visualizations of surface plots, two-dimensional profiles, and contour visualization further illustrate the dispersive dynamics and stability properties. Significance: This study shows that the truncated M-fractional derivative is a good operator to model memory-dependent nonlinear wave propagation. A new precise solution and reliable validation methods have been obtained for high-dimensional fractional nonlinear evolution equations in the hybrid analytical-numerical framework, which can be useful in plasma physics, nonlinear optics, and complex media. The present study contains restrictions for constant coefficients, a specific parametric regime, one fractional derivative definition, and experimental validation is not included. Future directions are limitations on constant coefficients, specific parametric regimes, one fractional derivative definition, and experimental validation is not included. The approach is to be extended in the future to variable coefficients, other fractional operators (Caputo, Riemann–Liouville), and to higher-order nonlinearities, and then to be experimentally tested in optical or plasma systems. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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22 pages, 373 KB

Open AccessArticle

Fractional Viscous–Resistive Magnetohydrodynamics at Critical Scales: Global Solutions and Gevrey Regularity

by Siyi Xie, Chengzhou Wei and Muhammad Zainul Abidin

Axioms 2026, 15(5), 372; https://doi.org/10.3390/axioms15050372 - 16 May 2026

Viewed by 254

Abstract

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on

R^{n}

with fractional diffusion

{(- Δ)}^{α}

, where

α \in (1 / 2, 1]

, and with positive viscosity and resistivity coefficients

μ, ν > 0

[...] Read more.

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on

R^{n}

with fractional diffusion

{(- Δ)}^{α}

, where

α \in (1 / 2, 1]

, and with positive viscosity and resistivity coefficients

μ, ν > 0

. The problem is treated at the scale-invariant regularity

s_{c} = \frac{n}{p} + 1 - 2 α .

For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space

{\dot{F}}_{p, r}^{s_{c}, q}

, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space

m_{m}^{'} {\dot{F}}_{p, r}^{s_{c}, q}

, built on Meyer wavelets and the parabolic gauge

t 2^{2 α j}

. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently,

e^{{(t {(- Δ)}^{α})}^{γ}} (u, b) \in {(m_{m}^{'} {\dot{F}}_{p, r}^{s_{c}, q})}^{2 n}

for some

γ > 0

depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction

α > \frac{1}{2}

enters through the factor

2^{j (1 - 2 α)}

, which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when

α = 1

. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

25 pages, 8764 KB

Open AccessArticle

A Comprehensive Study on the Applications of NTIM and OAFM in Analyzing Fractional Navier–Stokes Equations

by Siddiq Ur Rehman, Rashid Nawaz, Faisal Zia and Nick Fewster-Young

Axioms 2025, 14(7), 521; https://doi.org/10.3390/axioms14070521 - 7 Jul 2025

Viewed by 964

Abstract

This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. By comparing the solutions derived from these methods to exact solutions, it is clear that they provide accurate and efficient outcomes. These findings highlight the straightforward yet effective use of these methodologies in modeling engineering systems. Navier–Stokes equations have numerous practical uses, including analyzing fluid flow in pipelines and channels, predicting weather patterns, and constructing aircraft and vehicles. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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18 pages, 293 KB

Open AccessArticle

Existence and Controls for Fractional Evolution Equations

by Ying Chen and Yong Zhou

Axioms 2025, 14(5), 329; https://doi.org/10.3390/axioms14050329 - 24 Apr 2025

Viewed by 1488

Abstract

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals is studied, and the existence of optimal control pairs is proven. Finally, by constructing a corresponding Gramian controllability operator using the solution operator, a sufficient condition is provided for the existence of approximate controllability of the corresponding problem. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

13 pages, 272 KB

Open AccessArticle

Existence and Attractivity of Mild Solutions for Fractional Diffusion Equations Involving the Regularized ψ-Hilfer Fractional Derivatives

by Luyao Wang, Yuhang Jin, Wenchang He and Jia Mu

Axioms 2025, 14(2), 79; https://doi.org/10.3390/axioms14020079 - 22 Jan 2025

Cited by 1 | Viewed by 1031

Abstract

The regularized

ψ

-Hilfer derivative within the sense of Caputo is an improved version of the

ψ

-Hilfer fractional derivative, primarily because it addresses the issue where the initial conditions of problems involving the

ψ

-Hilfer fractional derivative lack clear physical significance unless [...] Read more.

The regularized

ψ

-Hilfer derivative within the sense of Caputo is an improved version of the

ψ

-Hilfer fractional derivative, primarily because it addresses the issue where the initial conditions of problems involving the

ψ

-Hilfer fractional derivative lack clear physical significance unless

p = 1

. This article’s main contribution is the use of the

ψ

-Laplace transform, which is the first to provide an explicit expression for mild solutions to the fractional diffusion equations with the regularized

ψ

-Hilfer derivative. Additionally, we investigate the existence and attractivity of mild solutions for fractional diffusion equations involving the regularized

ψ

-Hilfer fractional derivatives. Finally, we provide two examples to illustrate our main results. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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