The main aim of this paper is to analyze the behavior of time-fractional Emden–Fowler (EF) equation associated with Caputo–Katugampola fractional derivative occurring in mathematical physics and astrophysics. A powerful analytical approach, which is an amalgamation of q-homotopy analysis approach and generalized Laplace transform with homotopy polynomials, is implemented to obtain approximate analytical

Let [d1(x),d2(x),…,dn(x),…] be the Lüroth expansion of x∈(0,1], and let Ln(x)=max{d1(x),…,dn(x)}. It is shown that for any α≥0, the level set x∈(0,1]:limn→∞Ln(x)loglognn=α has Hausdorff dimension one. Certain sets of points for which the sequence {Ln(x)}n≥1 grows more rapidly are also investigated.

The examination of brain responses in individuals with a pornography addiction compared to those without sheds light on the neurobiological aspects associated with this behavior. Neuroscientific studies utilizing techniques such as electroencephalography (EEG) have shown that porn-addicted individuals may exhibit alterations in neural pathways related to reward processing and impulse control. In this

In this research, we introduce a new and generalized family of convex functions, entitled the α-convex functions in the second sense with respect to a parameter and examine their important algebraic properties. Based on this novel convexity concept, we explore a new class of fractional integral inequalities for functions that are twice differentiable. These results are derived from fundamental identities

In this paper, the gas-water two-phase flow characteristics of rock media are studied based on fractal theory and the relative roughness model, and the analytical model of gas-water relative permeability of rock pores with relative roughness is derived. Through numerical simulation, it is found that the maximum flow velocity in the rough microchannel is greater than the maximum flow velocity in the

The primary aim of this work is to investigate the nonlinear fractional Schrödinger equation, which is adopted to describe the ultra-short pulses in optical fibers. A variety of new soliton solutions and periodic solutions are constructed by implementing three efficient mathematical approaches, namely, the improved fractional F-expansion method, fractional Bernoulli (G′/G)-expansion method and fractional

Based on the pseudo-order relation, we introduce the concept of left and right ℏ-preinvex interval-valued functions (LR-ℏ-PIVFs). Further, we establish the Hermite–Hadamard and Hermite–Hadamard–Fejér-type estimates for LR-ℏ-PIVFs using generalized fractional integrals. Finally, an example of interval-valued fractional integrals is provided to illustrate the validity of the results derived herein. Our

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation

In gas-bearing coal seam mining projects, the pivotal considerations encompass the assessment of gas migration, emission trends, and coal seam stability, which are crucial for ensuring both the safety and efficiency of the project. The accurate evaluation of the nonlinear evolution of the fracture network, acting as the primary conduit for gas migration and influenced by mining disturbances, coal seam

This study investigates the nonlinear time-fractional Harry Dym (𝕋𝔽ℍ𝔻) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym (ℍ𝔻) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative

In this paper, we explore the price dynamics of 16 representative records of USA Education Group stocks, encompassing two non-overlapping periods (before, during, and after COVID-19). Based on information theory and cluster analysis techniques, our study provides insights into disorder, predictability, efficiency, similarity, and resilience/weakness, considering the most diverse financial stakeholders

By means of He’s fractal derivative, a new fractal (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation is extracted in this paper. The semi-inverse method is employed to establish the generalized fractal variational principle. The generalized fractal variational principle can show the conservation laws through the energy form in the fractal space. Moreover, some semi-domain solutions

Micro/nanoscale lubrication must take into account the fractal profile of the shaft and bearing surfaces. A new fractal rheological model is proposed to describe the properties of the non-Newtonian fluid, and a fractal variational principle is established by the semi-inverse method, and finally the Lagrange multipliers can be found in the obtained variational formulation. This work provides a new fractal

One of the major challenges in population economics is accurately predicting population size. Incorrect predictions can lead to ineffective population control policies. Traditional differential models assume a smooth change in population, but this assumption is invalid when measuring population on a small-time scale. To address this change, we developed two-scale fractal population dynamics that can

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency

In recent years, fuzzy and fractional calculus are utilized for simulating complex models with uncertainty and memory effects. This study is focused on fuzzy-fractional modeling of (2+1)-dimensional Wu–Zhang (WZ) system. Caputo-type time-fractional derivative and triangular fuzzy numbers are employed in the model to observe uncertainties in the presence of non-local and memory effects. The extended

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed

In this paper, we calculate the Hausdorff dimension of the fractal set x∈𝕋d:∏1≤i≤d|Tβin(xi)−xi|<ψ(n) for infinitely many n∈ℕ, where Tβi is the standard βi-transformation with βi>1, ψ is a positive function on ℕ and |⋅| is the usual metric on the torus 𝕋. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence

The short-term traffic flow prediction can help to reduce flight delays and optimize resource allocation. Using chaos dynamics theory to analyze the chaotic characteristics of en-route traffic flow is the basis of short-term en-route traffic flow prediction and ensuring the orderly and smooth state of the en-route. This paper takes the time series of en-route traffic flow extracted from Automatic-Dependent

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also

To achieve in situ condition-preserved coring of the lunar surface and deep lunar rocks and a return mission, it is necessary to explore the mechanical properties and failure modes of simulated lunar rocks that have physical and mechanical properties approximately equivalent to those of mare basalt under simulated lunar temperature environments (−120∘C to 200∘C). To this end, real-time uniaxial compression

In this paper, we investigate solutions of telegraph, Laplace and wave equations within the local fractional derivative operator by using local fractional Sumudu decomposition method. This method is coupled by the Sumudu transform and decomposition method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability

The edge-Wiener index is an important topological index in Chemical Graph Theory, defined as the sum of distances among all pairs of edges. Fractal structures have received much attention from scientists because of their philosophical and aesthetic significance, and chemists have even attempted to synthesize various types of molecular fractal structures. The level-3 Sierpinski triangle is constructed

In this work we study the two-dimensional local fractional Boussinesq equation. Based on the basic definitions and properties of the local fractional derivatives and bilinear form, we studied the soliton solutions of non-differentiable type with the generalized functions defined on Cantor sets by using bilinear method. Meanwhile, we discuss the result when fractal dimension is 1, and compare it with

The morphological characteristics of core disking can reflect the in-situ stress field characteristics to a certain extent, but a quantitative description method for disking-induced fracture surfaces is needed. The fractal geometry was introduced to refine the three-dimensional characteristics of the core disking fracture surfaces, and the disking mechanism was explored through morphological characteristics

In this paper, we present a new encryption method based on discrete wavelet transform (DWT). This method provides a number of advantages as a pseudo randomness and sensitivity due to the variation of the initial values. We start by decomposing the image with spatial reconstruction by DWT, followed by preformation by fractional chaotic cryptovirology and Henon map keys for space encryption. Bearing

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative

In this paper, we propose a tempered xi function obtained by the recombination of the decomposable functions for the Riemann xi function for the first time. We first obtain its functional equation and series representation. We then suggest three equivalent open problems for the zeros for it. We finally consider its behaviors on the critical line.

In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss–Ostrogradsky-like, Stokes-like and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The strong and

Wave–wave interaction occurs in the propagation deformation of nonlinear long waves in shallow-water. In order to further study the propagation mechanism of shallow-water long waves interaction, the exact traveling wave solutions of the local fractional generalized Hirota–Satsuma coupled Korteweg–de Vries (HS-KdV) equations defined by the Cantor sets are obtained. The non-differentiable solutions with

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized M set with different kinds of exponent k (k−M set), properties are hard to understand when k is a complex number. In this paper, fractal property of generalized M set with complex exponent z is studied. First, a relation is constructed between generalized

This work is devoted to the subtrigonometric and subhyperbolic functions in terms of theWiman class for the first time. The conjectures for the subsine and subcosine functions are considered in detail. The Weierstrass–Mandelbrot function is represented as the hyperbolic subsine, and hyperbolic subcosine functions to get new results for the nondifferentiable functions.

Optical soliton is a physical phenomenon in which the waveforms and energy of optical fibers remain unchanged during propagation, which has important application value in information transmission. In this paper, the coupled nonlinear Schrödinger equations describe the propagation of optical solitons with different frequencies in sense of local fractional derivative is analyzed. The exact traveling

In this paper, the anomalous diffusion models are studied in the framework of the scaling-law calculus with the Mandelbrot scaling law. A analytical technology analogous to the Fourier transform is proposed to deal with the one-dimensional scaling-law diffusion equation. The scaling-law series formula via Kohlrausch–Williams–Watts function is efficient and accurate for finding exact solutions for the

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine

In this paper, we use the local fractional variational iteration transform method LFVITM to solve a class of linear and nonlinear partial differential equations (PDEs), as well as a system of PDEs which are involving local fractional differential operators (LFDOs). The technique combines the variational iteration transform approach and the Yang–Laplace transform. To show how effective and precise the

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional (2+1)-dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported

In this paper, a new fractional derivative model for the non-Darcian seepage within the exponential decay kernel is addressed for the first time. The new fractional derivative model is for high-speed non-Darcian and low-speed non-Darcian seepage, in which the applied zone is enlarged.

In this paper, we first propose a method, which is originated from coupling local fractional Yang–Laplace transform with the Daftardar–Gejji–Jafaris method (LFYLTDGJM). The proposed method is successfully applied to solve the local time fractional nonlinear modified Korteweg–de Vries (TFNMKDV) equation. The approximate solution presented here illustrates the efficiency and accuracy of the proposed

In this paper, we propose a new frequency amplitude formula for the local fractional nonlinear oscillation via local fractional calculus. It is more general than the He’s frequency amplitude formula. Several test cases of local fractional nonlinear oscillations are given to prove the feasibility of the improved formula.

In this paper, based on Yang’s fractal theory, the Hermite–Hadamard’s inequalities for generalized h-preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional

In this paper, a new fractal nerve impulses modeling is successfully described via the Yang’s local fractional derivative in a microgravity space, and its approximate analytical solution is obtained by a new Adomian decomposition method. The efficiency and accuracy analysis of the proposed method is elucidated according to the graphs. The result shows that our method is excellent and accurate in dealing

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible

In order to describe the self-similarity and long-range dependence of financial asset prices, this paper adopts a new fractional-type process, i.e, the generalized mixed weighted fractional Brownian motion to describe the dynamic change process of risky asset prices. A European option pricing model driven by the generalized mixed weighted fractional Brownian motion is constructed, and explicit solutions

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional boundary value problems.

The stability properties of certain probability distribution functions under the combined effects of cascading and “dressing” in a binary multiplicative cascade are contemplated and proven herein. Their main importance for applications resides in parameterizing the multiplicative cascade generators of multifractal measures from single realizations, given the generic lack of distributional ergodicity

Studying the rough structure characteristics of rock fracture surfaces under microwave irradiation is of a great significance for understanding the rock-breaking mechanism. Therefore, this work takes fracture surface as the research object under three failure modes: microwave irradiation, uniaxial loading and microwave-uniaxial loading. The undulation and roughness are used to describe the morphological

Analytical solutions to the fractional Riccati equation are considered in this paper. Solutions to fractional differential equations are expressed in the form of fractional power series in the Caputo algebra. It is demonstrated that solutions to higher-order Riccati fractional equations inherit some solutions from lower-order Riccati equations under special initial conditions. Such nested and fractal-like

In this work, the (2+1)-dimensional Kadomtsev–Petviashvili model is investigated. A novel variational scheme, namely, the variational transform wave method (VTWM), is successfully established to seek the solitary wave solution of the Kadomtsev–Petviashvili model. Furthermore, the fractal solitary solution of fractal Kadomtsev–Petviashvili model is also studied based on the local fractional derivative

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of

Fractal pore structure exists widely in natural reservoir and dominates its transport property. For that, more and more effort is devoted to investigate the control mechanism on mass transfer in such a complex and multi-scale system. Apparently, effective characterization of the fractal structure is of fundamental importance. Although the newly emerged concept of complexity assembly clarified the complexity

In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized affine fractal interpolation surfaces (FISs). By using these matrices, we establish relationships between oscillation vectors of different levels, which enables us

An in-depth analysis of the attack vulnerability of fractal scale-free networks is of great significance for designing robust networks. Previous studies have mainly focused on the impact of fractal property on attack vulnerability of scale-free networks under static node attacks, while we extend the study to the cases of various types of targeted attacks, and explore the relationship between the attack

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered

The challenges of modeling shale oil transport are numerous and include strong solid-fluid interactions, fluid rheology, the multi-scale nature of the pore structure problem, and the different pore types involved. Until now, theoretical studies have not fully considered shale oil transport mechanisms and multi-scale pore structure properties. In this study, we propose a fractal-based oil transport

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic

In this paper, we explore upper box dimension of continuous functions on [0,1] and their Riemann–Liouville fractional integral. Firstly, by comparing function limits, we prove that the upper box dimension of the Riemann–Liouville fractional order integral image of a continuous function will not exceed 2−υ, the result similar to [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of