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The proposed model can emphasize the priority of new information and can extract messages from the first pair of original data. The comparison results show that the proposed model can improve the traditional grey model.

The grey multivariate model with fractional Hausdorff derivative is firstly put forward to enhance the forecasting accuracy of traditional grey model.

The proposed model is used to predict the air quality composite index (AQCI) in ten cities respectively.

The effect of population density on AQCI in cities with poor air quality is not as significant as that of the cities with better air quality.

The classical advection-dispersion equation (ADE) model cannot accurately depict the gas transport process in natural geological formations. This paper aims to study the behavior of CO₂ transport in fractal porous media by using an effective Hausdorff fractal derivative advection-dispersion equation (HFDADE) model.

Anomalous dispersion behaviors of CO₂ transport are effectively characterized by the investigation of time and space Hausdorff derivatives on non-Euclidean fractal metrics. The numerical simulation has been performed with different Hausdorff fractal dimensions to reveal characteristics of the developed fractal ADE in fractal porous media. Numerical experiments focus on the influence of the time and space fractal dimensions on flow velocity and dispersion coefficient.

The physical mechanisms of parameters in the Hausdorff fractal derivative model are analyzed clearly. Numerical results demonstrate that the proposed model can well fit the history of gas production data and it can be a powerful technique for depicting the early arrival and long-tailed phenomenon by incorporating a fractal dimension.

To the best of the authors’ knowledge, first time these results are presented.

The purpose of this paper is to propose a new grey prediction model, GOFHGM (1,1), which combines generalised fractal derivative and particle swarm optimisation algorithms. The aim is to address the limitations of traditional grey prediction models in order selection and improve prediction accuracy.

The paper introduces the concept of generalised fractal derivative and applies it to the order optimisation of grey prediction models. The particle swarm optimisation algorithm is also adopted to find the optimal combination of orders. Three cases are empirically studied to compare the performance of GOFHGM(1,1) with traditional grey prediction models.

The study finds that the GOFHGM(1,1) model outperforms traditional grey prediction models in terms of prediction accuracy. Evaluation indexes such as mean squared error (MSE) and mean absolute error (MAE) are used to evaluate the model.

The research study may have limitations in terms of the scope and generalisability of the findings. Further research is needed to explore the applicability of GOFHGM(1,1) in different fields and to improve the model’s performance.

The study contributes to the field by introducing a new grey prediction model that combines generalised fractal derivative and particle swarm optimisation algorithms. This integration enhances the accuracy and reliability of grey predictions and strengthens their applicability in various predictive applications.

The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set.

The fractal integrodifferential equations of resistor-inductor, resistor-capacitor, inductor-capacitor and resistor-inductor-capacitor circuits subjected to zero mean additive white Gaussian noise defined on fractal set have been formulated. The fractal time component has also been considered. The closed form expressions for crucial stochastic parameters of circuit responses have been derived from these equations. Numerical simulations of power spectral densities based on the derived autocorrelation functions have been performed. A comparison with those without fractal time component has been made.

We have found that the Hausdorff dimension of the middle b Cantor set strongly affects the power spectral densities; thus, the average powers of noise induced circuit responses and the inclusion of fractal time component causes significantly different analysis results besides the physical measurability of electrical quantities.

For the first time, the noise analysis of electrical circuit on fractal set has been performed. This is also the very first time that the fractal time component has been included in the fractal calculus-based circuit analysis.

Nizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept.

A two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system.

The NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems.

The research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones.

To the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

In this paper, we study a Cauchy-type problem for Hilfer fractional integrodifferential equations with boundary conditions. The existence of solutions for the given problem is proved by applying measure of noncompactness technique in an abstract weighted space. Moreover, we use generalized Gronwall inequality with singularity to establish continuous dependence and uniqueness of ϵ-approximate solutions.

The object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a bounded open domain in the n-dimensional Euclidean space.

The underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.

The theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.

Direct approach that considerably simplifies the analysis and computations.

It was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.

The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems.

In the presence of heterogeneity the exponent in the diffusion time scale can become anomalous, exhibiting values that differ from the expected value of 1/2. Here the author investigates, through directed numerical simulation, the two-dimensional melting of a phase change material (PCM) contained in a pattern of cavities separated by a non-PCM matrix. Under normal circumstances we would expect that the progress of melting F(t) would exhibit the normal diffusion time exponent, i.e., F∼t1/2. The author’s intention is to investigate what features of the PCM cavity pattern might induce anomalous phase change, where the progress of melting has a time exponent different from n=1/2.

When the PCM cavity pattern has an internal length scale, i.e., when there is a sub-domain pattern which, when reproduced, gives us the full domain pattern, the direct simulation recovers the normal ∼t1/2 phase change behavior. When, however, there is no internal length scale, e.g., the pattern is a truncated fractal, an anomalous super diffusive behavior results with melting going as t n; n > 1/2. By studying a range of related fractal patterns, the author is able to relate the observed sub-diffusive exponent to the cavity pattern’s fractal dimension. The author also shows, how the observed behavior can be modeled with a non-local fractional diffusion treatment and how sub-diffusion phase change behavior (F∼t n; n < 1/2) results when the phase change nature of the materials in the cavity and matrix are inverted.

Although the results clearly demonstrate under what circumstances anomalous phase change behavior can be practically produced, the question of an exact theoretical relationship between the cavity pattern geometry and the observed anomalous time exponent is not known.

The clear role of the influence of heterogeneity on heat flow behavior is illustrated. Suggesting that modeling heat and fluid flow in heterogeneous systems requires careful consideration.

The novel direct simulation of melting in a two-dimensional PCM cavity pattern provides a clear illustration of anomalous behavior in a classic heat and fluid flow system and by extension provides motivation to continue the numerical investigation of anomalous and non-local behaviors and fractional calculus tools.

The purpose of this paper is to employ a fractional approach to predict the permeability of nonwoven fabrics by simulating diffusion process.

The method described here follows a similar approach to anomalous diffusion process. The relationship between viscous hydraulic permeability and electrical conductivity of porous material is applied in the derivation of fractional power law of permeability.

The presented power law predicted by fractional method is validated by the results obtained from simulation of fluid flow around a 3D nonwoven porous material by using the lattice-Boltzmann approach. A relation between the fluid permeability and the fluid content (filling fraction), namely, following the power law of the form, was derived via a scaling argument. The exponent n is predominantly a function of pore-size distribution dimension and random walk dimension of the fluid.

The fractional scheme by simulating diffusion process presented in this paper is a new method to predict wicking fluid flow through nonwoven fabrics. The forecast approach can be applied to the prediction of the permeability of other porous materials.

The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus.

The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited.

The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead.

For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.