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This paper aims to investigate the conformable fractal approaches of unsteady natural convection in a partial layer porous H-shaped cavity suspended by nano-encapsulated phase change material (NEPCM) by the incompressible smoothed particle hydrodynamics method.

The partial hot sources with variable height L_Hot are in the H-cavity’s sides and center. The performed numerical simulations are obtained at the variations of the following parameters: source of hot length L_Hot = (0.4–1.6), conformable fractal parameter α (0.97–1), fusion temperature θ_f (0.05–0.9), thermal radiation parameter Rd (0–7), Rayleigh number Ra (10³–10⁶), Darcy parameter Da (10⁻² to 10⁻⁵) and Hartmann number Ha (0–80).

The main outcomes showed the implication of hot source length L_Hot, Rayleigh number and fusion temperature in controlling the contours of a heat capacity within H-shaped cavity. The presence of a porous layer in the right zone of H-shaped cavity prevents the nanofluid flow within this area at lower Darcy parameter. An increment in the thermal radiation parameter declines the heat transfer and changes the heat capacity contours within H-shaped cavity. The velocity field is strongly enhanced by an augmentation on Rayleigh number. Increasing the Hartmann number shrinks the velocity field within H-shaped cavity.

The novelty of this work is solving the conformable fractal approaches of unsteady natural convection in a partial layer porous H-shaped cavity suspended by NEPCM.

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 main aim of this paper is to investigate the fractional coupled nonlinear Helmholtz equation by two new analytical methods.

This article takes an inaugural look at the fractional coupled nonlinear Helmholtz equation by using the conformable derivative. It successfully finds new fractional periodic solutions and solitary wave solutions by employing methods such as the fractional method and the fractional simple equation method. The dynamics of these fractional periodic solutions and solitary wave solutions are then graphically represented in 3D with appropriate parameters and fractal dimensions. This research contributes to a deeper comprehension and detailed exploration of the dynamics involved in high dimensional solitary wave propagation.

The proposed two mathematical approaches are simple and efficient to solve fractional evolution equations.

The fractional coupled nonlinear Helmholtz equation is described by using the conformable derivative for the first time. The obtained fractional periodic solutions and solitary wave solutions are completely new.

On a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM).

The author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM.

He’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution.

The author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.

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.

This manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.

The generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.

The results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.

The work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.

The generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.

Public health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.

The use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.

The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.

Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.

This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative.

The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs).

This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance.

The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

Understanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.

The results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.

For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered.

This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian.

To the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.