Fractional Order Research Articles

Numerical investigation of two-dimensional fractional Helmholtz equation using Aboodh transform scheme

Purpose This paper aims to present a numerical investigation for two-dimensional fractional Helmholtz equation using the Aboodh integral homotopy perturbation transform scheme (AIHPTS). Design/methodology/approach The proposed scheme combines the Aboodh integral transform and the homotopy perturbation scheme (HPS). This strategy is based on an updated form of Taylor’s series that yields a convergent series solution. This study analyzes the fractional derivatives in the context of Caputo. Findings This study illustrates two numerical examples and calculates their approximate results using AIHPTS. The derived findings are also presented in tabular form and graphical representations. Research limitations/implications In addition, He’s polynomials are calculated using HPS, so the minimal computational outcome is a defining feature of this method and gives a competitive advantage over other series solution techniques. Originality/value Numerical data and graphical illustrations for different fractional order levels confirm the proposed method’s successful performance. The results show that the proposed approach is speedy and straightforward to execute on fractional-ordered models.

Study of fractional order rabies transmission model via Atangana–Baleanu derivative

In this work, we aim at disentangling the theoretical contribution through mathematical modeling approach to advance the understanding of rabies dynamics and control in livestock population. A fractional order model of rabies, using Atangana–Baleanu fractional operator is created. The analysis of suggested system and its application are both conducted. The capacity of proposed model to forecast the disease can help researchers and livestock health care agencies to take preventive actions.

Thermoelastic bending wave propagation of FG hybrid nanocomposite microbeam reinforced by GPLs and CNTs under fractional order nonlocal elasticity theory

Thermoelastic bending wave propagation of FG hybrid nanocomposite microbeam reinforced by GPLs and CNTs under fractional order nonlocal elasticity theory

Fractional order calculus enhanced dung beetle optimizer for function global optimization and multilevel threshold medical image segmentation

Fractional order calculus enhanced dung beetle optimizer for function global optimization and multilevel threshold medical image segmentation

1:2 Strong Resonance and Hybrid Control of Discrete Conformable Fractional Order Bacteria Population Model

1:2 Strong Resonance and Hybrid Control of Discrete Conformable Fractional Order Bacteria Population Model

Performance analysis of DC-DC Buck converter with innovative multi-stage PIDn(1+PD) controller using GEO algorithm

Power electronic converters are widely used in various fields of electrical equipment. Due to their fast dynamics and non-linear nature, controlling them requires dealing with various complexities. Therefore, having a well-designed, high-speed, and robust controller is critical to ensure the effective operation of these devices. In a DC-DC converter, steady-state performance with minimum error and fast dynamic response relies on controller design. This paper presents the design of a multi-stage PID controller with an N-filter combined with a one plus proportional derivative (1+PD) controller. This controller illustrates fast tracking reference voltage; additionally, it shows incredible results when the DC-DC converter operates in different modes. The parameters of the proposed controller are effectively determined using the golden eagle optimization (GEO) algorithm. Furthermore, a comprehensive comparison between the proposed controller, proportional–integral–derivative (PID), and fractional order PID (FOPID) controllers, as well as different metaheuristic optimization methods in various conditions, has been conducted to demonstrate the effectiveness of the proposed controller. The behavior of the closed-loop system under different conditions has been thoroughly investigated. The superior time and frequency domain characteristics of the closed-loop system with the PIDn(1+PD) controller highlight its superiority over other controllers. The demonstrated enhancements in settling time, voltage regulation accuracy, and transient response emphasize the potential applicability of the proposed control strategy in real-world power electronics systems, particularly in scenarios requiring high efficiency, stability, and dynamic performance.

Frequency stabilization of interconnected diverse power systems with integration of renewable energies and energy storage systems

Recent improvements in renewable energy sources (RESs) and their extensive use in the power industry have created significant issues for their operation, security, and management. Due to the ongoing reduction of power system inertia, maintaining operational frequency at its nominal value and minimizing tie-line power variations constitute essential variables for these improvements. A novel improved frequency stabilization approach based on modified fractional order tilt controller is presented for interconnected diverse power systems with integration of sea wave energy, photovoltaic, wind, energy storage system and biodiesel generator. The fractional order tilt integral double derivative (FOTIDD2) is a novel controller that is developed to address the frequency stabilization problems of a hybrid power structure that is integrated with sources of clean energy and devices for storing energy. A recent optimization algorithm inspired by human behavior called mother optimization algorithm (MOA) is applied to adjust the coefficient of the proposed cascaded controller. The FOTIDD2 controller was compared with other control methods in terms of its transient performance, in order to determine its effectiveness. Further, the mother optimization algorithm results are compared with those of sine cosine algorithm (SCA), fox optimization algorithm (FOA), and grey wolf optimization (GWO). FOTIDD2 controller was found to be more effective than PID controller in respect of reducing overshoot (Osh) by 79.31%, 83.78% and 67.99%, improving time settlement by 33.22%, 29.87%, and 22.45% and reducing undershoot (Ush) by 79.13%, 89.34%, and 86.90% for the (∆F1), (∆F2), and (∆Ptie), respectively.

A PROBLEM IN FRACTIONAL ORDER THERMO-VISCOELASTICITY THEORY FOR A POLYMER MICRO-ROD WITH AND WITHOUT ENERGY DISSIPATION

A new study has been developed that considers the size-dependent interaction between viscoelastic deformation and thermal fields, incorporating the fractional heat conduction law with and without energy dissipation. The model is used for a particular one-dimensional problem involving a polymer micro-rod of arbitrary length experiencing three different types of thermal loading without the presence of any heat source. The study uses Laplace transforms and numerical inversion to examine how fractional order, nonlocal elasticity, and nonlocal thermal conduction impact thermal dispersion and thermo-viscoelastic response. Comparative numbers demonstrate the effects of various parameters. Findings demonstrate that nonlocal thermal and viscoelastic characteristics have a significant impact on all recorded field values, offering possible suggestions for the creation and assessment of thermal-mechanical attributes in nanoscale devices.

Galerkin approximation of Burgers-Huxley equation with fractional order arising in reaction mechanism and diffusion transport

Abstract Burgers-Huxley model depicts a prototype model of the interaction of convection effect, reaction mechanisms and diffusion transport, used to study the liquid crystal and nerve fibers. This study introduces Galerkin approximation for time-fractional Burgers-Huxley equation (TFBHE) . The Caputo derivative is used to evaluate the temporal part using the $L_1$ formula. The Galerkin approach employs cubic B-spline as a shape and test function, resulting in a symmetric matrix that is easily convergent. In addition, the three-point quadrature rule is implemented to evaluate the integration of complex function . The Von Neumann analysis is used to discuss stability of the scheme. The performance and robustness of the technique is measured using various error norms. The results are compared with the exact solution, demonstrating effectiveness of the proposed method.

Fractional Einstein–Gauss–Bonnet Scalar Field Cosmology

Our paper introduces a new theoretical framework called the Fractional Einstein–Gauss–Bonnet scalar field cosmology, which has important physical implications. Using fractional calculus to modify the gravitational action integral, we derived a modified Friedmann equation and a modified Klein–Gordon equation. Our research reveals non-trivial solutions associated with exponential potential, exponential couplings to the Gauss–Bonnet term, and a logarithmic scalar field, which are dependent on two cosmological parameters, m and α0=t0H0 and the fractional derivative order μ. By employing linear stability theory, we reveal the phase space structure and analyze the dynamic effects of the Gauss–Bonnet couplings. The scaling behavior at some equilibrium points reveals that the geometric corrections in the coupling to the Gauss–Bonnet scalar can mimic the behavior of the dark sector in modified gravity. Using data from cosmic chronometers, type Ia supernovae, supermassive Black Hole Shadows, and strong gravitational lensing, we estimated the values of m and α0, indicating that the solution is consistent with an accelerated expansion at late times with the values α0=1.38±0.05, m=1.44±0.05, and μ=1.48±0.17 (consistent with Ωm,0=0.311±0.016 and h=0.712±0.007), resulting in an age of the Universe t0=19.0±0.7 [Gyr] at 1σ CL. Ultimately, we obtained late-time accelerating power-law solutions supported by the most recent cosmological data, and we proposed an alternative explanation for the origin of cosmic acceleration other than ΛCDM. Our results generalize and significantly improve previous achievements in the literature, highlighting the practical implications of fractional calculus in cosmology.

Mathematical modelling, energy consumption, and quality evaluation of wheat seeds subjected to intermittent drying

AbstractThis work aims to evaluate the kinetic profile of intermittent drying of wheat seeds using traditional models from the literature and the fractional calculus technique. Furthermore, it aims to verify the application of the intermittent drying process on the amount of antioxidant compounds, protein, and lipid content of the grain, in addition to energy consumption to obtain the desired final moisture content of the wheat. It was verified that the prediction of drying kinetics by Page and fraction order models were similar (modelling efficiency varying between the range of 0.917–0.995, varying the drying condition and wheat cultivar). Regarding antioxidant compounds for the three wheat cultivars, it can be seen that the higher fraction of ethanol (74.0% and 90.36%) used for extraction had greater process efficiency. Regarding the protein content in the three wheat cultivars, lower drying temperatures and intermittency periods result in lower quality losses of material (12.3% for BRS‐Atobá wheat, 9.89% for BRS‐Jacana wheat, and 18.71% for BRS‐Sanhaço wheat). In terms of lipids, it was found that the influence of temperature was greater on the lipid content than on the protein content of the material (for the best drying condition, there were percentage decreases for the best condition of 36.16% for the BRS‐Atobá wheat, 34.9% for BRS‐Jacana, and 52.2% for BRS‐Sanhaço). Regarding energy consumption during the drying process, it can be seen that the conditions used for intermittency and drying time, in addition to the sample conditions, directly impact energy consumption.

A frequency domain fractional order tilted integral derivative controller design for fractional order time delay processes

AbstractIn real‐time processes, time delays are inherent due to various factors, such as delays in volume, mass, and information transfer, leading to a deterioration in process performance and stability. Therefore, this paper proposes a fractional‐order tilted integral derivative (TID) controller design for stable and unstable fractional‐order processes with time delay, where the optimal controller parameters are designed using a deterministic approach. An inner loop controller is designed using stability analysis and a graphical approach for unstable fractional‐order time delay processes. The proposed approach offers flexibility and fine‐tuning in designing the controller for a specific application. A systematic reference to the disturbance ratio is carried out to assess the disturbance handling capacity of the proposed controller. It shows the sensitivity of the designed controller against disturbances in the frequency spectrum graphically. Numerous performance indices and time‐domain parameters are computed and compared with state‐of‐the‐art techniques to analyze the efficacy. Furthermore, it is validated on the experimental setup of a four‐tank system. Simulation and experimental results demonstrate that the proposed approach outperforms recent techniques in terms of process control and disturbance rejection.

Modal identification of wind turbine tower based on optimal fractional order statistical moments

AbstractIn vibration testing of civil engineering structures, the first two vibration modes are crucial in representing the global dynamic behavior of the structure measured. In the present study, a comprehensive method is proposed to identify the first two vibration modes of wind turbine towers, which is based on the analysis of fractional order statistical moments (FSM). This study offers novel contributions in two key aspects: (1) theoretical derivations of the relationship between FSM and vibration mode; and (2) successful use of 32/7‐order displacement statistical moment as the optimal FSM to identify wind turbine tower modes, by combining with noise resistance analysis, sensitivity analysis, and stability analysis, respectively. Using the proposed method, the FSM was first used to identify the modal vibration of wind turbine towers. By obtaining the response of the structure on the same vertical line, FSM was then calculated to estimate the corresponding structural modal vibration. Considering other influencing factors in the field test, the modal identification results of this index under different excitation forms and noise conditions were analyzed based on numerical simulation and verified with field wind tower test data. The results of the evaluation show that the proposed statistical moments of can accurately identify the first two vibration modes of wind turbine towers. This presents a new robust method for modal vibration identification, that is, simple and effective in its implementation.

Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag–Leffler kernel

This emerging infectious disease poses one of the most severe threats to public health in these locations, but there are not many reliable therapies yet. In this work, we developed the Ebola virus dynamics and control factors epidemic model with a piecewise hybrid fractional Operator in time scale measure insight of Mittag–Leffler kernel. Patterns and structures that repeat at various scales are the focus of fractal analysis, which has applications in complex systems such as biological ones. Both qualitatively and statistically, a proposed model with the Lipschitz criteria and linear growth is examined, considering positive solutions, boundedness, and uniqueness at equilibrium points with Leray-Schauder results under time scale ideas. The regulation for linear responses approach will be used by Chaos Control to stabilize the system after its equilibrium points. A fractional-order framework with a controlled design will be considered, where solutions are bounded in the feasible domain of relations of different compartments. Ulam–Hyers stability results in the solution are treated when function (constant or rising) for the component of qualitative inquiry in generalized form. The dynamical behaviors of the suggested model are discussed with the Newton polynomial approach used to implement on model in the sense of classical piecewise and Mittag–Leffler kernel at different fractional order values. The model shows that solutions are stable and confined within a feasible range, ensuring reliability. Through detailed simulations, it effectively captures how different interventions and infection rates influence Ebola spread. This fractional-order model enhances understanding of Ebola transmission, providing a strong basis for predicting outbreaks and planning effective control measures, with practical applications for analyzing real-world data.

Bayesian approach for identifying fractional order and time-dependent source in a fractional pseudo-parabolic equation

Bayesian approach for identifying fractional order and time-dependent source in a fractional pseudo-parabolic equation

Fractional Viscoelastic Analysis of Transversely Isotropic Surrounding Rock Along Shallow Buried Elliptical Tunnel

ABSTRACTThis study introduces the fractional order Merchant model to analytically solve the stress and displacement fields of the transversely isotropic viscoelastic surrounding rock along shallow elliptical tunnels. First, the stress and displacement solutions of fractional order viscoelastic and transversely isotropic half planes under arbitrary loads are obtained using the Laplace transform and the elastic‐viscoelastic correspondence principle. Second, based on the above half plane solution and the solution of the deep buried elliptical tunnel problem, the Schwarz alternating method is introduced to obtain the analytical solution of the shallow buried elliptical tunnel. A MATLAB program is developed, and the accuracy of the theory and program in this study is verified by comparing it with the results of ABAQUS. Finally, the effects of transversely isotropic parameters, tunnel burial depth, and viscoelastic parameters on the stress and displacement of tunnel surrounding rock are analyzed.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis 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.Design/methodology/approachThe 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.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe 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.Practical implicationsThe 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.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe 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.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

Retraction Note: A study on the reflection of quasi Rayleigh waves in a magneto self reinforced thermoelastic medium with fractional order derivative

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.