Fractional Differential Models Research Articles

Novel Fractional Order Differential and Integral Models for Wind Turbine Power–Velocity Characteristics

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (α) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at α=0.9 gives the best accuracy of WTPCs, while the original linear model was the least accurate.

A Description of the Isothermal Ageing Creep Process in Polymethyl Methacrylate Using Fractional Differential Models.

Fractional differential viscoelastic models can describe complex material behaviours and fit experimental data well; however, the physical significance of model parameters is difficult to express. In this study, the fractional differential Maxwell, Kelvin, and Zener models were used to fit the short-term creep compliance curves of polymethyl methacrylate at different ageing times. The model fits were in good agreement with the experimental data. As the ageing time increased, the fractional differential Zener model showed a relative increase in the modulus parameter of the spring and a relative decrease in the modulus parameter reflecting the viscosity of the spring-pot, which indicated that physical ageing made the material more elastic. The relaxation time of the material increased, which indicated that the physical ageing reduced the free volume of the material, hindered the movement of molecules/segments, and increased the time required for the material to reach equilibrium. The fractional order of the model decreased, which reflected the phenomenon that physical ageing reduced the creep compliance of the material. Using the relaxation time as the time scale, the creep curves at different ageing times under the same stress level could be superimposed, naturally presenting the time-ageing time equivalence principle.

Solutions of fractional differential models by using Sumudu transform method and its hybrid

This paper presents the Sumudu transform method and its hybrid for the construction of solutions of differential equations, both with integer-order and fractional derivatives. The paper discusses the construction of solutions of fractional differential equations with varying delay proportional to the independent variable by using two methods. The paper considers the mathematical model for the decay of Iodine 135 as an application of fractional differential equations in nuclear physics. The application indicates that fractional differential equations with variable delay proportional to the independent variable are a useful tool for the modeling of many anomalous phenomena in nature and in the theory of complex systems.

Об идентификации характеристик неоднородных вязкоупругих тел в рамках модели дробного порядка

Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.

A Novel Semi-Analytical Scheme to Deal with Fractional Partial Differential Equations (PDEs) of Variable-Order

This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to approximate the solution of such equations. The coefficients of this combination are considered a function of time, and they are obtained using the collocation method. The theoretical aspects of the method are investigated, and then by solving some problems, the efficiency of the method is presented.

Exploring exact solutions for physical differential models through generalized derivatives

In this study, we develop three well-known fractional differential physical models with novel exact solutions. Liouville, Dodd-Bullough-Mikhailov (DBM), and Sinh-Gordon equations are the models under consideration. These models will be broken down into three nonlinear ordinary differential equations using a waveform transformation, which can be precisely solved using the approach of the simplest equation method. The suggested method is applicable to several categories of nonlinear physical models and allows us to extract numerous generalized solutions in soliton and periodic forms The resulting solutions may also be directly compared with a number of findings obtained in the literature. Additionally, representations in two and three dimensions are provided to show how changing the fractional parameter’s amount may impact how monotonic the solutions are obtained.

Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics.

Thermal–viscoelastic analysis of polymethyl methacrylate using a fractional differential viscoelastic model

Vacuum forming is used to manufacture large molded parts. As forming conditions have a significant effect on the dimensional accuracy, these should be determined accordingly. In this study, a geometric nonlinear creep analysis of polymethyl methacrylate (PMMA), which is a common thermoplastic resin, was carried out at the target temperature of 393.15[Formula: see text]K and target strain of approximately 50% for vacuum forming. The proposed fractional differential viscoelastic model was extended to a three-element model, consisting of a single hyperelastic spring and two fractional differential (FD) models. It was further extended by time–temperature superposition (TTS) for thermo-viscoelastic analysis. The model determined all material constants by measuring the temperature/frequency sweeps at small strain amplitudes of 0.01% using dynamic mechanical analysis (DMA). Numerical analysis confirmed the validity of the proposed method through creep and stress-relaxation tests by DMA at the target temperature/strain. The results demonstrated that the finite element analysis constructed using the proposed method could predict the mechanical properties during vacuum-forming-oriented creep tests. These results are expected to provide important insights into the complex mechanical behavior of PMMA, which varies with the temperature and strain rate.

Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives

Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Numerical simulation of time fractional Benjamin-Bona-Mahony-Burger equation describing propagation of long waves on the water surface

The unidirectional propagation of long waves in a certain nonlinear dispersive system is explained in the Benjamin-Bona-Mahony-Burger equation. The motivation of this manuscript is to examine the Benjamin-Bona-Mahony-Burger equation numerically and analytically by employing reproducing kernel functions. The implementation of reproducing kernel Hilbert space method produces an analytical solution in the form of a series representation, and after truncating the infinite series, an approximate solution is attained. With the support of numerical examples, the accuracy of this method has been shown through graphs and error tables. The convergence analysis of the introduced scheme has been explained by some useful theorems and lemmas. Finally, it has been observed that the approximate solution consistently converges to the exact solution with less computational overhead. The proposed method would be very effective in solving various fractional partial differential equation models.

Non classical interaction aspects to a nonlinear physical model

This study deals the dynamics of waves to the conformable fractional (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations. The (2+1)-dimensional NNV equations are the isotropic Lax integrable extension of the (1+1)-dimensional Korteweg–de Vries equations. Fractional differential models (FDMs) from the corresponding integer order model can describes more complex behavior and even cover all properties of integer order model. By the usage of different test approaches, the Hirota bilinear method (HBM) has been successfully applied. The Hirota method is a well-known and reliable mathematical tool for finding the soliton solutions of fractional nonlinear partial differential equations in many fields. However, it demands bilinearization of nonlinear fractional PDE. A class of results in the shapes of lump-periodic, breather-type and two wave solutions have been extracted. Numerical visualizations of the results are also used to demonstrate the implications of the fractionality and parameters. It is acceptable to assume that many experts in engineering models will learn from this research. The results prove the used algorithm is effective, quick, easy, and flexible in its application to various systems.

A method for solving Caputo–Hadamard fractional initial and boundary value problems

In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order , and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models.

Lie analysis and laws of conservation for the two-dimensional model of Newell–Whitehead–Segel regarding the Riemann operator fractional scheme in a time-independent variable

The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in 2 D -space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized. HIGHLIGHT The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived. The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time. The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions.

On the Solution of the Variable Order Time Fractional Schrödinger Equation

In the present paper, we used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional Schrödinger equations. Twelve illustrative applications have been presented. When the fractional order is unity, the results are the same as those obtained by the standard integer order derivative. Some of the obtained results are discussed graphically. Some of the obtained results 1st appear in the literature. This technique can be applied to other linear or nonlinear fractional differential models.
 HIGHLIGHTS
 
 We study a class of time variable-order fractional nonlinear Schrödinger equations (tVOFNLSEs)
 We introduce a new procedure for solving the tVOFNLSE and tVOFSE
 We used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional equations
 The calculations are performed in the frame of Caputo sense
 
 GRAPHICAL ABSTRACT

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

A scale-dependent hybrid algorithm for multi-dimensional time fractional differential equations

Time fractional differential model is an effective tool to characterize anomalous diffusion phenomena in hydrology and environmental science. Efficient numerical method is necessary to overcome the bottleneck of expensive computational cost for real-world application. Therefore, this paper proposes a scale-dependent hybrid algorithm to numerically solve multi-dimensional time fractional differential models. We employ the Hausdorff metric-based hybrid algorithm to discretize time terms of time fractional differential equations (FDEs) with variable step size, which is applicable to non-uniform time steps geological problems. Meantime, another advantage of the proposed algorithm is optimizing the computational process. The method only requires O( $$n_s$$ $$n_e$$ ) memory and O( $$n_s$$ $$n_t$$ $$n_e$$ ) computational cost while classical finite difference method relatively demands O( $$n_s$$ $$n_t$$ ) and O( $$n_s$$ $$n_t^2$$ ), where $$n_s$$ , $$n_t$$ and $$n_e$$ are the number of space nodes, time steps and exponentials, respectively. Furthermore, we adopt a meshless generalized finite difference method to discretize space terms of FDEs. Robustness and accuracy of the new algorithm are verified by two examples and a set of experimental data.

Efficient high-order exponential time differencing methods for nonlinear fractional differential models

Efficient high-order exponential time differencing methods for nonlinear fractional differential models

Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

We examine in this paper some new problems on coincidence point and fixed point theorems for multivalued mappings in metric space. By applying the characterizations of a modified M T ~ -function, under the name D -function, a few novel fixed point results different from the existing fixed point theorems are launched. It is well-known that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to any differential equation inherits all the regularity properties of the mapping involved and of the solution itself. This does not hold in the case of differential inclusions. In particular, fractional-order differential inclusion models are more suitable for describing epidemics. Thus, as a generalization of a newly launched existence result for fractional-order model for COVID-19, using Banach and Shauder fixed point theorems, we investigate solvability criteria of a novel Caputo-type fractional-order differential inclusion model for COVID-19 by applying a standard fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the framework of Ulam-Hyers is also discussed. Nontrivial comparative illustrations are constructed to show that our ideas herein complement, unify and, extend a significant number of existing results in the corresponding literature.

An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices

Purpose:In the last decades, many researchers have performed their best in order to find the solution of the nonlinear evolution equations, nonlinear partial differential equations (NLPDEs), fractional nonlinear partial differential equations and nonlinear optical models by using Kudryashov methods, which have various forms. However, most researchers have used different methods under the same name or applied the same method under the different names. This situation has created confusion for researchers and readers, such as choosing the appropriate method, algorithm and the right article title. In order to eliminate this problem and to clarify the methods, we have conducted rigorous research and wrote a certain standard in this field by bringing together the Kudryashov methods, which are widely used under different names in the literature. Besides, it has been aimed to give the general algorithm of the methods, to make some addendums on the methods and to present new approaches to these methods for the researchers. We strongly believe that the study will be of great benefit not only for those who are interested in nonlinear evolution problems, but also for studies in the field optics. Methodology:Regarding the methods, first of all, the articles of the researchers who suggested the methods and the studies of the respected researchers in this field have been examined and tried to gather under one framework and give a general name with appropriate algorithm for each Kudryashov method. The algorithms and solution functions of each method have been checked on the related Riccati and Bernoulli equations with different softwares (Mathematica, Maple, Matlab, etc.), and confirmed every results carefully. Afterwards, all methods have been applied on (2+1)-dimensional Zoomeron equation for the global test purpose. With the usual standard approach, the nonlinear partial differential form of Zoomeron equation has been transformed into the nonlinear ordinary differential (NODE) form by applying the wave transformation, then each method applied to the NODE form of Zoomeron equation. After these operations, we have obtained a system of linear algebraic equations and their appropriate solutions. By obtaining the appropriate solution sets, the soliton solutions of the Zoomeron equation have been constructed and some selected graphical presentations given. Findings:The results have been obtained from the applications that applied on the Zoomeron equation via Kudryashov methods shows us they can be used widely and effectively in the solution of not only nonlinear evolution problems, but also highly dispersive and higher order nonlinear partial differential equations. It has been seen that all the applied Kudryashov methods give effective results. Besides, controlling and using the parameters in the Kudryashov methods play important role, especially obtaining the effective and desired solution sets. Originality:Based on our detailed research, we could not find any summative study in this area like this one. In addition to all these, some of the approaches and addendums have been presented for the first time in this paper. Besides, (2+1)-dimensional Zoomeron equation has also not been previously solved by some of these methods. It is thought that this study will be used as a reference by many researchers in the future and it will be of great benefit to researchers.