Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

A Hybrid ESA-CCD Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

Study of Fractional Order Dynamical System of Viral Infection Disease under Piecewise Derivative

This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the Caputo sense to investigate the crossover behavior of the considered dynamical system.We establish some qualitative results about the existence and uniqueness of the solution to the proposed problem.By utilizing the Newtonian polynomials interpolation technique,we recall a powerful algorithm to interpret the numerical findings for the aforesaid model.Here,we remark that the said viral infection is caused by an RNA type virus which can transmit from animals and also from an infected person to person.Fruits bats which are also known as flying foxes are one of the sources of transmission of NiV disease.Here in this work,we investigate its transmission mechanism through some new concepts of fractional calculus for further analysis and prediction.We present the approximate results for different compartments using different fractional orders.By using the piecewise derivative concept,we detect the crossover ormulti-steps behavior in the transmission dynamics of the mentioned disease.Therefore,the considered form of the derivative is used to deal with problems exhibiting crossover behaviors.

An incommensurate fractional discrete macroeconomic system:Bifurcation,chaos,and complexity

This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order.The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically.In particular,the zero equilibrium point stability is investigated to demonstrate that the discrete macroeconomic system exhibits chaotic behavior.Through using bifurcation diagrams,phase attractors,the maximum Lyapunov exponent and the 0–1 test,we verified that chaos exists in the new model with incommensurate fractional orders.Additionally,a complexity analysis is carried out utilizing the approximation entropy(ApEn)and C_(0)complexity to prove that chaos exists.Finally,the main findings of this study are presented using numerical simulations.

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Many constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one potential instrument for modelling memory-dependent phenomena. In this paper, the physical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail. Then a modified elasto-plastic constitutive model is proposed for evaluating the long-term deformation of granular soils under cyclic loading by incorporating the concept of fac- tional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow potential consider- ing particle breakage is used. Test results of several types of granular soils are used to validate the model performance.

Transfer function modeling and analysis of the open-loop Buck converter using the fractional calculus

Based on the fact that the real inductor and the real capacitor are fractional order in nature and the fractional calculus,the transfer function modeling and analysis of the open-loop Buck converter in a continuous conduction mode（CCM） operation are carried out in this paper.The fractional order small signal model and the corresponding equivalent circuit of the open-loop Buck converter in a CCM operation are presented.The transfer functions from the input voltage to the output voltage,from the input voltage to the inductor current,from the duty cycle to the output voltage,from the duty cycle to the inductor current,and the output impedance of the open-loop Buck converter in CCM operation are derived,and their bode diagrams and step responses are calculated,respectively.It is found that all the derived fractional order transfer functions of the system are influenced by the fractional orders of the inductor and the capacitor.Finally,the realization of the fractional order inductor and the fractional order capacitor is designed,and the corresponding PSIM circuit simulation results of the open-loop Buck converter in CCM operation are given to confirm the correctness of the derivations and the theoretical analysis.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

ON THE FRACTIONAL CALCULUS FUNCTIONS OF A FRACTAL FUNCTION

Based on the combination of fractional calculus with fractal functions, a new type of functions is introduced; the definition, graph, property and dimension of this function are discussed.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Fractional Order Modeling of Predicting COVID-19 with Isolation and Vaccination Strategies in Morocco

In this work,we present a model that uses the fractional order Caputo derivative for the novel Coronavirus disease 2019(COVID-19)with different hospitalization strategies for severe and mild cases and incorporate an awareness program.We generalize the SEIR model of the spread of COVID-19 with a private focus on the transmissibility of people who are aware of the disease and follow preventative health measures and people who are ignorant of the disease and do not follow preventive health measures.Moreover,individuals with severe,mild symptoms and asymptomatically infected are also considered.The basic reproduction number(R0)and local stability of the disease-free equilibrium(DFE)in terms of R0 are investigated.Also,the uniqueness and existence of the solution are studied.Numerical simulations are performed by using some real values of parameters.Furthermore,the immunization of a sample of aware susceptible individuals in the proposed model to forecast the effect of the vaccination is also considered.Also,an investigation of the effect of public awareness on transmission dynamics is one of our aim in this work.Finally,a prediction about the evolution of COVID-19 in 1000 days is given.For the qualitative theory of the existence of a solution,we use some tools of nonlinear analysis,including Lipschitz criteria.Also,for the numerical interpretation,we use the Adams-Moulton-Bashforth procedure.All the numerical results are presented graphically.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

A New Medical Image Enhancement Algorithm Based on Fractional Calculus

The enhancement of medical images is a challenging research task due to the unforeseeable variation in the quality of the captured images.The captured images may present with low contrast and low visibility,which might inuence the accuracy of the diagnosis process.To overcome this problem,this paper presents a new fractional integral entropy(FITE)that estimates the unforeseeable probabilities of image pixels,posing as the main contribution of the paper.The proposed model dynamically enhances the image based on the image contents.The main advantage of FITE lies in its capability to enhance the low contrast intensities through pixels’probability.Initially,the pixel probability of the fractional power is utilized to extract the illumination value from the pixels of the image.Next,the contrast of the image is then adjusted to enhance the regions with low visibility.Finally,the fractional integral entropy approach is implemented to enhance the low visibility contents from the input image.Tests were conducted on brain MRI,lungs CT,and kidney MRI scans datasets of different image qualities to show that the proposed model is robust and can withstand dramatic variations in quality.The obtained comparative results show that the proposed image enhancement model achieves the best BRISQUE and NIQE scores.Overall,this model improves the details of brain MRI,lungs CT,and kidney MRI scans,and could therefore potentially help the medical staff during the diagnosis process.

On the Connection between the Order of Riemann-Liouvile Fractional Calculus and Hausdorff Dimension of a Fractal Function

This paper investigates the fractal dimension of the fractional integrals of a fractal function. It has been proved that there exists some linear connection between the order of Riemann-Liouvile fractional integrals and the Hausdorff dimension of a fractal function.

Image Enhancement Using Adaptive Fractional Order Filter

Image enhancement is an important preprocessing task as the contrast is low in most of the medical images,Therefore,enhancement becomes the mandatory process before actual image processing should start.This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images,the proposed operators are based on Grunwald-Letnikov(G-L),Riemann-Liouville(R-L)and Caputo(Li&Xie),which are the definitions of fractional order calculus.In this fractional-order,differentiation is well focused on the enhancement of echocardiographic images.This provoked for developing a non-linear filter mask for image enhancement.The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features,particularly in smooth areas.The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions,details,and edges.Thereafter,a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image.It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis.The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement,histogram equalization,integer-order differential methods have been used.The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods.The values obtained are very close(almost equal to 99.9%)to the original values of the average gradient and entropy of the images.The results of the simulation validate the effectiveness of the proposed algorithm.

Adomian Decomposition Method for Solving Fractional Time-Klein-Gordon Equations Using Maple

Adomian decomposition is a semi-analytical approach to solving ordinary and partial differential equations. This study aims to apply the Adomian Decomposition Technique to obtain analytic solutions for linear and nonlinear time-fractional Klein-Gordon equations. The fractional derivatives are computed according to Caputo. Examples are provided. The findings show the explicitness, efficacy, and correctness of the used approach. Approximate solutions acquired by the decomposition method have been numerically assessed, given in the form of graphs and tables, and then these answers are compared with the actual solutions. The Adomian decomposition approach, which was used in this study, is a widely used and convergent method for the solutions of linear and non-linear time fractional Klein-Gordon equation.

An Image Enhancement Method Based on Fractional Calculus and Retinex

In order to resolve the problem of the image degradation, an image enhancement method based on fractional calculus and Retinex is proposed, which can preserve or enhance texture information and remove the noise of images. The fractional differential is used to preprocess the input image to enhance texture information, and using guided filter to estimate the illumination component, so it has less halo phenomena. The reflection component, obtained according to the Retinex theory, is denoised by fractional integral to remove the noises. The image is equalized by the contrast limited adaptive histogram equalization to improve the image contrast, and a final enhanced image is obtained. The experimental results show that the method can effectively achieve image enhancement, and the enhanced image has better visual effects.

Application of Fractional Calculus to a Linear Third Order (Nonhomogeneous and Homogeneous) Ordinary Differential Equation

In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-1)(β-2α-2)φ=f.

Particular Solutions of Generalized Linear Second Differential Equations by Fractional Calculus

In this parer, applications of the fractional calculus to the form (Az 2+Bz+C)ψ 2+(Dz+G)ψ 1+Eψ=f and the partial differential equation 2μz 2(Az 2+Bz+C)+(Dz+G)μz+δμ(z,t)=M 2μT 2+NμT, where ψ 1= d ψ d z and ψ 2= d 2ψ d z 2 are presented.