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In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new fractional derivative operator in which the generalized cardinal sine function is used as a non-singular analytic kernel. In addition, we provided the corresponding fractional integral operator. We expressed the new fractional derivative and integral operators as sums in terms of the Riemann–Liouville fractional integral operator. Next, we introduced an efficient extension of the new fractional operator that includes integrable singular kernel to overcome the initialization problem for related differential equations. We also proposed a numerical approach for the numerical simulation of IVPs incorporating the proposed extended fractional derivatives. The proposed fractional operators, the developed relations and the presented numerical method are expected to be employed in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

This paper introduces a deterministic fractional-order epidemic model (FOEM) for studying the transmission dynamics of the human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS). The model highlights the substantial role of unaware and undetected HIV-infected individuals in spreading the disease. Control strategies, such as wielding condoms, level of preventive measures to avoid infection, and self-strictness of susceptibles in sexual contact, have been incorporated into the study. The basic reproduction number ℛ 0 α has been derived, which suggests the conditions for ensuring the persistence and elimination of the disease. Further, to validate the model, actual HIV data taken from Mexico and India separately have been used. The disease dynamics and its control in both countries are analyzed broadly. The values of biological parameters are estimated at which numerical solutions better match the actual data of HIV patients in the case of fractional-order (FO) instead of integer-order (IO). Moreover, in the light of ℛ 0 α , our findings forecast that the disease will abide in the population in Mexico, and at the same time, it will die out from India after a long time. [ABSTRACT FROM AUTHOR]

In this paper, a special class of fractional Sturm–Liouville problems of order α (3 < α ≤ 4) is discussed. The method is based on utilizing the shifted Legendre polynomials on the interval [ 0 , L ] to find numerical approximations for eigenvalues and corresponding eigenfunctions. The proposed method is implemented on several numerical examples. The numerical results indicate the high performance of the method, its effectiveness, and its simplicity in use. [ABSTRACT FROM AUTHOR]

Stefan melting problems involve the tracking of a sharp melt front during the heat conduction controlled melting of a solid. A feature of this problem is a jump discontinuity in the heat flux across the melt interface. Time fractional versions of this problem introduce fractional time derivatives into the governing equations. Starting from an appropriate thermodynamic balance statement, this paper develops a new sharp interface time fractional Stefan melting problem, the memory-enthalpy formulation. A mathematical analysis reveals that this formulation exhibits a natural regularization in that, unlike the classic Stefan problem, the flux is continuous across the melt interface. It is also shown how the memory-enthalpy formulation, along with previously reported time fractional Stefan problems based on a memory-flux, can be derived by starting from a generic continuity equation and melt front condition. The paper closes by mathematically proving that the memory-enthalpy fractional Stefan formulation is equivalent to the previous memory-flux formulations. A result that provides a thermodynamic consistent basis for a widely used and investigated class of time fractional (memory) Stefan problems. • A new time fractional Stefan problem is presented, the memory-enthalpy formulation. • The problem is obtained from an appropriate thermodynamic balance statement. • We prove that this formulation exhibits a natural regularization of the problem. • A comparison with the previous memory-flux formulation is made. • We prove that the memory-enthalpy formulation is equivalent to the memory-flux one. [ABSTRACT FROM AUTHOR]

In this paper, a method based on radial basis function finite difference(RBF-FD) is developed for solving the time fractional convection-diffusion-wave equation(TFCDWE). We first approximate the equation by a scheme of order O (τ + h 2) , where τ , h are the time step size and spatial step size, respectively. We prove the stability and convergence of the discrete scheme, then the multiquadric RBF-FD approach is used to approximate the spatial derivatives. The aim of this paper is to show that the RBF-FD method is useful for solving our mentioned equation when the shape parameter selection is appropriate. The proposed method can be applied to complex domain, and has the advantages of mesh-free and simple procedure. Finally, numerical examples are proposed to verify the correctness of our previous theoretical analysis and to demonstrate the superiority of the RBF-FD method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the scaling invariance of survival probability in the Caputo fractional standard map of the order 1 < α < 2 considered on a cylinder. We consider relatively large values of the nonlinearity parameter K for which the map is chaotic. The survival probability has a short plateau followed by an exponential decay and is scaling invariant for all considered values of α and K. • The Caputo Fractional Standard Map is derived by a kicked differential equations of motion considering Caputo derivatives. • A formal mathematical derivation of fractional map is obtained by a generalization of Volterra integral for Caputo derivative. • The Caputo fractional standard map survival probability is scaling invariant for all control parameters. [ABSTRACT FROM AUTHOR]

The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial–temporal homogeneous and isotropic random fields and their spherical restrictions are studied. The spectral representations of these fields are derived and the associated angular spectrum is analysed. The obtained mathematical results are illustrated by numerical examples. In addition, the numerical investigations assess the dependence of the covariance structure and other properties of these fields on the orders of fractional derivatives. • The hyperbolic diffusion equation is studied with two diverse fractional derivatives. • The solutions are given as spatial–temporal homogeneous and isotropic random fields. • The obtained mathematical results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents the study of fractional Emden–Fowler (FEF) equations by utilizinga new adequate procedure, specifically the q- homotopy analysis transform method (q- HATM). The EF equation has got greater significance in both physical and mathematical investigation of capillary and nonlinear dispersive gravity waves. The projected technique is tested by considering four illustrations of the time-fractional EF equations. The q -HATM furnish ℏ , known as an auxiliary parameter, by the support of ℏ we can modulate the various stages of convergence of the series solution. Additionally, to certify the resolution and accurateness of the proposed method we fitted the suitable numerical simulations. The redeem results guarantee that the proposed process is more convincing and scrutinizes the extremely nonlinear issues emerging in the field of science and engineering. • q -HATM is applied to obtain the solution to fractional Emden-Fowler equations. • Accuracy of the proposed method is validated through suitable numerical simulations. • The proposed method is efficient to study nonlinear issues in the areas of science. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in (0 , 1). If its solution u (x , t) has strong regularity, for example u (⋅ , t) ∈ C 2 [ 0 , T ] for a given time T , then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u (x , t) likely behaves a certain regularity at the starting time, i.e., ∂ u ∂ t and ∂ 2 u ∂ 2 t can blow up as t → 0 + albeit u (⋅ , t) ∈ C [ 0 , T ] for a given time T. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The authors' concern of the present paper is to conduct a systematic study on a time-fractional nonlinear water wave equation which is an evolutionary version of the Boussinesq system. The study goes on by adopting a new analytical method based on the Laplace transform and the homotopy analysis method to the governing model and obtaining its approximate solutions in the presence of the Caputo fractional derivative. To analyze the influence of the Caputo operator on the dynamical behavior of the approximate solutions, some graphical illustrations in two- and three-dimensions are formally presented. Furthermore, several numerical tables are given to support the performance of the new analytical method in handling the time-fractional nonlinear water wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the numerical computation for a class of fourth-order fractional sub-diffusion equations with spatially variable coefficient under the first Dirichlet boundary conditions. Two finite difference schemes with second order accuracy are derived by applying L 2 − 1 σ formula and FL 2 − 1 σ formula respectively to approximate the time Caputo derivative. The main novelty is that a novel technique is introduced to deal with the first Dirichlet boundary conditions, which is compatible with the main equation with spatially variable coefficient. The solvability, unconditional stability and convergence of both schemes are proved by using the discrete energy method and mathematical induction. A difference scheme for such problem with two dimensions is also proposed and analyzed. Numerical results show that the suggested schemes have the almost same accuracy and the FL 2 − 1 σ scheme can reduce the storage and computational cost significantly. • The variable coefficient time fractional fourth-order sub-diffusion equation is considered. • A novel technique is introduced to deal with the first Dirichlet boundary conditions. • The resulting schemes proved to be unconditionally stable and convergent. • The fast scheme can reduce the computational complexity significantly. • The unconditionally stability and convergence of suggested schemes are proved. [ABSTRACT FROM AUTHOR]

Heartwater is a tick-borne illness that affects ruminants and is carried by the amblyomma ticks. The condition may sometimes be deadly. This paper studies the Caputo fractional version of the disease spread in domestic ruminants and amblyomma ticks. We obtain the positivity and boundedness condition through the Laplace transform. The stability state of the proposed model is obtained using the Ulam–Hyers and Ulam–Hyers–Rassias stability conditions. The heartwater-free equilibrium point is obtained. The fractional model is fitted to the heartwater incidence data from 2006 to 2019. The heartwater reproduction number, R 0 τ , from the parameter estimation is R 0 τ = 1. 9345 with a fitting fractional order, τ , of 0.6990. The residuals from the data fitting were randomly distributed, indicating that the proposed model could be used for further predictions. Furthermore, we observed the dynamic effect of varying the fractional order and noticed that changing the fractional order produces crisscrossed behaviour in infected compartments of domestic ruminants and infected adult amblyomma ticks. Finally, we showed the relative impact of varying the transmission rates and the infectivity potential of peractive, active, and recovered carrier ruminants on the overall dynamics of the disease. Thus, a reduction in the rate of transmission from nymph and adult amblyomma ticks to vaccinated ticks will increase the number of healthy domestic ruminants. [ABSTRACT FROM AUTHOR]

The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein–Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein–Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to analyze the modified epidemiological Susceptible-Infected-Removed model including an antidotal population compartment A (SIRA). The fractional natural decomposition method (FNDM) and variational iteration method (VIM) are applied to the governing model. This model is used to explain the wave behaviors of the infection virus arising in computer science. The SIRA model uses the Caputo derivative satisfying the initial conditions. Moreover, numerical investigations and strain conditions for the optimal values of parameters to minimize the effect of computer virus are also reported. The Lipschitz condition theorem and the Banach space are considered to present the uniqueness with the Caputo operator. Furthermore, various wave distributions of the nature of virus's are also extracted in plots. [ABSTRACT FROM AUTHOR]

• A new fractional order mathematical model for the assessment of treatment on the dynamics of COVID-19 is proposed. • Parameters estimation procedure from cumulative cases in Saudi Arabia is presented. • Rigorous theoretical analysis of the model is presented. • Numerical Simulations are performed. In this paper, a mathematical model is formulated to study the transmission dynamics of the novel coronavirus infection under the effect of treatment. The compartmental model is firstly formulated using a system of nonlinear ordinary differential equations. Then, with the help of Caputo operator, the model is reformulated in order to obtain deeper insights into disease dynamics. The basic mathematical features of the time fractional model are rigorously presented. The nonlinear least square procedure is implemented in order to parameterize the model using COVID-19 cumulative cases in Saudi Arabia for the selected time period. The important threshold parameter called the basic reproduction number is evaluated based on the estimated parameters and is found R 0 ≈ 1.60. The fractional Lyapunov approach is used to prove the global stability of the model around the disease free equilibrium point. Moreover, the model in Caputo sense is solved numerically via an efficient numerical scheme known as the fractional Adamas-Bashforth-Molten approach. Finally, the model is simulated to present the graphical impact of memory index and various intervention strategies such as social-distancing, disinfection of the virus from environment and treatment rate on the pandemic peaks. This study emphasizes the important role of various scenarios in these intervention strategies in curtailing the burden of COVID-19. [ABSTRACT FROM AUTHOR]

This paper explores the globally β -Mittag-Leffler stability in Lagrange sense for the fractional-order complex-valued neural network (FOCVNN) with impulsive effects. By Lyapunov method and matrix inequalities, some novel sufficient conditions are obtained to guarantee the globally β -Mittag-Leffler stability in Lagrange sense for two class of complex-valued (CV) activation functions. The convergent rate is also given, which is controlled by the parameters of the addressed system. The existence and uniqueness of the solution for this system do not require consideration. To show the validity and usefulness of the results, two numerical stimulations are provided. [ABSTRACT FROM AUTHOR]