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• A comparison of Bank data in different fractional operators are proposed. • Parameter estimations using the least square method is presented. • Different fractional operators such as Caputo, Caputo–Fabrizio and Atangana–Baleanu versus real data is shown. • A novel numerical approaches for the solution of fractional and parameter fractional model are given. • Comparison of all the operators with real data is presented. The present paper investigate the dynamics of the bank data through a competition model with real field data for the year 2004–2014. Initially, we formulate a competition model for the bank data and then use different fractional approaches to simulate the model with the real data for many fractional order parameters α. Then, we present a novel approach for each fractional model and provide a graphical illustration with real data. We show that all these fractional approaches have good resemblance to each other and can be used to model such real data case. We prove in general that the results of the fractional approaches utilized here are good for modeling purpose but also we prove the results of the fractional Atangana–Baleanu operator is more accurate and flexible and can be used confidently to modeled such real case problem. [ABSTRACT FROM AUTHOR]

The Optimal Control Problem (OCP) for variable-order fractional differential systems with time delay is considered. The fractional time derivative is Atangana–Baleanu derivatives in a Caputo sense. The existence and the uniqueness results are derived. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as an integral function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable-order and time delay. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Atangana–Baleanu derivative, we obtain an optimality system for the optimal control. To obtain the optimality conditions for the given problem, the generalization of the Dubovitskii–Milyutin Theorem was applied. To illustrate the results we introduce some examples. [ABSTRACT FROM AUTHOR]

Highlights • We present a numerical algorithm for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. • Tedious manipulation of the series appearing in the implementation of the method have been simplified to obtain a system of algebraic equations for the coefficients. • Several examples are discussed to demonstrate the efficiency and accuracy of the present scheme. Abstract In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

Highlights • A fractional TB model with Caputo derivative is presented. • The real data of TB since 2002–2017 of Khyber Pakhtunkhwa, is considered. • The model is locally and globally stable with fractional Lyapunov function. • Numerical results with long term behavior of the infected individuals is presented. Abstract In this paper, we present a nonlinear fractional order model in the Caputo sense to explore and simulate the TB dynamics. Using the TB confirmed notified cases from the year 2002 to 2017 in Khyber Pakhtunkhwa, Pakistan, we estimate the model parameters and demonstrate that the proposed fractional model provides a good fit to the real data. Initially, we compute the basic reproduction number and the model equilibria. Then, the existence and uniqueness of the model are shown via generalized mean value theorem. Further, we explore the local and global stability of the disease free equilibria in fractional environment. Finally, numerical results are obtained in order to validate the importance of the arbitrary order derivative and theoretical results. We conclude that the fractional epidemic model is more generalized than the classical model, which give more information about the disease dynamics and give a good agreement to the real data of TB infection. [ABSTRACT FROM AUTHOR]

Highlights • Mathematical formulation for a fractional phase change problem by using memory fluxes. • Caputo and Riemann–Liouville derivatives in time. • Jumping formulas due to the variable starting time (depending on the free boundary). • Derivation of an integral condition equivalent to the fractional Stefan condition. Abstract A mathematical formulation for a one-phase change problem in a form of Stefan problem with a memory flux is obtained. The hypothesis that the integral of weighted backward fluxes is proportional to the gradient of the temperature is considered. The model that arises involves fractional derivatives with respect to time both in the sense of Caputo and of Riemann–Liouville. An integral relation for the free boundary, which is equivalent to the "fractional Stefan condition", is also obtained. [ABSTRACT FROM AUTHOR]

The latent potentialities and applications of fractional calculus present a mathematical challenge to establish its theoretical framework. One of these challenges is to have a compact and self-contained fractional power series representation that has a wider application scope and allows studying analytical properties. In this letter, we introduce a new more general form of fractional power series expansion, based on the Caputo sense of fractional derivative, with corresponding convergence property. In order to show the functionality of the proposed expansion, we apply the corresponding iterative fractional power series scheme to solve several fractional (integro-)differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form D α i x i = f i ( x 1 , x 2 , x 3 ) , 0 ≤ α i ≤ 1 , ( i = 1 , 2 , 3 ) ; and showed the existence of chaos in case of fractional Lorenz system when Σ = α 1 + α 2 + α 3 ≤ 3 . Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter. In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS. [ABSTRACT FROM AUTHOR]

By considering the Caputo fractional derivative and Riemann-Liouville integral, in the present work, we are concerned with a nonlinear sequential fractional differential system of Duffing oscillator type. The considered system has neither the commutativity nor the semi group properties, since the sum of the two orders of derivatives, of the left hand side of the problem, are outside the interval [0,1]. With the absence of these two properties, we have to find other arguments to obtain the integral representation of the problem, to be able thereafter to present the other main results. Then, using the contraction mapping principle and Scheafer theorem, two main theorems on the uniqueness and existence of solutions are proved. Finally, some examples are given to illustrate the proposed main results. [ABSTRACT FROM AUTHOR]

• Fractional order model with Caputo operator provides more flexible outcomes than the corresponding integer order. • Sensitive parameters essential for the control of the infection. • Fractional operators gives more biologically feasible behavior about the dynamic of pinewilt disease. In this work, we investigate the transmission dynamics of pine wilt disease infection and developed a new model utilizing Caputo fractional-order derivative. Moreover, with the use of fixed point theorem, the existence and uniqueness of the pine wilt disease model are obtained under Caputo operator. Using forward normalized sensitivity index, we determine the most sensitive parameters essential for the control of the infection and the results show that, decreasing the values of contact rate of a susceptible vector with infected pine trees and progression rate play a significant role in controlling the spread of pine wilt disease infection. On the other hand, we obtain different numerical simulations results of the model using the appropriate parameter values. Hence, from the graphs, it can be concluded that Caputo fractional operator gives more biologically observable behavior of the proposed disease model thanks to the changed fractional order. Compared to the previously built integer order model, the non-integer order derivative provided more efficient and flexible information about the complexity of the model's dynamics. [ABSTRACT FROM AUTHOR]

In this article, a predator-prey model has been evolved in the form of a system of fractional order differential equations incorporating two important factors, namely, fear factor and prey refuge factor. Here, the fractional calculus has been taken into consideration to investigate the dynamical behaviour of the solutions of the proposed model system as the changes in life cycle of prey species are of memory bound. Biological validation and well-posedness such as positivity and boundedness of solutions of the model system have been proved analytically. Stability analysis of all the feasible equilibrium points of the model system has been performed in a systematic way. Some important dynamical features of the model system (such as transition of stability of the system) have been demonstrated through rigorous numerical simulation. It is observed that our proposed model system experiences Hopf-bifurcation around the interior equilibrium point with respect to both the parameters f and m 1 , which are linked with amount of predator induced fear and rate of prey refuge, respectively. The system dynamics is more likely to be stable in the framework of fractional order derivative in comparison to integer-order derivative. The high amount of predator induced fear f and prey refuge rate m 1 are independently capable to make the system dynamics to be stable in integer order model system. On the other hand, the dynamics of the model system shifts towards the stability from its unstable behaviour when we continuously reduce the order of the model system; especially under the scenario of low level of predator induced fear and prey refuge rate. Thus, our comprehensive mathematical findings reveal the fact that fading memory can play a contributory role towards stable coexistence of the predator-prey system whereas strong memory of the species deteriorates the stable coexistence of the model system. [ABSTRACT FROM AUTHOR]