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This paper investigates a novel version for the nonlinear 2D telegraph equation involving variable-order (V-O) time fractional derivatives in the Atangana–Baleanu–Caputo sense with Mittag–Leffler non-singular kernel. A meshfree method based on the moving least squares (MLS) shape functions is proposed for the numerical solution of this class of problems. More precisely, the V-O fractional derivatives in this model are approximated by the finite difference scheme at first. Then, the θ-weighted method is utilized to derive a recursive algorithm. Next, the solution of the problem is expanded in terms of the MLS shape functions with undetermined coefficients. Eventually, by substituting this expansion and its partial derivatives into the original equation, solution of the problem in each time step is reduced to the solution of a linear system of algebraic equations. Several numerical examples are investigated to show the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the established method is high accurate in solving such V-O fractional models.

In this paper, the third order partial differential equation defined by Caputo fractional derivative with Atangana–Baleanu derivative has been investigated. The stability estimates are proved for the exact solution. Difference schemes for Crank–Nicholson finite difference scheme method is constructed. The stability of difference schemes for this problem is shown by Von Neumann method (Fourier analysis method). Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the technique. The reproducing kernel function for the problem has been found.

In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves., Comment: 15 pages, 4 figures

In this paper, the classical Duffing oscillator under dual frequency excitations is studied by the homotopy analysis method(HAM). Analytical study of the low-order approximations is firstly conducted and the saddle node(SN) bifurcation boundary for the initial guess solution is obtained. The maximum value bifurcation plot of the high order approximations with the bifurcation parameters f1 and λ1 are obtained and compared with the numerical solutions based on the Runge–Kutta method. The results show that the initial guess solution can qualitatively reflect the trend of the numerical solution, and the high order approximations agree well with the numerical solutions. The maximum value bifurcation plots of high order approximations show periodic and quasi-periodic solutions, which agree well with the numerical ones.

In this paper, we propose a fractional form of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps involving the fractional conformable derivative of Khalil’s and Atangana’s type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order. We obtain new chaotical behaviors considering numerical schemes based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Also, the dynamics of the proposed maps are investigated numerically through phase plots considering combinations of these derivatives and mixed integration methods for each map. The numerical simulations show very strange and new behaviors for the first time in this manuscript.

In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters.

In this paper, a mathematical model which considers population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen and the body’s natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville–Caputo sense were considered. Special solutions using an iterative scheme via Laplace transform were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana–Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful information about the complexity of the dynamics of the breast cancer competition model.

In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ*. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced.

Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem.

This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana–Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana–Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana–Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight.

Suggested independently with different definitions, fractal derivative and conformable derivative are α proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations.

An epidemic system of HIV/AIDS transmission is examined in this paper. The classical time derivative is modelled with the Atangana-Baleanu nonlocal and nonsingular fractional operator in the Caputo sense. Mathematical analysis which shows that both the disease free equilibrium state and endemic equilibrium are locally asymptotically stable. A viable numerical approximation technique of Atangana-Baleanu operator is also given. Some numerical simulation results obtained for different instances of fractional order γ are reported to justify the theoretical results.

In this review paper, we are mainly concerned with the numerical methods for solving fractional equations, which are divided into the fractional differential equations (FDEs), time-fractional, space-fractional and space-time-fractional partial differential equations (FPDEs), fractional integro-differential equations (FIDEs) and delay fractional differential and/or fractional partial differential equations (DFDE/DFPDEs). The concept of the variable-order fractional operators will also be reviewed. At the same time, the techniques for improving the accuracy and computation and storage are also introduced.

In this paper we investigate an investment problem with partial reversibility proposed by Abel and Eberly [4] in a finite horizon. In this model, a firm can purchase capital at a given price and sell capital at a lower price. This problem can be categorized into a singular control problem and can be formulated as a Hamilton–Jacobi–Bellman(HJB) equation. Based on theoretical results in [10] and the Mellin transform techniques, we derive the coupled integral equations satisfied by the optimal investment and disinvestment boundaries, respectively. By using the recursive integration method, we solve numerically the integral equations and present the optimal investment boundary and disinvestment boundary.

In the present paper we analysis the Modified Chua attractor using new concept of fractional differentiation with non-local and non-singular kernel. A new numerical scheme that was recently suggested was used for the Volterra equation with Atangana–Baleanu fractional integral, Caputo–Fabrizio integral and finally Riemann–Liouvile integral. The numerical solution obtained from the new numerical scheme let no doubt than to believe that the new numerical scheme is very efficient and converges toward exact solution very rapidly. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic system with fractional order. Existence and uniqueness of the Volterra type is presented. We presented some numerical simulations for different values of fractional order.

In this research paper, we plan to study the relation among concentration of greenhouse gases, human population and environmental tax. For this purpose, a deterministic nonlinear mathematical model is proposed and analyzed in regard to the boundedness and persistence of its solutions, equilibria and their stabilities. Whenever, a policy is implemented, an automated delay comes into existence because outcomes of application of any policy always take some time to become visible. To see the effects of delay, analyses for the stability and direction of Hopf bifurcation for delay system are done. Critical value of the delay parameter is calculated theoretically and numerically, and then verified graphically. For verifications and descriptions of analytical findings, numerical simulations are performed. Graphical comparisons for different values of various parameters provides us some realistic and interesting results. Overall, model analysis shows that the increasing level of greenhouse gases can be controlled by applying environmental tax and imposing some restrictions on corruption.

The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed.

In this paper, a range of chaotic systems with some interesting behaviors such as multi-scroll attractors, self-excited and hidden attractors, period-doubling to chaos, periodic and chaotic bursting oscillations, and different multiple coexisting attractors have been considered and modelled with the new Atangana–Baleanu fractional derivative operator in time. Existence and uniqueness of general system as well as local stability analysis are examined. In the simulation framework, a range of chaotic patterns examined through time series were obtained for different instances of fractional orders. Comparison between the integer (with p = 1 ) and noninteger (0

Fractional-order systems are utilized widely to model material or physical systems with the properties of memory, non-locality and history-dependent, thus massive studies have been drawn on their dynamics during the past decade. In this paper, we are motivated to study stochastic dynamics in a fractional-order system subjected to Gaussian noise. The statistical complexity measure (SCM), an approach of information theory, has been well defined and calculated in the fractional-order Langevin model based on its Shannon entropy, by which the stochastic resonance (SR) response has been detected and measured. To compare, we further derive the equivalent system by means of the minimum mean square error rule to estimate the signal-to-noise ratio (SNR). It has been found that the SCM and the SNR depict analogous profile and exhibit peaks almost at the same level of noise. Varying the signal frequency, the SCM even quantifies Multiple-SR that SNR does not measure in the fractional-order Langevin model. Thus, the SCM may be utilized as an alternative candidate to measure SR response in bistable fractional-order systems.

This paper investigates the coexistence of multiple attractors in Sprott B system. Two independent chaotic attractors and two independent periodic attractors are numerically found in the system with different parameters and initial values. Multistability is also generated from the Sprott B system by using the sign function. A nonlinear controller is designed for forcing the chaotic motion of Sprott B system to an equilibrium point and enabling the system trajectories to switch between different chaotic attractors. The effectiveness of the controller is illustrated by some numerical experiments.

In this paper we study the generalized Gause model, with a logistic growth rate for the prey in absence of the predator, a constant death rate for the predator and for several different classes of functional response, all non-analytical. First we consider the piecewise-linear functional response of Holling type I, which essentially has a linear functional response on a bounded interval and a constant functional response for large enough prey density. Next we consider differentiable modifications of this type of functional response, one being a concave down function, the other one being a sigmoidal function. Our main interest is the number of closed orbits of the systems under consideration and the global stability of the system. We compare the generalized Gause model with a functional response that is non-analytical with the generalized Gause model with a functional response that is analytical (e.g., Holling type II or III) and show that the behaviour in the first case is more complicated. As examples of this more complicated behaviour we mention: the co-existence of a stable equilibrium with a stable limit cycle and the existence of a family of closed orbits.

This paper is concerned with a fractional Caputo-difference form of Ikeda map. The dynamics of the proposed map is investigated numerically through phase plots and bifurcation diagrams considered from different perspectives. In addition, a stabilization controller is proposed and the asymptotic convergence of the states is established using the stability theory of linear fractional-order discrete systems. Furthermore, a new synchronization scheme is introduced whereby a new 2D fractional-order chaotic map is considered as the master system and the fractional-order Ikeda map is considered as the response system. Experimental investigations and numerical simulations are also provided to confirm the main findings of the study.

The purpose of this paper is to present two essentially different schemes for deriving the partial differential equations (PDE) for the price of the so-called defaultable derivatives. In the first one the asset price is represented as a solution of a stochastic differential equation (SDE), stopped at a stochastic time. The second one explores the idea of adding a jump process assuming that the stopping time is the moment of its first jump. We investigate also the role of the loss rate, which represents the loss of the asset at the default moment. In both cases we examine various assumptions and dependencies between the underlying asset, the stopping time, and the loss rate. We examine separately the cases when the underlying asset price is driven by a Brownian motion or by a Levy process. We give a method to solve the PDEs for the derivative prices by the use of the so-called default premium. As an example we derive a closed form formula for the price of a contingent convertible bond.

The intrinsic metric formula on the code set of the Sierpinski Gasket is explicitly given in Definition 1.1. In this paper, we obtain the intrinsic metric formula on the code set of the Sierpinski tetrahedron and we investigate some geometrical properties of this structure. Moreover, we define a dynamical system on the Sierpinski tetrahedron and then we give an algorithm to compute the periodic points of this dynamical system. Finally, we show that this dynamical system is both Devaney chaotic and Li-Yorke chaotic.

In this paper, we propose and study a Logistic model with feedback control of fractional order that is extended directly from a system of ordinary differential equations. Uniform asymptotic stability of this model is established based on an appropriate Lyapunov function and an important consequence of this result; we present a simple proof for the global stability of the original system of ordinary differential equations. Besides, to numerically solve and simulate the proposed fractional model, unconditionally positive nonstandard finite difference schemes are constructed and analyzed. Finally, the numerical simulations obtained by the constructed nonstandard finite difference schemes are compared with the Grunwald–Letnikov scheme to reveal that the proposed scheme is convenient for solving the proposed model and confirm the validity of the established results.

In this paper, tumor-immune system interaction has been considered by two fractional order models. The first and the second model consist of system of fractional order differential equations with Caputo and conformable fractional derivative respectively. First of all, the stability of the equilibrium points of the first model is studied. Then, a discretization process is applied to obtain a discrete version of the second model where conformable fractional derivative is taken into account. In discrete model, we analyze the stability of the equilibrium points and prove the existence of Neimark-Sacker bifurcation depending on the parameter σ. Moreover, the dynamical behaviours of the models are compared with each other and we observe that the discrete version of conformable fractional order model exhibits chaotic behavior. Finally, numerical simulations are also presented to illustrate the analytical results.

The present paper is devoted to the study of Renyi entropy in the fuzzy case. We define the Renyi entropy of a fuzzy partition and its conditional version and derive basic properties the suggested entropy measures. In particular, it was shown that the Renyi entropy of a fuzzy partition is monotonically decreasing. Consequently, using the proposed concept of Renyi entropy, the notion of Renyi entropy of a fuzzy dynamical system is introduced. Finally, it is proved that the Renyi entropy of a fuzzy dynamical system is invariant under isomorphism of fuzzy dynamical systems.

With the development of nonlinear science, it is found that the fractional differential equation can more accurately describe the variation of natural phenomena.Therefore, the study of fractional differential equations and their boundary value problems is of great significance for solving nonlinear problems.The Dirichlet function, as an abstract mathematical model, has many unique properties in calculus.It points out special circumstances when describing many mathematical concepts, It can also be used to construct counterexamples in calculus and to clarify many fuzzy concepts to deepen the understanding of mathematical concepts.Therefore, this paper mainly studies the necessary and sufficient conditions for the controllability of fractional linear differential systems in abstract space.The finite difference decomposition method of fractional calculus in the abstract space for the Dirichlet function equation is also studied.The existence of solutions for boundary value problems of fractional differential equations with Dirichlet boundary value condition in the abstract space is discussed by using the critical point theory.

This paper develops a three-factor valuation model of vulnerable European options incorporating stochastic yield spreads and interest rates, which extends a constant yield spread and deterministic interest rate proposed in Hull and White [12]. The dynamics of the short-term interest rate are represented implicitly by a stochastic bond price process. Therefore, the stochastic factors in the model are the spot price of the underlying asset that follows a geometrical Brownian motion process, the yield spreads and the default-free unit discount bond that are modeled by a mean-reverting Ornstein-Uhlenbeck stochastic process. Furthermore, we exploit Mellin transform techniques to derive a closed-form solution for vulnerable European options under the Black-Scholes model, which is simply computed using the standard normal cumulative distribution function so that the pricing and hedging of vulnerable European options can be computed very accurately and rapidly. Numerical experiments demonstrate how credit risk and interest rate risk affect the prices of European options.

Marketing researchers and practitioners have noticed that the demand of many retail items are proportional to the amount of inventory displayed. In this paper, a new supply chain model is proposed assuming that demand of a product does not increase monotonically with the increase of inventory. In this model it is assumed that the demand has a saturation level and it does not increase monotonically with the inventory. The stationary solutions (fixed points) of the model are determined and their stability natures are determined using Routh–Hurwitz criteria. The variation of the model dynamics for different parameter values are presented numerically by drawing time evolution diagrams and phase diagrams. Bifurcation diagram of the model with respect to the demand saturation parameter is presented to note the importance of the parameter. Next, synchronization behavior of two coupled identical supply chain models under both unidirectional and bidirectional coupling are discussed. Sufficient conditions for synchronization in case of bidirectional coupling are derived. Numerical simulation results are presented to validate the analytical findings.

The logarithmic time evolution is widely observed in laboratory and field measurements. However, logarithmic stability has not been well considered till now. In this paper, the definition of Lyapunov stability, including logarithmic and inverse Mittag-Leffler stabilities, is proposed. And via the Lyapunov direct method, the stability of nonlinear dynamical systems based on the structural derivative is investigated. Furthermore, the comparison principle based on the structural derivative is presented in order to obtain the stability conditions for nonlinear dynamical systems. Finally, two demonstrative examples are given to test the proposed stability concept.

In this paper, we use Atangana–Baleanu derivative which is defined with the Mittag–Leffler function and has all the properties of a classical fractional derivative for solving the system of fractional differential equations. The classical model of polluted lakes system is modified by using the concept of fractional differentiation with nonsingular and nonlocal fading memory. The new numerical scheme recommended by Toufik and Atangana is used to analyze the modified model of polluted lakes system. Some numerical illustrations are presented to show the effect of the new fractional differentiation.

In this paper, a new stochastic method combining the stochastic averaging of quasi-conservative and energy loss is developed to analyze the probabilistic response of a class of nonlinear vibro-impact oscillator with bilateral barriers driven by colored noise. According to the relationships between the total energy and the potential energy of impact positions of the nonlinear system, the energy loss of every movement period of the unperturbed nonlinear vibro-impact system with given total energy is analyzed in detail. Based on this analysis, an averaged It o ^ stochastic differential equation is derived through the stochastic method which combining a nonlinear transformation and the stochastic averaging of quasi-conservative, in which the averaged It o ^ equation has the typical characteristics of piecewise smooth. Next, the stationary probability density function (SPDF) is obtained by solving the averaged drift and diffusion terms of the corresponding piecewise smooth Fokker-Planck equation for the averaged It o ^ equation with the Fourier series expansion. Finally, an example is given to validate the effectiveness of the proposed method. The effects of four physical quantities of the stochastic vibro-impact system: the positions of impact, restitution coefficient, noise intensity and correlation time of the colored noise, on the probabilistic response are discussed by the SPDFs in detail, and the results are verified through the Monte Carlo simulation.

In this paper, we have studied the dynamical behaviour of a Leslie-Gower model with Holling type-IV functional response and nonlinear prey harvesting where target prey for harvesting are above a particular age which is called maturation delay. The model system has been studied using qualitative analysis, bifurcation theory, bionomic equilibrium, MSY and optimal harvesting policy. We explain that the interior equilibrium is delay dependent and locally asymptotically stable. The age-selective harvesting system under goes a Hopf-bifurcation with respect to the maturation delay. Conditions for the MSY and existence of bionomic equilibrium are analyzed and Pontryagin's maximum principle has been used to find the path of a singular control. Difficult analytical results are featured by numerical examples to better understand the system in a simple way.

The aim of this paper to present some new fractals constructions based on a fixed point approach for Meir–Keeler type operators in complete metric spaces. The concept of coupled fixed point is considered and coupled fractals are generated. Our results extend some recent theorems for single-valued Banach contractions, as well as some other results concerning the dynamics of the self-similar sets generated by iterated function systems.

Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker–Planck–Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.

This paper is concerned with the boundary control of the fractional wave equation when the boundary is subject to persistent external disturbances. By developing the sliding mode control approach to infinite-dimensional fractional order systems, the fractional order sliding mode boundary control law is designed for the infinite dimensional setting. Moreover, based on the fractional asymptotical stability theorem, the asymptotical stability for the fractional wave equation under the control strategies proposed is addressed. Finally, numerical examples are provided to illustrate the viability of the theoretical results.

In this paper, we studied a diffusive predator-prey model with competitive interference and Crowley-Martin type functional response. The conditions for local stability of coexisting equilibrium are given by analyzing the eigenvalue spectrum. By using delay as bifurcation parameter, conditions for occurrence of Hopf bifurcation are also given. The property of bifurcating period solutions is investigated by calculating the normal form. Some numerical simulations are performed to support our theoretical result. Our conclusions show that diffusion and delay are two factors that should be considered in establishing the predator-prey model, since they can induce spatially bifurcating period solutions.

In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained.

In this paper, we introduce a separation property for self-similar sets which is necessary to reach the equality between the similarity dimension and the Hausdorff dimension of these spaces. The similarity boundary of a self-similar set is investigated from the viewpoint of that property. In this way, the strong open set condition (in the self-similar set setting) posed by Keesling and Krishnamurthi has been weakened leading to a Moran type theorem. Moreover, both a result based on a conjecture posed by Deng and Lau as well as an improved version of a theorem due to Bandt and Rao have been contributed regarding the size of the overlaps among the pieces of a self-similar set. Several (equivalent) conditions leading to the equality between the similarity dimension and a new Hausdorff type dimension for attractors described in terms of finite coverings are also provided. Finally, we list some open questions.

In this paper we present new methods to generate fractal interpolation surfaces which generalize the existing fractal interpolation surfaces. We ensure that attractors of nonlinear iterated function systems constructed by Geraghty contractions are graphs of some continuous functions which interpolate the given data. In particular, we give two explicit illustrative examples to demonstrate the effectiveness of obtained results.

In this paper, we consider a class of nonlinear nonlocal problem involving an anisortropic operator and an external potential. We show the existence of positive and sign-changing solutions of this problem via the variational methods.

In the present paper, the synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative is presented in detail taking into consideration the effects of the very slow intracellular exchange of calcium ion (Ca 2 + ) occurring between cytoplasm and its store. That is, we approach the synchronization by the Ge–Yao–Chen partial region stability theory. The method of resolution used is the powerful Adam–Bashforth–Moulton method which converges quickly to the exact solution. Numerical simulations of the error dynamics proved the effectiveness of the control method. The synchronization time is evaluated for different orders of the derivative and its analysis shows that, it is closely related to the order of the derivative and that fractional order induces quick synchronization compared to integer order.

In this paper, we present an upwind finite volume method to solve the convection-diffusion equations with Dirichlet boundary on rectangular mesh. By utilizing the technique of element-by-element analysis, the stability of the method has been proved and the H1-norm error estimate is presented. Furthermore, we provide the proofs of the maximum principle and L∞-norm error estimate. Finally, some numerical experiments are provided to confirm our theoretical results.

In this paper we construct the higher order Heisenberg supermagnet models in (2+1)-dimensions by introducing different auxiliary matrix variables. We investigate their structure and integrability with respect to two constraints. By virtue of the gauge transformation, we derive their gauge equivalent counterparts which are the super and fermionic higher order nonlinear Schrodinger equations in (2+1)-dimensions. Furthermore, we obtain their corresponding Backlund transformations.

The dynamics of the language competition model is considered in this paper. The classical system is converted to non-integer order case by replacing the second-order partial derivative with the Riesz fractional derivative. A well-known numerical approximation methods based on the Fourier spectral algorithm in space and the third-order exponential time-differencing scheme are formulated to numerically simulate the three component fractional-in-space reaction-diffusion system in one and high dimensions for different values of α. Numerical results indicate α ∈ (1, 1.5] as the key control parameter that can influence the coexistence of various speakers over a period of time.

Movement of individuals within metapopulations is characterised by individuals frequenting their home ranges. This not only constitutes memory but also nonlocal property of the resulting system making it plausible to be modelled by Fractional order differential equations. In this paper, we propose a fractional order metapopulation model for transmission of cholera between communities with differing standards of living. Important basic properties of the model such as non-negativity of solutions as well as boundedness are tested. The solutions to the model are shown to exist and the steady state is unique whenever it exists. The model is numerically integrated using the iterative Adams-Bashforth-Mouton method. Our results show that, there is increase synchronous fluctuation in the population of infected individuals in connected communities with either restricted movement or with unrestricted movement of susceptible and infected individuals. In communities with movement restricted to only susceptible individuals, synchronous fluctuation of the infected population in the two communities is more pronounced at lower orders of the fractional derivatives. In unrestricted communities however, the infected population in the two adjacent communities synchronously regardless of the order of the fractional derivative.

This paper deals with the complex dynamical behavior of a discrete-time two-prey one-predator system with Holling Type-III functional response, coupled with inter-specific competition among the prey due to dietary overlap and intra-specific competition among the predators. The existence and local stability criteria of the steady states of the system are analyzed and the conditions for permanent co-existence of the three species are obtained. Finally, using a suitable example, some numerical simulations are performed to illustrate the rich dynamics of the system.

In this paper, we discuss the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana–Baleanu fractional derivative. Benefiting from the Gronwall inequality in the frame of Riemann–Liouville fractional integral, we establish a Gronwall inequality in the frame of Atangana–Baleanu fractional integral. Then, we study the stability of such equations in the sense of Ulam.

Human African trypanosomiasis has been a major problem in sub-Saharan countries including Ghana. In this paper, a fractional human African trypanosomiasis model with fractional time dependent control is studied. The existence and uniqueness of the model were obtained. Collected data from Ghana were considered to fit the derived model. Fractional optimal control via Atangana–Baleanu derivative in Liouville–Caputo sense is used to determine the best strategy for minimizing the spread of human African trypanosomiasis. The numerical result shows that the combination of three controls is effective, however the combination of prevention and treatment can also offer a good strategy since the cost for implementing two controls is lower than three controls.