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Highlights • Nonlocality is an essential feature in fractional calculus. • Methods neglecting nonlocality lead to wrong results. • Numerical evidence of poor results obtained by neglecting nonlocality is given. Abstract In the paper titled "New numerical approach for fractional differential equations" by Atangana and Owolabi (2018) [1], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the stochastic differential delay equations with Poisson jump (SDDEsPJ). As stochastic differential equations, most SDDEsPJ cannot be solved explicitly. Therefore, numerical solutions have become an important issue in the study of SDDEsPJ. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJ when the drift and diffusion coefficients are Taylor approximations. [Copyright &y& Elsevier]

Abstract: In this paper, we shall examine the convergence of semi-implicit Euler approximation for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching. Here, the main ideas from the papers Ronghua et al. (2009) and Wang and Wang (2010) are successfully developed to the more general cases. Finally, a numerical example is provided to illustrate the theoretical result of convergence. [ABSTRACT FROM AUTHOR]

Abstract: The above mentioned paper contains some fundamental mistakes and misinterpretations along with a false conclusion. Applying the optimal homotopy asymptotic method (OHAM) in an incorrect manner, Niu and Wang have drawn the false conclusion that this approach is not efficient in practice because it is time-consuming for high-order of approximation. We emphasized the presence of some evident mistakes and misinterpretations in their paper and we proved that OHAM is very efficient in practice since we solved all three examples analyzed by Niu and Wang using only the first-order of approximation, which yields accurate results. We demonstrate that OHAM does not need high-orders of approximation as Niu and Wang suggests and we show that the main strength of OHAM is its rapid convergence, contradicting Niu and Wang’s assumption. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when . However, the HPM solution is divergent for all x and t except . It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results. [Copyright &y& Elsevier]

In this paper, we will introduce the reconstruction of variational iteration method (RVIM) to solve multi-order fractional differential equations (M-FDEs), which include linear and nonlinear ones. We will easily obtain approximate analytical solutions of M-FDEs by means of the RVIM based on the properties of fractional calculus. Moreover, the convergence of proposed method will be shown. Our scheme has been constructed for the fully general set of M-FDEs without any special assumptions, and is easy to implement numerically. Therefore, our method is more practical and helpful for solving a broad class of M-FDEs. Numerical results are carried out to confirm the accuracy and efficiency of proposed method. Several numerical examples are presented in the format of table and graphs to make comparison with the results that previously obtained by some other well known methods. [ABSTRACT FROM AUTHOR]

Recently, Abdeljawad and Baleanu (2011) introduced Caputo q-fractional derivatives and used it to solve Caputo q-fractional initial value problem. For this purpose, they applied successive approximation method to obtain an explicit solution; but did not clarify under which conditions that this method will be convergence. In this paper, we propose q-Krasnoselskii–Krein type condition to investigate the convergence of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the convergence of a numerical algorithm that switches in some deterministic or random manner, the control parameter of a class of continuous-time nonlinear systems while the underlying initial value problem is numerically integrated. The numerically obtained attractor is a good approximation of the attractor obtained when the control parameter is replaced with the average of the switched values. In this way, a generalization of Parrondo’s paradox can be obtained. As an application, the Lorenz and Rabinovich–Fabrikant systems are used for illustration. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with model-free control of the Lorenz chaotic system, where only the online input and output are available while the mathematic model of the system is unknown. The problem is formulated from an optimal control perspective and solved using an iterative method. The convergence of the iteration and the stability of the control law are proven in theory. Simulations validate theoretical conclusions and demonstrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved. [Copyright &y& Elsevier]

Abstract: The numerics of the Kuramoto–Tsuzuki equation is dealt with in this paper. We propose a split-step Fourier pseudospectral discretization for solving the problem, which is split into one linear subproblem and one nonlinear subproblem. The nonlinear subproblem is integrated exactly via solving the equations for the amplitude and phase angle of the unknown complex-valued function respectively. The linear subproblem is first approximated by Fourier pseudospectral discretization to the spatial derivative, and then integrated exactly in phase space via solving the equations for the Fourier coefficients analytically. We apply a fourth-order splitting integration in time advances, and therefore the overall error in space discretization is of spectral order and the overall error in time discretization is of fourth order which merely comes from the splitting. The scheme is fully explicit, easy to implement and quite efficient thanks to FFT. Moreover, it is time reversible and gauge invariant which are two properties in the continuous problem. Extensive numerical results are reported, which are geared towards testing the convergence and demonstrating the efficiency and accuracy. [Copyright &y& Elsevier]

Abstract: This paper employs the homotopy analysis method (HAM) to derive analytical approximate solutions for the nonlinear problem with high-order nonlinearity. Such a problem corresponds to the large-amplitude vibration of electrostatically actuated microbeams. The HAM is also optimized to accelerate the convergence of approximate solutions. To verify the accuracy of the present approach, illustrative examples are provided and compared with other analytical and exact solutions. [Copyright &y& Elsevier]

Abstract: Stochastic age-dependent population equations, one of the important classes of hybrid systems are studied. In general most equations of stochastic age-dependent population do not have explicit solutions. Thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical scheme and show the convergence of the numerical approximation solution to the analytic solution. In the last section a numerical example is given. [Copyright &y& Elsevier]

Abstract: In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms. [Copyright &y& Elsevier]

Abstract: In this paper two common collocation approaches based on radial basis functions (RBFs) have been considered; one is computed through the differentiation process (DRBF) and the other one is computed through the integration process (IRBF). We investigate these two approaches on the Volterra’s Population Model which is an integro-differential equation without converting it to an ordinary differential equation. To solve the problem, we use four well-known radial basis functions: Multiquadrics (MQ), Inverse multiquadrics (IMQ), Gaussian (GA) and Hyperbolic secant (sech) which is a newborn RBF. Numerical results and residual norm show good accuracy and rate of convergence of two common approaches. [Copyright &y& Elsevier]

Abstract: In this paper, radial basis functions (RBFs) approximation method is implemented for time fractional advection–diffusion equation on a bounded domain. In this method the first order time derivative is replaced by the Caputo fractional derivative of order α ∈(0,1], and spatial derivatives are approximated by the derivative of interpolation in the Kansa method. Stability and convergence of the method is discussed. Several numerical examples are include to demonstrate effectiveness and accuracy of the method. [Copyright &y& Elsevier]

Abstract: This paper presents approximate analytical solutions for a system of singular Volterra integral equations of convolution type by using the fractional differential transform method. The solutions are calculated in the form of convergent series with easily computable terms and also the exact solutions can be achieved by well-known series solutions. Several examples are given to demonstrate reliability and performance of the presented method. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method is applied to obtain the solution of fractional partial differential equations with spatial and temporal fractional derivatives in Riesz and Caputo senses, respectively. Some properties of Riesz fractional derivative utilized in obtaining the series solution are proved. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter ℏ on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders α and β on the solution behavior. [Copyright &y& Elsevier]

Abstract: In this paper, the approximation technique proposed in Breda et al. (2005) for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu’s equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The “spectral accuracy” convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) and Sun and Song (2009) . [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a study of the relationship between the homotopy analysis method (HAM) and harmonic balance (HB) method. The HAM is employed to obtain periodic solutions of conservative oscillators and limit cycles of self-excited systems, respectively. Different from the usual procedures in the existing literature, the HAM is modified by retaining a given number of harmonics in higher-order approximations. It is proved that as long as the solution given by the modified HAM is convergent, it converges to one HB solution. The Duffing equation, the van der Pol equation and the flutter equation of a two-dimensional airfoil are taken as illustrations to validate the attained results. [Copyright &y& Elsevier]

Abstract: In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity. [Copyright &y& Elsevier]

Abstract: In this paper, an optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity. [Copyright &y& Elsevier]

Abstract: In this paper, we propose the cubic semiorthogonal compactly supported B-spline wavelets as a basis functions for solution of nonlinear Fredholm–Hammerstein integral equations of the second kind. Properties of these wavelets and some operational matrices are first presented. These properties are then used to reduce integral equations to some algebraic equations. The exponential convergence rate of the method, , is proved. The method is computationally attractive, and applications are demonstrated through illustrative examples. [Copyright &y& Elsevier]

Abstract: In this paper, an analytic technique, namely the homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of the solution series. [Copyright &y& Elsevier]

Abstract: In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods. [Copyright &y& Elsevier]