A New Approach to Computational Mathematics: Fractional Ratio Derivative (FRD)

Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence. References Boris Baeumer, Mihaly Kovacs, and Harish Sankaranarayanan. Higher order grunwald approximations of fractional derivatives and fractional powers of operators. Transactions of the American Mathematical Society, 367(2):813–834, 2015. doi:10.1090/S0002-9947-2014-05887-X E. Barkai, R. Metzler, and J. Klafter. From continuous time random walks to the fractional fokker-planck equation. Physical Review E, 61(1):132, 2000. doi:10.1103/PhysRevE.61.132 Z. Hao, Z. Sun, and W. Cao. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 281:787–805, 2015. doi:10.1016/j.jcp.2014.10.053 Ch Lubich. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3):704–719, 1986. doi:10.1137/0517050 M. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77, 2004. doi:10.1016/j.cam.2004.01.033 H. M. Nasir, B. L. K. Gunawardana, and H. M. N. P. Aberathna. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 3(4):237, 2013. doi:10.7763/IJAPM.2013.V3.212 H. M. Nasir and K. Nafa. A new second order approximation for fractional derivatives with applications. SQU Journal of Science, 23(1):43–55, 2018. doi:10.24200/squjs.vol23iss1pp43-55 W. Tian, H. Zhou, and W. Deng. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 84(294):1703–1727, 2015. doi:10.1090/S0025-5718-2015-02917-2 Y. Yu, W. Deng, and Y. Wu. Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations. arXiv preprint arXiv:1408.6364, 2014. doi:10.4310/CMS.2017.v15.n5.a1 Y. Yu, W. Deng, Y. Wu, and J. Wu. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 112:126–145, 2017. doi:10.1016/j.apnum.2016.10.01 L. Zhao and W. Deng. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 31(5):1345–1381, 2015. doi:10.1002/num.21947

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

Fractional calculus was formulated in 1695, shortly after the development of classical calculus. The earliest systematic studies were attributed to Liouville, Riemann, Leibniz, etc. [1,2]. For a long time, fractional calculus has been regarded as a pure mathematical realm without real applications. But, in recent decades, such a state of affairs has been changed. It has been found that fractional calculus can be useful and even powerful, and an outline of the simple history about fractional calculus, especially with applications, can be found in Machado et al. [3]. Now, fractional calculus and its applications is undergoing rapid developments with more and more convincing applications in the real world [4,5]. This Theme Issue, including one review article and 12 research papers, can be regarded as a continuation of our first special issue of European Physical Journal Special Topics in 2011 [4], and our second special issue of International Journal of Bifurcation and Chaos in 2012 [5]. These selected papers were mostly reported in The Fifth Symposium on Fractional Derivatives and Their Applications (FDTA'11) that was held in Washington DC, USA in 2011. The first paper presents an overview of chaos synchronization of coupled fractional differential systems. A list of coupling schemes are presented, including one-way coupling, Pecora–Carroll coupling, active–passive decomposition coupling, bidirectional coupling and other unidirectional coupling configurations. Meanwhile, several extended concepts of synchronizations are introduced, namely projective synchronization, hybrid projective synchronization, function projective synchroni- zation, generalized synchronization and generalized projective synchronization. Corresponding to different kinds of synchronization schemes, various analysis methods are presented and discussed [6]. The rest of the papers can be roughly grouped into three parts: three papers for fundamental theories of fractional calculus [7–9], five papers for fractional modelling with applications [10–14] and four papers for numerical approaches [15–18]. In the theory part, three papers focus on the existence of the solutions to the considered classes of nonlinear fractional systems, the equivalence system of the multiple-rational-order fractional system, and the reflection symmetry with applications to the Euler–Lagrange equations [7–9]. Baleanu et al. [7] use fixed-point theorems to prove the existence and uniqueness of the solutions to a class of nonlinear fractional differential equations with different boundary-value conditions. Li et al. [8] apply the properties of the fractional derivatives to change the multiple-rational-order system into the fractional system with the same order. Such a reduction makes it convenient for stability analysis and numerical simulations. The reflection symmetry and its applications to the Euler–Lagrange equations in fractional mechanics are investigated in Klimek [9], where an illustrative example is presented. The part on fractional modelling with applications consists of five papers [10–14]. Chen et al. [10] establish a fractional variational optical flow model for motion estimation from video sequences, where the experiments demonstrate the validity of the generalization of derivative order. Another fractional modelling in heat transfer with heterogeneous media is studied in Sierociuk et al. [11]. In the following paper, two-particle dispersion is explored in the context of the anomalous diffusion, where two modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes [12]. The last two papers in this part emphasize the applications of fractional calculus [13,14], where a novel method for the solution of linear constant coefficient fractional differential equations of any commensurate order is introduced in the former paper, and where the CRONE control-system design toolbox for the control engineering community is presented in the latter paper. The last four papers in part three are attributed to numerical approaches [15–18]. Sun et al. [15] construct a semi-discrete finite-element method for a class of temporal-fractional diffusion equations. On the other hand, an implicit numerical algorithm for the spatial- and temporal-fractional Bloch–Torrey equation is established, where stability and convergence are also considered [16]. In Fukunaga & Shimizu [17], a high-speed scheme for the numerical approach of fractional differentiation and fractional integration is proposed. In the last paper, Podlubny et al. [18] further develop Podlubny's matrix approach to discretization of non-integer derivatives and integrals, where non-equidistant grids, variable step lengths and distributed orders are considered. We try our best to organize this Theme Issue in order to offer fresh stimuli for the fractional calculus community to further promote and develop cutting-edge research on fractional calculus and its applications.

Continued mortality from Covid‐19 disease and environmental considerations resulting from the cemetery, landfill, wastewater treatment plants, and mass personal protective equipment (PPE) disposal sites demonstrate the importance of carefully studying how microorganisms spread in and across the soil, surface water, and groundwater. In this research, fractional diffusion equation for different functions has been solved using fractional derivative form with variable‐order (VO) to predict the diffusion of the SARS‐CoV‐2 virus. Comparison of the obtained results with the available experimental data and mathematical calculations shows that the use of VO form can predict the viral behavior more accurately. The main advantage of considering a VO fractional derivatives instead of a constant‐order fractional derivative or classical derivative is accurately predicting the behavior of propagation of microorganisms over large time and dimensional scales. In addition, to the specified SARS‐CoV‐2 virus, the method presented in this study will be able to investigate the diffusion pattern of all environmental pollutants at the nanoscale/microscale, including carcinogenic compounds, such as biogenic amines in aqueous and porous media. The obtained results manifest a high accuracy of mathematical approach which provides helpful information to urban and health planners/managers worldwide to manage other possible outbreaks.

We propose a novel mathematical framework to examine the free damped transverse vibration of a nanobeam by using the nonlocal theory of Eringen and fractional derivative viscoelasticity. The motion equation of a nanobeam with arbitrary attached nanoparticle is derived by considering the nonlocal viscoelastic constitutive equation involving fractional order derivatives and using the Euler-Bernoulli beam theory. The solution is proposed by using the method of separation of variables. Eigenvalues and mode shapes are determined for three typical boundary conditions. The fractional order differential equation in terms of a time function is solved by using the Laplace transform method. Time dependent behavior is examined by observing the time function for different values of fractional order parameter and different ratios of other parameters in the model. Validation study is performed by comparing the obtained results for a special case of our model with corresponding molecular dynamics simulation results found in the literature.

Steady state response of fractionally damped nonlinear viscoelastic arches by residue harmonic homotopy

On the basis of simple two-component nonlinear incommensurate fractional-order systems with positive and negative feedbacks, some general properties of fractional auto-oscillation systems are established. By linear stability analysis and numerical simulation, it is shown that fractional derivative orders and ratio between them can substantially change the stability conditions of the system and lead to appearing of complex oscillations and attractors, which cannot be found in their integer counterparts.

The complex mechanical behavior of materials are characterized by fluid and solid models with fractional calculus differentials to relate stress and strain fields. Fractional derivatives have been shown to describe the viscoelastic stress from polymer chain theory for molecular solutions [Rouse and Sittel, J. Appl. Phys. 24, 690 (1953)]. Here the propagation of infinitesimal waves in one dimensional horns with a small cross-sectional area change along the longitudinal axis are examined. In particular, the linear, conical, exponential, and catenoidal shapes are studied. The wave amplitudes versus frequency are solved analytically and predicted with mathematical computation. Fractional rheology data from Bagley [J. Rheol. 27, 201 (1983); Bagley and Torvik, J. Rheol. 30, 133 (1986)] are incorporated in the simulations. Classical elastic and fluid ‘‘Webster equations’’ are recovered in the appropriate limits. Horns with real materials that employ fractional calculus representations can be modeled to examine design trade-offs for engineering or for scientific application.

The purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functions, while for negative α it can be interpreted in terms of Liouville fractional integrals. Many of the main properties of the above Stirling functions are established; they generalize those, well known, for the classical Stirling numbers S(n, k) (). Several new applications are presented. Thus, the sum , for any complex and , is represented as a finite sum involving the S(α + 1, k). Whereas the case α = n is a classical result, even the particular case α = −n gives a new application. Further, Hadamard-type fractional integrals and derivatives, basic in Mellin transform theory on (0, ∞), are represented in terms of infinite series involving the S(α, k) and the powers of the operator . Its corresponding classical discrete version, involving the S(n, k), plays an important role in computational mathematics, combinatorial analysis and discrete mathematics.

Analyzing fractional effects on solutions of the generalized perturbed Zakharov-Kuznetsov equation using a residual series method with Caputo derivatives

Owing to integrating the dense range of distinct electric power sources, high volume of power generation units, abrupt and continuous changes in load demand, and rising utilization of power electronics, the electric power system (EPS) is striving for high-performance control schemes to counterwork the concerns depicted above. Additionally, it is highly creditable to have the controller structure as simple as possible from a viewpoint of practical implementation. Thus, this paper describes a virgin application of fractional order proportional integral–fractional order proportional derivative (FOPI–FOPD) cascade controller for load frequency control (LFC) of electric power generating systems. The proposed controller includes fractional order PI and fractional order PD controllers connected in cascade wherein orders of integrator ( $$\lambda$$ ) and differentiator ( $$\mu$$ ) may be fractional. The gains and fractional order parameters of the controller are concurrently tuned using recently proposed dragonfly search algorithm (DSA) by minimizing the integral time absolute error (ITAE) of frequency and tie-line power deviations. DSA is the mathematical model and computer simulation of static and dynamic swarming behaviors of dragonflies in nature, and its implementation in LFC studies is very rare, unveiling additional research gap to be bridged. Performance of the advocated approach is first explored on popular two-area thermal PS with/without governor dead band (GDB) nonlinearity and then on three-area hydrothermal PS with suitable generation rate constraints. To highlight the prominence and universality of our proposal, the work is extended to single-/multi-area multi-source EPSs. Several comparisons with DSA optimized FOPID controller and the relevant recent works for each test system indicate the contribution of proposed DSA optimized FOPI–FOPD cascade controller in alleviating settling time/undershoot/overshoot of frequency and tie-line power oscillations.

Nowadays, due to the fact that motor efficiency is strictly related to the diminution of emissions, researchers pay much attention to find a robust and efficient motor control technique for hybrid electric vehicles (HEVs). In this study, an optimal type-2 fuzzy fractional proportional plus integral derivative (IT2FOFP+ID) controller is applied to solve the throttle position and speed control problem of the HEVs. It is undeniable that the performance and effectiveness of the fuzzy-based proportional-integral-derivative (PID) controllers are dependent on its gains’ value. Hence, a novel improved heuristic technique, called IJAYA algorithm, is employed for the online tuning of the coefficients embedded in the specific controller structure. In contrast with the classical control methodologies that suffer from the lack of the self-regulating feature, the established controller has been adjusted online automatically. As another advantage of this control strategy, it is a model-free scheme and does not need the mathematical computation to identify the system model. To appraise the supremacy of the optimal IT2FOFP+ID controller than the other prevalent methodologies, a highly nonlinear EV model is utilized as a case study. In addition, the usefulness and robustness of the proposed method are tested by the experimental data, the EPA New York City Cycle. In the end, the new time-varying proposed technique is validated and implemented in hardware-in-the-loop real-time simulation based on OPAL-RT to study the feasibility of the designed IT2FP+ID controller with check outcomes on a physical platform.

In the present work, the natural transform iterative method (NTIM) is implemented to solve the biological population model (BPM) of fractional order. The method is tested for three nonlinear examples. The NTIM is a combination of a new iterative method and natural transform. We see that the solution pattern converges to the exact solution in a few iterations. The method handles an extensive range of differential equations of both fractional and integer order. The fractional order derivative is considered in Caputo’s sense. For mathematical computation, Mathematica 10 is used.

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.