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In this research, the Hadamard fractional derivative is used to define the time fractional coupled nonlinear Schrödinger–Hirota equations. The logarithmic Chebyshev cardinal functions, as a new category of cardinal functions, are introduced to build a numerical method to solve this system. To do this, the Hadamard fractional differentiation matrix of these functions is obtained. In the developed method, by considering a hybrid approximation of the problem's solution using the logarithmic Chebyshev cardinal functions (for the temporal variable) and classical Chebyshev cardinal polynomials (for the spatial variable), and employing the interpolation property of these basis functions, along with the expressed derivative matrix, solving the fractional system turns into obtaining the solution of an algebraic system of equations. Two numerical examples are investigated to acknowledge the high accuracy of the introduced procedure. • A new family of basis functions called logarithmic Chebyshev cardinal functions (LCCFs) is introduced. • Hadamard fractional differentiation matrix of these functions is obtained. • The Hadamard time fractional coupled nonlinear Schrödinger–Hirota equations are introduced. • A hybrid method based on LCCFs and classical Chebyshev cardinal polynomials is developed for this system. • The accuracy of the developed method is checked by solving some examples. [ABSTRACT FROM AUTHOR]

• Establishing a relationship between Atangana-Baleanu-Caputo derivative and semilinear systems. • Extend the principle of application of the control function on different type of fractional systems. • Prove the existence of mild solutions by adopting Darbo's fixed point techniques. • Show the effect of impulses and delay on fractional differential equations with an Atangana-Baleanu-Caputo derivative. • Build a study on delay systems by establishing some spaces. We discuss the controllability of semilinear differential equations of fractional order with impulses and delay. We make use of the Atangana-Baleanu derivative. Our main tools are semigroup theory, the fixed point theorem due to Darbo and their combination with the properties of measures of noncompactness. Our abstract results are well supported by an illustrative example. [ABSTRACT FROM AUTHOR]

Highlights • The index concept is generalized for categorizing system of fractional differential algebraic equations and analyzing the solvability of such systems. • A model of a simple pendulum in a Newtonian fluid is obtained and described by using nonlinear fractional differential algebraic equations. • An application in an electrical circuit is introduced and the solvability of the obtained linear system is studied. • A simple and efficient numerical method for solving a system of FDAEs with different fractional derivatives including Liouville-Caputo's definition, Caputo-Fabrizio's definition and a definition with Mittag-Leffler kernel are introduced. Abstract One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo's definition, Caputo–Fabrizio's definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method. [ABSTRACT FROM AUTHOR]

Highlights • System of fractional differential equations involving non-singular Mittag-Leffler kernel is solved by collocation methods on discontinuous piecewise polynomial space. • Existence and regularity of the solutions are investigated. • The convergence and super-convergence properties of the introduced methods are derived on graded meshes. • Non-singular fractional diffusion equation is solved by transforming to the system of fractional differential equations. Abstract A system of fractional differential equations involving non-singular Mittag-Leffler kernel is considered. This system is transformed to a type of weakly singular integral equations in which the weak singular kernel is involved with both the unknown and known functions. The regularity and existence of its solution is studied. The collocation methods on discontinuous piecewise polynomial space are considered. The convergence and superconvergence properties of the introduced methods are derived on graded meshes. Numerical results provided to show that our theoretical convergence bounds are often sharp and the introduced methods are efficient. Some comparisons and applications are discussed. [ABSTRACT FROM AUTHOR]

A new operational matrix based approach is studied for numerical solution of 2D-integro-differential equations with non-local (integral) boundary conditions whose arise in some physical problems. Some important theoretical results are presented to reduce complexity and computational costs of the proposed method. We also give an error estimation which will be useful in estimating the error of approximate solution for the problems that we do not have any information about their exact solution. Illustrative numerical examples are also given to clarify the performance and accuracy of the new method. [ABSTRACT FROM AUTHOR]

This paper proposes a new strategy to study some useful properties of growth rates of functions in C α ∈ R spaces in order to analyze the Leibniz rule for fractional derivatives. The differential operators are taken in the Riemann–Liouville sense. Moreover, stability analysis of the proposed strategy is investigated. The results demonstrate that the proposed theoretical analysis is accurate. [ABSTRACT FROM AUTHOR]

In this work, the distributed-order time fractional version of the Schrödinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss–Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples. • The distributed-order time fractional version of the Schrödinger equation is defined. • The orthonormal piecewise Jacobi functions as a new class of basis functions are defined. • A new formulation for the Caputo fractional derivative of these piecewise functions is derived. • The Jacobi polynomials and the piecewise Jacobi functions are employed mutually to solve this problem. • The accuracy of the proposed method is investigated in some numerical examples. [ABSTRACT FROM AUTHOR]

Glioblastoma is the most common and deadly primary brain tumor in adults. To optimize the treatment strategies, it is essential to understand the tumor growth dynamics in different periods. In this study, we use image processing techniques to combine the available early-stage imaging data and applied a fractional reaction–diffusion equation to predict the human glioblastoma multiforme tumor growth. We consider the heterogeneity of the brain tissue by assigning different diffusion coefficients for the three regions of the human brain. A meshfree method based on the thin plate spline radial basis function is used for the numerical solution of nonlinear time fractional Proliferation-Invasion equation. The results show that the proposed model has a better fit with the experimental data. The prediction of tumor growth at any desired time with no need to repeated imaging is another advantages of the model which could reduce the side effects and cost of diagnostic and therapeutic methods. The model can also incorporate the effects of various treatment modalities such as hyperthermia, radiation, and surgery on tumor growth. Furthermore, it can enable the use of patient-specific characteristics in diagnosis and treatment and facilitate the development of personalized medicine approaches. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some fixed point theorems for generalized contractions in cone metric spaces. Our theorems extend some results of Suzuki (2008) [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc Amer Math Soc 136(5) (2008), 1861–1869] and Kikkawa and Suzuki (2008) [M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal 69(9) (2008), 2942–2949]. [Copyright &y& Elsevier]

Abstract: In this paper the authors suggest a new method of detection of possible differences between similar near infrared (NIR) spectra based on the self-similar (fractal) property. This property is a general characteristic that belongs to a wide class of the strongly-correlated systems. As an example we take a set of NIR spectra measured for three systems: (1) glassy carbon (GC) electrodes, (2) GC electrodes affected by azobenzene (AB) substance and finally (3) films (AB-FILM). Besides the physical model that should describe the intrinsic properties of these substances we found the fitting function that follow from the linear principle for the strongly-correlated variables. This function expressed in the form of linear combination of 4 power-law functions describes with the high accuracy the integrated curves that were obtained from the averaged values of the initially measured spectra. The nine fitting parameters can be considered as the quantitative “finger prints” for detection of the differences between similar spectra. Besides this result we established the self-similar behavior of the remnant functions. In other words, the difference between the initially integrated function and its fitting function can be expressed in the form of linear combinations of periodical functions having a set of frequencies following to relationship ω(k)= ω 0 ξ k , where the initial frequency ω 0 and scaling factor ξ are determined by the eigen-coordinates method. This behavior in the NIR spectra was discovered in the first time and physical reasons of such behavior merit an additional research. [Copyright &y& Elsevier]

Abstract: In this paper the new prediction method based on analysis of the integrated (cumulative) curves is suggested. This method includes the procedure of the optimal linear smoothing (POLS) for the finding of optimal trends, independent “reading” of relative fluctuations in terms of β-distribution function that are formed after subtraction of the calculated trend and the recognition of the proper fitting hypothesis for the integrated optimal trends by the eigen-coordinates method. The combined noninvasive approach was applied to analysis of temperature data obtained from the site http://data.giss.nasa.gov/gistemp/ related to the global warming (GW) phenomenon. These data are considered as nontrivial examples of verification of new forecasting method. The available data were combined into six files covering the mean/anomalous temperature 1546month’s points covering the period from the January of 1880 up to October of 2008. Besides the global registered points the combined files included in themselves the north/south data points measured independently for both the Earth’s hemispheres. The combined new method (preliminary verified on mimic data) applied to these files predicts the changing of the GW period by the global cooling (GC) period that will happen during the years 2038–2136. Besides this important result a new method helps to discover the influence of a small but stable oscillating process with a set of self-similar periods Ω n = Ω 0 ξ n , n=0, ±1, ±2, ±3, ±4 with mean period 〈T〉=12.55year. This fact should present interest for ecologists and meteorologists working in this field. [Copyright &y& Elsevier]

• A new stochastic fractional Coronavirus (2019-nCov) mathematical model with modified parameters are presented. • Milstein's higher order method is constructed to study the model problem • Mean square stability of Milstein algorithm is proved. • Comparative studies and numerical simulations are implemented In this paper, a new stochastic fractional Coronavirus (2019-nCov) model with modified parameters is presented. The proposed stochastic COVID-19 model describes well the real data of daily confirmed cases in Wuhan. Moreover, a novel fractional order operator is introduced, it is a linear combination of Caputo's fractional derivative and Riemann-Liouville integral. Milstein's higher order method is constructed with the new fractional order operator to study the model problem. The mean square stability of Milstein algorithm is proved. Numerical results and comparative studies are introduced. [ABSTRACT FROM AUTHOR]

In this paper we consider multi variable orders differential equations (MVODEs) with non-local and no-singular kernel. The derivative is described in Atangana and Baleanu sense of variable order. We use the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices. We transfer the original equations to a system of algebraic equations using operational matrices and collocation method. The convergence analysis of the presented method is discussed. Few examples are presented to show the efficiency of the presented method. [ABSTRACT FROM AUTHOR]

In this article, we examine the impact of leakage delays on robust synchronization for fractional order multi-weighted complex dynamical networks(MFCDN) under non-linear coupling via non-fragile control. By employing the fractional order comparison principle, suitable Lyapunov method, and some fractional order inequality techniques, we ensured the robust asymptotical synchronization for MFCDN. In addition to common findings, we have done some specific research in order to get reliable synchronization for multi-weighted complex dynamical network(MCDN) without leakage delay. Additionally, our findings gained are applicable to single weighted FCDN and integer order complex dynamical networks, regardless of whether they have a single weight or many weights. Our suggested approach is shown to be more effective and practical in this article by providing a numerical simulation. [ABSTRACT FROM AUTHOR]

• To generalize truncated M -fractional derivative as a novel and effective derivative under interval uncertainty. • To investigate the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative. • To present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w -increasing. • To develop the significant and applicable classical properties of this novel fractional derivative under interval uncertainty. Abstract In the recent years some efforts were made to propose simple and well-behaved fractional derivatives that inherit the classical properties from the first order derivative. In this regards, the truncated M -fractional derivative for α -differentiable function was recently introduced that is a generalization of four fractional derivatives presented in the literature and has their important features. In this research, we aim to generalize this novel and effective derivative under interval uncertainty. The concept of interval truncated M -fractional derivative is introduced and some of the distinguished properties of this interesting fractional derivative such as Rolle's and mean value theorems, are developed for the interval functions. In addition, the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative are also investigated. Finally, we present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w -increasing (w -decreasing) by solving a number of test problems. [ABSTRACT FROM AUTHOR]

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin–Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed. [ABSTRACT FROM AUTHOR]

This paper focuses on providing a new scheme to find the fuzzy approximate solution of fractional differential equations (FDEs) under uncertainty. The Caputo-type derivative base on the generalized Hukuhara differentiability is approximated by a linearization formula to reduce the corresponding uncertain FDE to an ODE under fuzzy concept. This new approach may positively affect on the computational cost and easily apply for the other types of uncertain fractional-order differential equation. The performed numerical simulations verify the proficiency of the presented scheme. [ABSTRACT FROM AUTHOR]

In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes. [ABSTRACT FROM AUTHOR]

In this article, we examine a fractional-order biological population model with carrying capacity. The blended homotopy techniques pertaining to the Sumudu transform are utilized to explore the solutions of a nonlinear fractional-order population model with carrying capacity. The fractional derivative of the Caputo type is utilized in the proposed investigation. The numerical computations indicate the sufficiency of the iterations for the improved estimations of the solutions of this fractional-order biological population model which exemplifies the potency and soundness of the utilized schemes. The analysis explored through the utilization of the projected methods reveals that both of the schemes are in a great agreement with each other. The variations of the prey and predator populations with respect to time and fractional order of the Caputo derivative are presented and graphically illustrated. [ABSTRACT FROM AUTHOR]

This article presents a variable order nonlinear mathematical model and its optimal control for a Tumor under immune suppression. The formulation generalizes the one proposed by Kim et. al. consisting of eleven integer order differential equations. The new approach adopts a variable-order fractional model with the derivatives defined in the Caputo sense. Two control variables, one for immunotherapy and one for Chemotherapy, are proposed to eliminate or reduce the Tumor cells. A simple numerical technique called the nonstandard generalized Euler method is developed to solve the proposed optimal control problem. Moreover, the stability analysis and the truncation error are studied. Numerical simulations and comparative studies are implemented. Our findings disclose that the proposed scheme used here has two main advantages: it is faster than the generalized Euler scheme and it can reduce the number of Tumor cells in a proper process better than the second scheme. [ABSTRACT FROM AUTHOR]