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In this paper we establish new L p -estimates for solutions of anisotropic fractional equations, as the ones governed by the anisotropic fractional laplacian Δ β , s . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the time-fractional generalized thin film equation (TFGTFE) with two arbitrary functions and perform the group classification with respect to these arbitrary functions. Specifically, all vector fields admitted by the considered equations are provided utilizing Lie symmetry analysis. Then the corresponding symmetry reductions are carried out and exact solutions to some special equations are obtained. In particular, we construct the power series solutions to one type of TFGTFE by means of the combination of the Erd e ´ lyi-Kober (EK) operator with the analytic power series method and verify the convergence of the power series solutions using implicit function theorem. In addition, taking advantage of Matlab software, the three-dimensional diagrams and two-dimensional graphs of some obtained solutions are demonstrated for the purpose of visualization. [ABSTRACT FROM AUTHOR]

This paper concentrates on providing a new approach to arrive at approximate solution of a fractional nonlinear system of weakly singular integro-differential equations. This approach has the ability to calculate several undetermined values of the problem at several grid points out of considering the starting values. The convergence of the adopted approach is investigated and it is shown that the new strategy has a high order of convergence. Some examples are considered to observe the accuracy of the established scheme. [ABSTRACT FROM AUTHOR]

In this paper, an implicit robust difference method with graded meshes is constructed for the modified Burgers model with nonlocal dynamic properties. The L1 formula on graded meshes for the fractional derivative in the Caputo sense is employed. The Galerkin method based on piece-wise linear test functions is used to handle the nonlinear convection term u u x implicitly and attain a system of nonlinear algebraic equations. The Taylor expansion with integral remainder is used to deal with the fourth-order term u xxxx and the second-order term u xx . Then the existence and uniqueness of numerical solutions for the proposed implicit difference scheme are proved. Meanwhile, the unconditional stability is also derived. By introducing a new discrete Gronwall inequality, we improve the L 2 -stability to the α -robust stability, that is, when α → 1 - , the bound will not blow up. And we also yield the optimal convergence order in the L 2 energy norm. Finally, we give three numerical experiments to illustrate and compare the efficiency of the proposed robust method on uniform meshes and graded meshes. It shows that the results are consistent with our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and (β , γ k) -resolvent family { S β , γ k (t) } t ≥ 0 . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this present paper, we investigate a new class of singular double phase p-Laplacian equation problems with a ψ -Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold. [ABSTRACT FROM AUTHOR]

In this paper, Darbo's fixed point theorem is generalized and it is applied to find the existence of solution of a fractional integral equation involving an operator with iterative relations in a Banach space. Moreover, an example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we discuss the uniqueness and existence of solutions to these initial-boundary value problems. Under some suitable conditions on the problem data, we then prove positivity of the solutions. Based on these results, several comparison principles for the solutions to the initial-boundary value problems for the linear time-fractional diffusion equations are derived. [ABSTRACT FROM AUTHOR]

This paper presents new consistent methods for continuous-time Multiple-Input-Single-Output (MISO) Errors-In-Variables (EIV) systems by fractional models. The proposed idea is to use Higher-Order Statistics (HOS), such as fourth-order cumulants (foc), instead of noisy input and output measurements to obtain unbiased estimates. Firstly, all differentiation orders are assumed to be known a priori and linear coefficients are estimated. The developed estimator is based on minimizing the equation error and it is called fractional fourth-order based-least squares estimator ( f r a c - f o c - l s ). Secondly, the global commensurability of the fractional MISO system is considered. The f r a c - f o c - l s is combined with a non linear technique to estimate the global commensurate order along with linear coefficients. The developed algorithm is based on minimizing the output error and called fractional fourth-order cumulants based-least squares combined with global commensurate order optimization ( f r a c - f o c - g c o o l s ). The consistency of the developed estimators, in presence of high levels of noise corrupting both the input and output measurements, is assessed through a numerical example with the help of Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an inverse problem of determining the time-dependent source coefficient in a semilinear reaction–diffusion equation involving the Caputo fractional time derivative of order 0 < α < 1 in the case of nonlocal boundary and integral overdetermination conditions. We prove the existence and uniqueness of the classical solution by Fourier analysis and the iteration method. Moreover, we show the continuous dependence of this solution upon the additional data of the inverse problem. [ABSTRACT FROM AUTHOR]

We introduce and study fractional Sobolev spaces of functions taking their values in a Banach space. Our approach is based on Riemann-Liouville derivative. In this regard, paper is a continuation of the paper [D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. of Function Spaces and Appl. 2013 (2013), Art. ID 128043, 15 pp.], where real-valued functions are investigated. Moreover, we study the relation between the fractional derivative of an abstract function of one variable and partial fractional derivative of a real-valued function of two variables. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the problem of recovering the historical distribution for diffusion equations in which the diffusion operators are described by the coupling of local and nonlocal type. The problem essentially arises in many real-world circumstances including the biological population dynamic where a population competes for the resources and diffuses by a combination of the Brownian and Lévy processes. We first design a typical example to illustrate the ill-posed nature of the problem. A fractional filter method is then proposed to achieve reliable approximations of the problem. The stability and convergence of the proposed method are gingerly analyzed. Four numerical examples, with the support from the finite difference method and the fast Fourier transform, are implemented to validate the theoretical results including the ill-posedness and the effect of regularization. The numerical results agree with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the space-time Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: ∂ β + ν 2 (- Δ) α / 2 u (t , x) = I t γ σ u (t , x) W ˙ (t , x) , t > 0 , x ∈ R d , where W ˙ is the space-time white noise, α ∈ (0 , 2 ] , β ∈ (0 , 2) , γ ≥ 0 and ν > 0 . The existence/uniqueness of a random field solution has been obtained in [9] under the condition that 2 (β + γ) - 1 - d β / α > 0 . The Hölder regularity of the solution has been obtained in the same reference, but only for the case β + γ ≤ 1 . In this paper, we use the idea from the local fractional derivative to establish the Hölder regularity of the solution for all possible cases – β ∈ (0 , 2) , which in particular recovers the special case in [9] when β ∈ (0 , 1 - γ ] . As a rather surprising consequence, when γ = 0 , α = 2 and β is close to 2, the space and time Hölder exponents are both to 1 - , which is different from the known Hölder exponents for the stochastic wave equation which are (1 / 2) - . [ABSTRACT FROM AUTHOR]

In this paper, we initially derive the equivalent fractional integral equation to Ψ -Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The paper's primary objective is to obtain estimates on Ψ -Hilfer fractional derivative and utilize it to derive the hybrid fractional differential inequalities involving Ψ -Hilfer fractional derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions and comparison theorems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, some exact controllability results are investigated for an integral solution of a non-densely defined abstract fractional differential system involving Hilfer fractional derivative. We tend to implement semigroup theory, fractional calculus, and measure of noncompactness to obtain the main results by fixed point technique. Finally, an application is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen's inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area. [ABSTRACT FROM AUTHOR]

Many natural (biological, physical, etc.) and social systems possess power-law memory, and their mathematical modeling requires application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic behavior, which may be very different from the behavior of memoryless systems. Finding periodic solutions is essential for understanding regular and chaotic behavior of nonlinear systems. Fractional systems do not have periodic solutions except fixed points. Instead, they have asymptotically periodic solutions which, in the case of stable regular behavior, converge to the periodic sinks (similar to regular dissipative systems) and, in the case of unstable/chaotic behavior, act as repellers. In one of his recent papers, the first author derived equations which allow calculations of asymptotically periodic points for a wide class of discrete maps with memory. All fractional and fractional difference maps of the orders 0 < α < 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1$$\end{document} belong to this class. In this paper we derive the equations that allow calculations of the coordinates of the asymptotically periodic points for a wider class of maps which include fractional and fractional difference maps of the arbitrary positive orders α > 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document}. The maps are defined as convolutions of a generating function - G K (x) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-G_K(x)$$\end{document} , which may be the same as in a corresponding regular map x n + 1 = - G K (x n) + x n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1}=-G_K(x_n)+x_n$$\end{document} , with a kernel U α (k) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_\alpha (k)$$\end{document} , which defines the type of a map. In the case of fractional maps, it is U α (k) = k α \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_\alpha (k)=k^\alpha $$\end{document} , and it is U α (k) = k (α) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_\alpha (k)=k^{(\alpha)}$$\end{document} , the falling factorial function, in the case of fractional difference maps. In this paper we define the space of kernel functions which allow calculations of the periodic points of the corresponding maps with memory. We also prove that in fractional maps of the orders 1 < α < 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\alpha <2$$\end{document} the total of all physical momenta of period–l points is zero. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the convolution polynomials and series that are a far reaching generalization of the conventional polynomials and power series with both integer and fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution polynomials and series generated by the Sonine kernels. These kernels were recently employed for construction of the general fractional integrals and derivatives, which include the single-, the multi-order, and the distributed order fractional derivatives as their particular cases. In the first part of the paper, we formulate and prove the second fundamental theorem for the n-fold general fractional integrals and the n-fold general sequential fractional derivatives of both the Riemann-Liouville and Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula. It provides a representation of a function as a convolution polynomial with a remainder in form of a composition of the n-fold general fractional integral and the n-fold general sequential fractional derivative in the Riemann-Liouville and the Caputo sense, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the explicit formulas for their coefficients in terms of the n-fold general sequential fractional derivatives evaluated at the point zero. [ABSTRACT FROM AUTHOR]

This paper proposes a new operational numerical method based on Touchard wavelets for solving fractional pantograph differential equations. First, we present an operational matrix of fractional integration as well as the fractional derivative of the Touchard wavelets. Then, by approximating the fractional derivative of the unknown function in terms of the Touchard wavelets and also by using collocation method, the original problem is reduced to a system of algebraic equations. Finally, to show the accuracy and the validity of the proposed technique, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional time-fractional chiral non-linear Schrödinger equation in physical media is considered in this paper. At the outset, the Lie group analysis is applied to build a set of infinitesimal generators for this equation with the aid of the Riemann–Liouville fractional derivatives. Consequently, the reduction for the considered equation into an ordinary differential equation of fractional order is obtained by using these generators and the ErdLélyi–Kober fractional operator. As a result of this reduction, we use power series analysis to get an analytical solution provided by a convergence analysis of the obtained solution. Furthermore, we construct a numerical solution based on hyperbolic functions using the fractional reduced differential transform method in the sense of Caputo fractional derivatives. Also, we detect absolute errors by performing a comparison between the exact and numerical solutions of the equation under study, while investigating the effect of fractional order α on the numerical solution. Finally, conservation laws are derived using the the formal Lagrangian and new conservation theorem. [ABSTRACT FROM AUTHOR]

In recent decades, there has been a growing interest in generalizing differential equations of arbitrary order by replacing integer derivatives with non-integer derivatives. This allows for the construction of new models that can replicate memory effects due to the non-locality of fractional operators. However, this approach can lead to models that suffer from physical misinterpretation, do not maintain mass balance, and alter the original units of model parameters. This paper introduces an approach based on the generalization of exponential by Mittag-Leffler decays, which is associated with waiting time distributions. The model remains physically interpretable and ensures that the dimensions are constructively corrected. We discuss the mean waiting time, the asymptotic behavior, and apply this theory to develop a novel fractional SIRS model. Additionally, we present numerical results. [ABSTRACT FROM AUTHOR]

This paper focuses on modeling and analyzing a nonlinear fractional Maxwell fluid across a vertical plate using a novel definition of the Caputo fractional derivative. Specifically, the newly formulated fractional definition is tailored for the finite difference method. Buoyancy effects are considered to bring the model closer to the natural occurrences of the magnetic field. The novel fractional derivative definition of the designed model has been implemented, and the results have been further analyzed using theoretical and numerical methods. The range of fractional parameters α and β is 0 < α ≤ 1 and 0 < β ≤ 1 , respectively. Significant theorems are also included in the study. A simulation was performed to determine which fractional derivative operator is more accurate than the others. The designed definition demonstrates a high convergence rate and decreased computational cost, both of which are primary objectives. Graphs have also been plotted to illustrate the velocity and temperature distributions as the physical parameters vary. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of parametric GF operators are described. Some properties of the parametric GFIs and the parametric GFDs are proved. Representation of the parametric GFIs and parametric GFDs via the GF integrals on the finite interval (a, b). Fundamental theorems of parametric GFC are proved. Applied mathematical interpretations of parametric general fractional derivatives for processes with memory are given. [ABSTRACT FROM AUTHOR]

In the current paper, an inverse boundary value problem so-called the sideways problem for the multi-term time-fractional diffusion equation is investigated. The problem of interest includes the recovering of the diffusion distribution from the boundary data. We prove that the problem is ill-posed as the solution does not continuously depend on the boundary data. We further propose a fractional filter method to regularize the problem. The stability and convergence of the proposed method are gingerly analyzed. Two numerical examples, with the support from the fast Fourier transform (FFT), are implemented to illustrate the theoretical results. The numerical results are consistent with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In the present paper, some inequalities regarding Csiszár divergence are generalized for diamond integrals by utilizing Green function and Fink’s identity. By selecting various kinds of convex functions, bound of different entropies and divergence measures are obtained. Furthermore, in seek of applications, bounds of different divergence measures are estimated on diverse fixed time scales. These divergence measures are fundamental in problems pertaining to source and channel coding, large deviations theory, and other sorts of problems. [ABSTRACT FROM AUTHOR]

The paper is designed to highlight the presence of an optimal solution to a differential system comprising fractional order derivatives, precisely the ψ-Caputo derivative, under the prescribed initial constraints. This is attained through a new variant of cyclic (noncyclic) condensing operators incorporating a pair of maps. The altered notion facilitates in evolving best proximity point (pair) results as well as coupled best proximity point results with supportive corollaries. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a coupled system which consists of a fractional differential hemivariational inequality (FDHVI, for short) combined with a fractional hemivariational inequality (FHVI, for short). By utilizing Rothe method and surjectivity result, the existence of coupled system is established. In addition, we apply the results to a new quasistatic contact problem, in which the constitutive equations are described by the fractional Kelvin-Voigt laws. [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results. [ABSTRACT FROM AUTHOR]