Discrete Gronwall’s inequality for Ulam stability of delay fractional difference equations

In this paper, the authors analyze a time-fractional advection–diffusion equation, involving the Riemann–Liouville derivative, with a nonlinear source term. They determine the Lie symmetries and reduce the original fractional partial differential equation to a fractional ordinary differential equation. The authors solve the reduced fractional equation adopting the Caputo’s definition of derivatives of non-integer order in such a way the initial conditions have a physical meaning. The reduced fractional ordinary differential equation is approximated by the implicit second order backward differentiation formula. The analytical solutions, in terms of the Mittag-Leffler function for the linear fractional equation and numerical solutions, obtained by the finite difference method for the nonlinear fractional equation, are used to evaluate the solutions of the original advection–diffusion equation. Finally, comparisons between numerical and exact solutions and the error estimates show that the proposed procedure has a high convergence precision.

The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalised fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.

The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalised fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.

<p>In this paper, we investigate the existence and uniqueness of $ L^p $-solutions for nonlinear fractional differential and integro-differential equations with boundary conditions using the Caputo-Hadamard derivative. By employing Hölder's inequality together with the Krasnoselskii fixed-point theorem and the Banach contraction principle, the study establishes sufficient conditions for solving nonlinear problems. The paper delves into preliminary results, the existence and uniqueness of $ L^p $ solutions to the boundary value problem, and presents the Ulam-Hyers stability. Furthermore, it investigates the existence, uniqueness, and stability of solutions for fractional integro-differential equations. Through standard fixed-points and rigorous mathematical frameworks, this research contributes to the theoretical foundations of nonlinear fractional differential equations. Also, the Adomian decomposition method ($ {\mathcal{ADM}} $) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.</p>

Deep neural network models are vulnerable to attacks from adversarial methods, such as gradient attacks. Evening small perturbations can cause significant differences in their predictions. Adversarial training (AT) aims to improve the model’s adversarial robustness against gradient attacks by generating adversarial samples and optimizing the adversarial training objective function of the model. Existing methods mainly focus on improving robust accuracy, balancing natural and robust accuracy and suppressing robust overfitting. They rarely consider the AT problem from the characteristics of deep neural networks themselves, such as the stability properties under certain conditions. From a mathematical perspective, deep neural networks with stable training processes may have a better ability to suppress overfitting, as their training process is smoother and avoids sudden drops in performance. We provide a proof of the existence of Ulam stability for deep neural networks. Ulam stability not only determines the existence of the solution for an operator inequality, but it also provides an error bound between the exact and approximate solutions. The feature subspace of a deep neural network with Ulam stability can be accurately characterized and constrained by a function with special properties and a controlled error boundary constant. This restricted feature subspace leads to a more stable training process. Based on these properties, we propose an adversarial training framework called Ulam stability adversarial training (US-AT). This framework can incorporate different Ulam stability conditions and benchmark AT models, optimize the construction of the optimal feature subspace, and consistently improve the model’s robustness and training stability. US-AT is simple and easy to use, and it can be easily integrated with existing multi-class AT models, such as GradAlign and TRADES. Experimental results show that US-AT methods can consistently improve the robust accuracy and training stability of benchmark models.

Fractional calculus, in the understanding of its theoretical and real-world presentations in numerous regulations, for example, astronomy and manufacturing problems, is discovered to be accomplished of pronouncing phenomena owning long range memory special effects that are challenging to handle through traditional integer-order calculus. Nearby an increasing concentration has been in the modification of fractional calculus as a successful modelling instrument for complicated systems, contributing to innovative viewpoints in their dynamical investigation and regulator. This improvement in the methodical knowledge is established by an enormous quantity of evens developing on the subject, manuscripts, and presentations in the past years. Nevertheless, countless singularities still pose significant confronts to the apprehensive population and fractional calculus appears to be plausibly contestant to incorporate larger exemplars through detaching graceful dependent on the explanation of involvedness. This special issue contains papers about recent theoretical development and methods and applications results on the topics in almost all branches of sciences and engineering. We have received 56 papers during the submission period. Five were withdrawn; 34 were rejected including the papers submitted to the member of our editorial board. Only 17 good papers were accepted for publication. The papers of this special issue cover some new algorithms and procedures designed to explore conventional, fractional, and time-scales differential equations of general interest. New understandings of existences and uniqueness theorems of some differential equations were also offered. In the following we give the brief summary of the content of the special issue. The existence and uniqueness theorems for impulsive fractional equations with the two-point and integral boundary conditions and sufficient condition on the fractional integral for the convergence of a function were presented. Besides the stability, boundedness, and Lagrange stability of fractional differential equation with initial time difference, stability of nonlinear Dirichlet BVPs governed by fractional Laplacian was proposed. A novel study on the singular perturbations fractional equations, analysis of a fractional-order couple model with acceleration in feelings, q-Sumudu transforms of q-analogues of Bessel functions, certain fractional integral formulas involving the product of generalized Bessel functions, and an expansion formula with higher-order derivatives for fractional operators of variable order were investigated in detail. A novel study underpinning construction of solution for fractional differential equations such as decomposition method for time fractional reactional-diffusion equation and high-order compact difference scheme for numerical solution of time fractional heat equation, a procedure to construct exact solutions of nonlinear fractional differential equations, were presented. An investigation on impulsive multiterm fractional differential equations and multiple positive solutions for nonlinear fractional boundary value problems were undertaken. The editorial board trust that the set of nominated papers will offer readers an opportune renovate of significant investigation subjects and may also operate as a policy for encouraging additional contribution in this fast evolving ground.

The classical Lane-Emden differential equation, a nonlinear second-order differential equation, models the structure of an isothermal gas sphere in equilibrium under its own gravitation. In this paper, the Mittag-Leffler function expansion method is used to solve a class of fractional LaneEmden differential equation. In the proposed differential equation, the polytropic term f(y(x)) = ym(x) (where m = 0,1,2,... is the polytropic index; 0 < x <=1) is replaced with a linear combination f(y(x)) = a0 + a1y(x) + a2y2(x) + ··· + amym(x) + ··· + aNyN(x),0 <=m <=N,N <= N_0. Explicit solutions of the fractional equation, when f(y) are elementary functions are presented. In particular, we consider the special cases of the trigonometric, hyperbolic and exponential functions. Several examples are given to illustrate the method. Comparison of the Mittag-Leffler function method with other methods indicates that the method gives accurate and reliable approximate solutions of the fractional Lane-Emden differential equation.

In quantum field theory, the fractional Kundu-Eckhaus and massive Thirring models are nonlinear partial differential equations under fractional sense inside nonlinear Schrödinger class. In this study, approximate analytical solutions of such complex nonlinear fractional models are acquired by means of conformable residual power series method. This method presents a systematic procedure for constructing a set of periodic wave series solutions based on the generalization of conformable power series and gives the unknown coefficients in a simple pattern. By plotting the solutions behavior of the models; the convergence regions in which the solutions coincide to each other are checked for various fractional values. The approximate solutions generated by the proposed approach are compared with the exact solutions -if exist- and the approximate solutions obtained using qHATM and LADM. Numerical results show that the proposed method is easy to implement and very computationally attractive in solving several complex nonlinear fractional systems that occur in applied physics under a compatible fractional sense.

Sustainable development is now engaged in many leading countries. The notion is connected with the activity of countries, their economies, social and political institutions. One of the criteria of sustainability is the economic sphere, the study of which will allow you to control and predict its activity. It is known that the description of continuous processes that arise in economic problems, using definite integrals. In the case when the studied continuous development is carried out in small jumps, when it is the description of the use of fractional integral and differential equations. However, their exact solution is possible only in special cases. Therefore, the construction of approximate methods for solving fractional integral and differential equations is of great importance. In this paper, we give an approximate solution of fractional integral equations of the first kind, where the fractional integral is constructed using the Weyl fractional integral. It turns out that the constructed operator of fractional integration is symmetrical and positively definite. These properties are built by the operator enables to introduce the scalar product on the basis of the specified operator and the corresponding energy space. Next, using the same methodology that is used in the finite element method, approximate solution is constructed, constructed a rough scheme of the method and rationale of the proposed method to construct the energy space. The integral equation of the standard form is given as a specific example, as well as an example solution to illustrate the effectiveness of the proposed method.

In this paper we will use a Liapunov functional to prove the stability of the trivial solution of the () order nabla (q, h)-fractional difference equation (0.1) where the operator in this equation is defined in Definition 2.3. When and this fractional difference equation is a fractional nabla quantum difference equation (see the monograph by Kac and Cheung) and when and this fractional equation is simply an order nabla difference equation.

An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10−14, while the L2 error is on the order of 10−13. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

Approximate analytical solutions of the Thomas–Fermi–Dirac (TFD) and Thomas–Fermi–Dirac–Gombás (TFDG) differential equations are obtained for neutral and positively charged atoms. The approximate solutions are of the form ψ = φ0 + η and Ψ = φ0 + η̃, respectively, where φ0 is a universal approximate analytical solution of the Thomas–Fermi (TF) differential equation for neutral atoms and η and η̃ are correction functions. The function φ0 has been obtained previously, by making use of a variational principle (the equivalent of the TF equation), and the functions η and η̃ are determined by solving second-order inhomogeneous linear differential equations with the boundary and subsidiary (normalization) conditions that are required in the TFD and TFDG models, respectively. The correction functions η and η̃ depend on the atomic or ionic radius, which, in turn, depends on the atomic number Z and electron number N of the system considered. The approximate solutions ψ and Ψ are tested by calculating the first ionization energies of Ar, Kr, and Xe. The values obtained are about the same as those calculated from the exact numerical solution of the TFD and TFDG equations. In three Appendices approaches for improved approximate analytical solutions of the TFD and TFDG equations are outlined. Appendix A proposes an iterative procedure for obtaining the correction functions η and η̃ in successively more refined approximations, which is expected to be of importance for multiply charged ions or for neutral atoms of low atomic number. Appendix B establishes a variational principle (which to the author's knowledge has not been formulated before) for obtaining approximate analytical solutions of the TFD and TFDG equations, or for achieving further refinements in the approximate solutions ψ and Ψ that are derived in this paper. Appendix C describes Jensen's modification of the TFD and TFDG models, which, with the technique developed in this paper, enables one to obtain approximate analytical solutions for negative ions.

The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense.