Important Lemmas Research Articles

Solving Multi‐Term Time‐Fractional Diffusion Equations Using a High‐Order Fractional Finite Difference Scheme Accompanied by Neural Network Techniques

ABSTRACTThis article is devoted to solving multi‐term time‐fractional diffusion equations with a high‐order finite difference scheme. We approximate the temporal fractional derivatives by the L1‐2‐3 formula, an expansion of the L1 formula with higher accuracy. We employ neural network strategies and automatic differentiation techniques for approximating integer‐order derivatives. In the present work, we propose very important lemmas and theorems to prove the unconditional stability and convergence of the proposed scheme. Extensive numerical experiments for one‐ and two‐dimensional problems, especially for curved boundaries, confirm the theoretical convergence orders. Comparisons with fractional physics‐informed neural networks and finite difference methods in solving the multi‐term time‐fractional diffusion equations demonstrate the superiority of the proposed method. It is also illustrated that the proposed scheme can efficiently and accurately extract the pattern of the solutions even when noise corruption up to ten percent is imposed on the forcing term.

General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions

In this paper, we consider a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions in domains with non‐locally reacting boundary. By potential well method, we get the global existence of solution and through several important lemmas, we prove the general energy decay estimate of the system.

Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system.

Analysis of an Integrated Pest Management Model with Impulsive Diffusion between Two Regions

This paper investigates an integrated pest management model with pulsed diffusion. As we all know, humans have been fighting against pests since they entered the age of farming. When pests are controlled, humans can achieve better harvests. We use the stroboscopic mapping of discrete dynamic system to obtain some important lemmas. Based on the lemmas, firstly, we give the conditions for the global asymptotic stability of the periodic solution of the pest eradication boundary; secondly, the conditions for the permanence of the investigated system are derived; thirdly, numerical simulations are used to verify our obtained theoretical results; finally, increased dispersal was found to have the opposite effect on integrated pest management. We conclude that a combination of impulsive diffusion, spraying pesticides, and releasing natural enemies can play a crucial role in integrated pest management.

Numerical approximation of the stochastic equation driven by the fractional noise

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT-θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests.

Extend Nearly Pseudo Quasi-2-Absorbing submodules(I)

The concept of a 2-Absorbing submodule is considered as an essential feature in the field of module theory and has many generalizations. This articale discusses the concept of the Extend Nearly Pseudo Quasi-2-Absorbing submodules and their relationship to the 2-Absorbing submodule, Quasi-2-Absorbing submodule, Nearly-2-Absorbing submodule, Pseudo-2-Absorbing submodule, and the rest of the other concepts previously studied. The relationship between them has been studied, explaining that the opposite is not true and that under certain conditions the opposite becomes true. This article aims to study this concept and gives the most important propositions, characterizations, remarks, examples, lemmas, and observations related to it. In the end, we will present a very important equivalent of our concept with the rest of the concepts presented previously.

JORDAN RIGHT DERIVATIONS ON SEMIPRIME Γ-RING

In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then . In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then .

Research on Some Problems for Nonlinear Operators in the Z-Z-B Space

In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven.

Finite-Time Output Synchronization for Output-Coupled Reaction-Diffusion Neural Networks With Directed Topology

For output-coupled reaction-diffusion neural networks (RDNNs), the finite-time output synchronization (FTOS) issue is researched under directed topology in this paper. By virtue of the Lyapunov approach, inequality skills and important lemmas, the feedback control protocol is designed to obtain the FTOS criterion for coupled RDNNs with output coupling. Furthermore, to ensure that the output-coupled RDNNs realize output synchronization in finite time, a novel adaptive coupling weights control strategy and output feedback controller are proposed. Finally, the relevant simulation results are presented to demonstrate the correctness of our obtained theorems respectively.

The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I)

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α∈(1,2]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order Oτ2+h4 for α∈(1,1.5), where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme.

Dynamical analysis of a stochastic three-species predator\u2013prey system with distributed delays

A stochastic two-prey one-predator system with distributed delays is proposed in this paper. Firstly, applying the linear chain technique, we transform the predator–prey system with distributed delays to an equivalent system with no delays. Then, by use of the comparison method and the inequality technique, we investigate the stability in mean and extinction of species. Further, by constructing some suitable functionals, using M-matrix theory and three important lemmas, we establish sufficient conditions assuring the existence of distribution and the attractivity of solutions. Finally, some numerical simulations are given to illustrate the main results.

Distributed Adaptive Consensus of Parabolic PDE Agents on Switching Graphs With Relative Output Information

This article mainly solves the consensus issue of parabolic partial differential equation (PDE) agents with switching topology by output feedback. A novel edge-based adaptive control protocol is designed to reach consensus under the condition that the switching graphs are always connected at any switching instants. Different from the existing adaptive protocol associated with partial differential dynamics, the proposed adaptive observer-type law relies on the relative output information rather than relative state information. A proper Lyapunov functional is constructed and some important lemmas are used, then a sufficient condition is obtained for the consensus of parabolic PDE agents on switching graphs. Besides, a corollary about the distributed adaptive consensus of parabolic PDE agents on fixed undirected communication networks is given. Finally, the theoretical results are demonstrated by two numerical simulations.

Mittag–Leffler stability and finite-time control for a fractional-order hydraulic turbine governing system with mechanical time delay: An linear matrix inequalitie approach

This article mainly studies the Mittag–Leffler stability and finite-time control of a time-delay fractional-order hydraulic turbine governing system. First, properties of the Riemann–Liouville derivative and some important lemmas are introduced. Second, considering the mechanical time delay of the main servomotor, the mathematical model of a fractional-order hydraulic turbine governing system with mechanical time delay is presented. Then, based on Mittag–Leffler stability theorem, a suitable sliding surface and finite-time controller are designed for the hydraulic turbine governing system. The system stability is confirmed, and the stability condition is given in the form of linear matrix inequalities. Finally, the traditional proportional–integral–derivative control method and an existing sliding mode control method are selected to verify the effectiveness and robustness of the proposed method. This study also provides a new approach for the stability analysis of the time-delay fractional-order hydraulic turbine governing system.

A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives.

The first symptomatic infected individuals of coronavirus (Covid‐19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid‐19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor‐Corrector scheme. For the proposed model of Covid‐19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non‐integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's‐Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non‐linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non‐linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non‐integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives

The first reported case of coronavirus disease (COVID-19) in Brazil was confirmed on 25 February 2020 and then the number of symptomatic cases produced day by day. In this manuscript, we studied the epidemic peaks of the novel coronavirus (COVID-19) in Brazil by the successful application of Predictor–Corrector (P-C) scheme. For the proposed model of COVID-19, the numerical solutions are performed by a model framework of the recent generalized Caputo type non-classical derivative. Existence of unique solution of the given non-linear problem is presented in terms of theorems. A new analysis of epidemic peaks in Brazil with the help of parameter values cited from a real data is effectuated. Graphical simulations show the obtained results to classify the importance of the classes of projected model. We observed that the proposed fractional technique is smoothly work in the coding and very easy to implement for the model of non-linear equations. By this study we tried to exemplify the roll of newly proposed fractional derivatives in mathematical epidemiology. The main purpose of this paper is to predict the epidemic peak of COVID-19 in Brazil at different transmission rates. We have also attempted to give the stability analysis of the proposed numerical technique by the reminder of some important lemmas. At last we concluded that when the infection rate increases then the nature of the diseases changes by becoming more deathly to the population.

Three-Dimensional Fourth-Order Time-Fractional Parabolic Partial Differential Equations and Their Analytical Solution

In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. The fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposed method are proved. The novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution when α = 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs. The obtained results revealed that the method is capable and simple to solve fractional partial differential equations. The software used for the calculations in this study is Mathematica 7.

Mixed finite element method for the nonlinear time-fractional stochastic fourth-order reaction–diffusion equation

The nonlinear time-fractional stochastic fourth-order reaction–diffusion equation perturbed by noises is considered by the mixed finite element method in this paper. Based on the mixed finite element in spatial direction and the generalized BDF2−θ in temporal discretization, the semi- and fully-discrete schemes are obtained. Further, the error estimates for the semi- and fully-discretizations and the regularity of solution are derived by some important lemmas. The numerical experiment is conducted to further confirm the theoretical results.

On the L∞ convergence of a conservative Fourier pseudo‐spectral method for the space fractional nonlinear Schrödinger equation

AbstractIn this paper, we take the space fractional nonlinear Schrödinger as an example to establish the L∞ convergence error analysis for the conservative Fourier pseudo‐spectral method, which has not been studied. We introduce a new fractional Sobolev norm to construct the discrete fractional Sobolev space, and also prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo‐spectral method is unconditionally convergent with order O(τ2 + Nα/2 − r) in the discrete L∞ norm, where τ is the time step and N is the number of collocation points used in the spectral method. Numerical examples are presented to verify the theoretical analysis.

Extension of fixed point results in intuitionistic fuzzy b metric space

The aim of this paper is to define the notion of intuitionistic fuzzy b metric space (in short, IFbMS) along with some useful results. We establish some important Lemmas in order to study the Cauchy sequence in IFbMS. To further develop the work, we establish some fixed point theorems and study the existence of unique fixed point of some self mappings in IFbMS. We also develop the concept of Ćirić quasi-Contraction theorem in IFbMS. Examples are provided to validate the non-triviality of the results.

A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R3

The purpose of this paper is to treat a nonlinear transmission-type problem for a generalized version of the Darcy- Forchheimer - Brinkman system and the classical Brinkman system in complementary Lipschitz domains in R^3 . First of all, we define the required spaces in which we seek our solution. Next, we describe the generalized Brinkman and the generalized Darcy- Forchheimer - Brinkman systems. Further, we give important lemmas that allow us to introduce the trace and conormal derivative operators that appear in the formulation of our transmission problem. We invoke a result regarding the well- posedness of the (linear) transmission problem for the generalized and classical Brinkman systems in complementary Lipschitz domains in R^3 . The above mentioned well- posedness result in the linear case combined with Banach's fixed point theorem will allow us to establish the main result of the paper, the well- posedness of the transmission problem for the Brinkman system and the nonlinear generalized Darcy- Forchheimer - Brinkman system in Lipschitz domains in R^3 .