Singular Differential Systems Research Articles

Distributed iterative learning consensus tracking for singular partial differential multi-agent systems under fixed and iteration-varying topologies

This paper investigates the consensus tracking problems of singular partial differential multi-agent systems (SPD-MASs) with initial state errors under fixed and iteration-varying network topologies, where only certain follower agents have access the trajectory defined by virtual leaders. To solve this problem, a distributed iterative learning consensus protocol with initial state learning is proposed, ensuring accurate tracking of the leader's trajectory by all follower agents with initial state errors. The convergence conditions for consensus tracking error are derived through mathematical analysis between adjacent agents. Theoretical analysis shows that, under convergence conditions, the consensus error between any two agents can converge to zero along the iterative axis. Finally, numerical simulations show validate the effectiveness of the proposed distributed control protocol with initial state learning compared to traditional protocols without state learning.

On a singular mathematical model for brain lactate kinetics

In this paper, we consider a singular nonlinear differential system that characterizes the intricate dynamics of brain lactate kinetics between cells and capillaries, as described by System (1.1) below. We begin by establishing the existence and uniqueness of nonnegative solutions for our system through the application of Schauder's fixed‐point theorem. Subsequently, we explore the behavior of these solutions as the viscosity term approaches zero, shedding light on the system's dynamic evolution in such scenarios. To provide empirical validation for our theoretical findings, we offer a series of numerical simulations. These simulations not only confirm the results we have obtained but also reinforce prior research, underscoring the model's efficiency in capturing the complexities of the brain's lactate kinetics. Our work contributes not only to the theoretical underpinning of this field but also to its practical implications, making it a valuable resource for both researchers and practitioners seeking to comprehend and manipulate these vital biological processes.

A design of novel Gudermannian neural networks for the nonlinear multi-pantograph delay differential singular model

In this paper, a new stochastic numerical platform through the Gudermannian neural network (GNN) based intelligent computing solver (GNNICS) is accessible for solving the nonlinear singular multi-pantograph delay differential (MP-DD) systems. In GNNICS, Gudermannian kernel is exploited to construct the neural network models of differential operators with different neurons for the nonlinear system along with the hybrid computing via global genetic algorithm (GA) and local refinements based active set (AS), i.e., GNN-GAAS method. A fitness function with GNN models is formulated for solving the MP-DD equation along with the optimization of design variables of the network using GAAS. To investigate the performance of the designed GNNICS based GNN-GAAS algorithm, three different variants of the MP-DD systems are used to assess the correctness, effectiveness, and robustness. The statistical investigations based on different performance are presented to authenticate the consistent accuracy, convergence, and stability of the designed GNN-GAAS algorithm. Furthermore, the negligible absolute error that are performed as 10−06 to 10−08 for solving the GNNICS based on GNN-GAAS algorithm.

Some stability results on non-linear singular differential systems with random impulsive moments

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well.

Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices

AbstractIn this article, we consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.

A Nonlinear Singular Differential System with Perturbed Integral Boundary Conditions

A Nonlinear Singular Differential System with Perturbed Integral Boundary Conditions

Lyapunov stability of singular planarsystems related to dispersion-managedsolitons in optical fiber

In this paper, we consider two singular planar differential systems which can describe the evolution of the optical pulse width and chirp for the so-called dispersion-managed solitons. Based on the method of third order approximation in combination with some quantitative information obtained by the upper–lower solutions method and the averaging method, some results on the existence and Lyapunov stability of the periodic solutions are obtained. Moreover, the formula of the first twist coefficient and a stability criterion of a nonlinear differential equation are also established.

A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model

In this study, the mathematical form of the second order perturbed singular delay differential system is presented. The comprehensive features using the singular-point, perturbed factor and pantograph term are provided together with the shape factor of the second order perturbed singular delay differential system. The novel model is simulated numerically through the artificial neural networks (ANNs) based on the global/local optimization procedures, i.e., genetic algorithm (GA) and sequential quadratic programming (SQP). An activation function is constructed by using the differential model based on the second order perturbed singular delay differential system. The optimization of fitness function is performed through the hybrid computing strength of the ANNs-GA-SQP to solve the second order perturbed singular delay differential system. The exactness, substantiation, and authentication of the novel system is observed to solve three different variants of the novel model. The convergence, robustness, correctness, and stability of the numerical approach is performed using the comparison procedures of the available exact solutions. For the reliability, the statistical performances with necessary processes are provided using the ANNs-GA-SQP.

On numerical methods to second-order singular initial value problems with additive white noise

In this work, we investigate the strong convergence of the Euler–Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler–Maruyama method is of (1/2−ɛ) order convergence in mean-square sense, where ɛ is an arbitrarily small positive number, which is different from the consensus that the Euler–Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergence order in mean-square sense will be increased to 1−ɛ. Our theoretical findings are well verified by numerical examples.

A novel computing stochastic algorithm to solve the nonlinear singular periodic boundary value problems

In this work, a class of singular periodic nonlinear differential systems (SP-NDS) in nuclear physics is numerically treated by using a novel computing approach based on the Gudermannian neural networks (GNNs) optimized by the mutual strength of global and local search abilities of genetic algorithms (GA) and sequential quadratic programming (SQP), i.e. GNNs-GA-SQP. The stimulation of offering this numerical computing work comes from the aim of introducing a consistent framework that has an effective structure of GNNs optimized with the backgrounds of soft computing to tackle such thought-provoking systems. Two different problems based on the SPNDS in nuclear physics will be examined to check the proficiency, robustness and constancy of the GNNs-GA-SQP. The outcomes obtained through GNNs-GA-SQP are compared with the true results to find the worth of designed procedures based on the multiple trials.

The positive solutions to the boundary value problem of a nonlinear singular impulsive differential system

The positive solutions to the boundary value problem of a nonlinear singular impulsive differential system

Existence of Positive Solution For a Fourth-order Differential System

Existence of Positive Solution For a Fourth-order Differential System 
 where µ > 0 is a constant, and the nonlinear terms f, g may be singular with respect to the time and space variables. By fixed point theorem in cones, the existence is established for singular differential system. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Positive Solution for a Singular Fourth-Order Differential System

In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system u(4) = φu + f (t, u, u”, φ), 0 t φ” = μg (t, u, u”), 0 t u (0) = u (1) = u”(0) = u”(1) = 0, φ (0) = φ (1) = 0; where μ > 0 is a constant, and the nonlinear terms f, g may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Positive solutions for singular semipositone nonlinear fractional differential system

In this paper, under suitable conditions we employ the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel?skii fixed point theorem to show the existence of positive solutions for a system of nonlinear singular Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to integral boundary conditions. Some examples are given to illustrate our main results.

Multilayer Perceptron and Chebyshev Polynomials Based Neural Network for Solving Emden–Fowler Type Initial Value Problems

Present study is devoted to solve singular non-linear differential equations and system of singular non-linear differential equations of Emden–Fowler type using the Multilayer perceptron and Chebyshev polynomials based functional link neural network techniques. We are emphasizing on obtaining accurate solutions of such problems with less computation time than the earlier neural network techniques. Comparisons between Multilayer perceptron and Chebyshev polynomials based network in aspects of approximation abilities and computational efficiency are presented with several examples.

Uniqueness of iterative positive solutions for the singular infinite-point p-Laplacian fractional differential system via sequential technique

By sequential techniques and mixed monotone operator, the uniqueness of positive solution for singular p-Laplacian fractional differential system with infinite-point boundary conditions is obtained. Green's function is derived, and some useful properties of Green' function are obtained. Based on these new properties, the existence of unique positive solutions is established, moreover, an iterative sequence and a convergence rate are given, which are important for practical application, and an example is given to demonstrate the validity of our main results.

Fractional Singular Differential Systems of Lane–Emden Type: Existence and Uniqueness of Solutions

A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.

Practical Stability and Integral Stability for Singular Differential Systems with Maxima

In this paper, we introduce various definitions of practical stability and integral stability for nonlinear singular differential systems with maxima and give criteria of stability for such systems via the Lyapunov method and comparison principle.

Control of bounded solutions for first-order singular differential equations with impulses

Abstract This paper is concerned with a kind of first-order singular differential system with impulses. Based on the Schaefer fixed-point theorem, some new verifiable algebraic criteria are given to ensure the controllability of bounded solutions for the considered system. The results obtained in this paper not only achieve the controllability of the singular differential system with impulses for the first time, but also complement the previous researches on singular differential system with impulses. Consequently, the results established are essentially new. Finally, the effectiveness of the obtained results are illustrated via a numerical example.

Formal reduction of singular linear differential systems using eigenrings: A refined approach

This paper provides a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. This allows us to establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple and examples are given in Saade (2018).