Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with p-Laplacian Operator

This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Study on Sex Ratio of Lampreys Based on Simulated Ecosystem-Food Web Model

Lampreys, as an important participant in the ecosystem, play an irreplaceable role in the stability of nature. A variety of models were used to simulate ecosystems and food webs, and the dynamic evolution of multiple populations was solved. The temporal changes of the biomass and the health of the ecosystem affected by the population of Lampreys in other ecological niches were solved. For problem 1, Firstly, a simple natural ecosystem is simulated based on the threshold model and BP neural network model. The dynamic change of the sex ratio of lampreys population and the fluctuation of ecosystem health value were found to generate time series maps. Lampreys overprey on low-niche animals, which damages the overall stability of the ecosystem. For problem 2, We used the Lotka-Volterra model to construct ecological competition between lampreys and primary consumers and predators. Then, the Lotka-Volterra equations were solved, and a control group without gender shift function was set up, which reflected the advantages and disadvantages of the sex-regulated characteristics of lampreys in the natural environment. For problem 3, The ecosystem model established in question 1 was further deepened, and the food web was simulated by the Beverton-Holt model and the Logistic time-dependent differential equations model. The parameters of the food web model were input into the neurons of the ecosystem model, and the two models were integrated to form an overall biosphere model. The output layer of the ecosystem neural network was input into the food web Beverton-Holt and Logistic differential equations, and finally, the three-dimensional analytical solution was obtained by numerical simulation. Then Euler method is used to obtain the exact value of the solution surface. The Random forest model was used to predict the future development of lampreys and other ecological niches. For problem 4, By investigating relevant literature, we normalized the populations of lampreys and a variety of fish as well as other ecological niche animals, plants and microorganisms in the same water area, set different impact weights of lampreys, constructed weight evaluation matrix, and obtained positive and negative ideal solution vectors and negative correlation proximity by using TOPSIS comprehensive evaluation method. It is concluded that many kinds of fish are greatly affected by the sex regulation of lampreys.

Oscillation of Systems of Parabolic Differential Equations with Deviating Arguments

To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.

EXPRESSION OF MEROMORPHIC SOLUTIONS OF SYSTEMS OF ALGEBRAIC DIFFERENTIAL EQUATIONS WITH EXPONENTIAL COEFFICIENTS

Using the Nevanlinna theory of the value distribution of meromorphic functions and theory of differential algebra, we investigate the problem of the forms of meromorphic solutions of some specific systems of generalized higher order algebraic differential equations with exponential coefficients and obtain some results.

EXISTENCE, UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions. Finally, we give examples to illustrate its applications.

OSCILLATION OF SOLUTIONS OF THE SYSTEMS OF QUASILINEAR HYPERBOLIC EQUATION UNDER NONLINEAR BOUNDARY CONDITION

Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.

On the Order of the Solutions of Systems of Complex Algebraic Differential Equations

This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.

Asymptotic Analysis of Linear and Interval Linear Fractional-Order Neutral Delay Differential Systems Described by the Caputo-Fabrizio Derivative

Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.

ON THE STABILITY OF DIFFERENTIAL SYSTEMS WITH TIME LAG

In this paper the inequality of Lemma 1 of [1] is extended. By means of our inequality and the method of Lyapunov function we study the stability of two kinds of large scale differential systems with time lag and the stability of a higher-order differential equation with time lag. The sufficient conditions for the stability (S. ), the asymptotic stability (A. S. ), the uniformly asymptotic stability (U. A. S. ) and the exponential asymptotic stability (E. A. S. ) of the zero solutions of the systems are obtained respectively.

Discretized Multisplitting AOR Waveform Relaxation Algorithms for Initial Value Problem of Systems of ODEs

The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived if the subsystems are solving exactly.Firstly,in theis paper,we deal with an extension of the waveform relaxation algorithm by us ing multisplittin AOR method based on an overlapping block decomposition. We restricted our selves to equidistant timepoints and dealed with the case that an implicit integration method was used to solve the subsystems numerically in parallel. Then we have proved convergence of multi splitting AOR waveform relaxation algorithm on a fixed window containing a finite number of timepoints.

Periodic Solutions of Singular Neutral Differential Systems with Infinite Delay

In this paper,we discuss the periodic solutions of the nonlinear singular neutral differential systems with infinite delay.By using matrix measure and Krasnoselskii＇s fixed point theorem,we obtained the suffcient conditions of the existence of periodic solutions.

THE ESTIMATION OF SOLUTION OF THE BOUNDARY VALUE PROBLEM OF THE SYSTEMS FOR QUASI-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.

Strong Stability Preserving IMEX Methods for Partitioned Systems of Differential Equations

We investigate strong stability preserving(SSP)implicit-explicit(IMEX)methods for partitioned systems of differential equations with stiff and nonstiff subsystems.Conditions for order p and stage order q=p are derived,and characterization of SSP IMEX methods is provided following the recent work by Spijker.Stability properties of these methods with respect to the decoupled linear system with a complex parameter,and a coupled linear system with real parameters are also investigated.Examples of methods up to the order p=4 and stage order q—p are provided.Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration,and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.

Periodic Solutions for Some Second-order Differential System with p（t）-Laplacian

In this article, we investigate the existence of periodic solutions for a class of nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained by using critical point theory, which extend some known results.

Bifurcation Behavior near T-B Point in Delay Differential Systems

In this paper, we investigate the description of T-B singularity and the bifurcation behavior near T-B point for delay differential systems with two parameters.

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

ON THE INITIAL VALUE PROBLEMS OF FIRST ORDERIMPULSIVE DIFFERENTIAL SYSTEMS

The purpose of this paper is to study the existence and iterative approximation of minimax quasi-solutions for a class of initial value problems of first order impulsive differential systems by using monotone iterative methods.

BOUNDARY VALUE PROBLEMS FOR A CLASS OF QUASILINEAR DIFFERENTIAL SYSTEMS WITH SINGULAR NONLINEARITIES

In this paper the existence and multiplicities of positive solutions for a class of quasiliear differential systems with singular nonlinearities via Leray Schauder degree theory are established.

Intelligent computing networks for nonlinear influenza-A epidemic model

The differential equations having delays take paramount interest in the research community due to their fundamental role to interpret and analyze the mathematical models arising in biological studies.This study deals with the exploitation of knack of artificial intelligence-based computing paradigm for numerical treatment of the functional delay differential systems that portray the dynamics of the nonlinear influenza-A epidemic model(IA-EM)by implementation of neural network backpropagation with Levenberg-Marquardt scheme(NNBLMS).The nonlinear IA-EM represented four classes of the population dynamics including susceptible,exposed,infectious and recovered individuals.The referenced datasets for NNBLMS are assembled by employing the Adams method for sufficient large number of scenarios of nonlinear IA-EM through the variation in the infection,turnover,disease associated death and recovery rates.The arbitrary selection of training,testing as well as validation samples of dataset are utilizing by designed NNBLMS to calculate the approximate numerical solutions of the nonlinear IA-EM develop a good agreement with the reference results.The proficiency,reliability and accuracy of the designed NNBLMS are further substantiated via exhaustive simulations-based outcomes in terms of mean square error,regression index and error histogram studies.