Dierential Equations Research Articles

Simplification of a System of Geodesic Equations by Reference to Conservation Laws

This paper is purposed to exploit prevalent premises for determining analytical solutions to di erential equations formulated from the calculus of variations. We realize this premises from the statement of Emmy Noether’s theorem; that every system in which a conservation law is observed also admits a symmetry of invariance (Olver, 1993, pp.242; Dresner, 1999, pp.60-62 ). As an illustration, the infinitesimal symmetries for Ordinary Di erential Equations (O.D.E’s) of geodesics of the glome are explicitly computed and engaged following identification of a relevant conservation law in action. Further prospects for analysis of this concept over the same manifold are then presented summarily in conclusion.

Motion Analysis of Multibody Systems Subject to Friction Based on the Null Space Matrix Method of Dierential Equation Type

Motion Analysis of Multibody Systems Subject to Friction Based on the Null Space Matrix Method of Dierential Equation Type

Conjunto de alcanzabilidad de un sistema mecánico controlable y condiciones de estabilidad robusta

<p>En este trabajo se determina de forma numérica la frontera del conjunto de alcanzabilidad de una ecuación diferencial de segundo orden con una perturbación externa, empleando la solución del problema de la variación máxima de las amplitudes de oscilación de sus soluciones. El método consiste en determinar en un conjunto de funciones dado, la perturbación externa que provoque amplitudes de oscilación máxima en las soluciones de la ecuación diferencial que describe, como caso particular, la dinámica de un sistema mecánico controlable con impactos. Con ayuda de esta perturbación, se determina la existencia de una trayectoria cerrada que describe la frontera del conjunto de alcanzabilidad, lo cual permite establecer condiciones suficientes sobre la estabilidad robusta de las soluciones de la ecuación diferencial. Los resultados se ilustran de forma numérica en casos particulares.</p>

Existence of Nonoscillation Solutions of Second-order Neutral Dierential Equations

This paper is concerned with existence of nonoscillation solution for a family of second-order neutral differential equations with positive and negative coefficients. A sufficient conditions for existence of nonoscillation solution is obtained by contraction fixed point theorem, special case of the equation has also been studied.

QUANTUM FIELD THEORY FROM FIRST PRINCIPLES

The mathematical description of quantum systems univocally identies their nature. In other words we treat a system as quantum if we describe its behaviour adopting Hilbert spaces and structures thereof, as prescribed by the postulates of quantum theory. The choice of using quantum systems as the elementary systems of physics can be justied in terms of informational principles, thanks to results of the last decade. Such results come as the conclusion of a research program that lasted almost one century, with the aim of reformulating quantum theory in terms of operational principles. This achievement now poses a new challenge, that we face here. If the systems of quantum theory are thought of as elementary information carriers in the rst place, rather than elementary constituents of matter, and their connections are logical connections within a given algorithm, rather than space-time relations, then we need to nd the origin of mechanical concepts—that characterise quantum mechanics as a theory of physical systems. To this end,we will illustrate howphysical laws can be viewed as algorithms for the update of memory registers that make a physical system. Imposing the characteristic properties of physical laws to such an algorithm, i.e. homogeneity, reversibility and isotropy, we will show that the physical laws thus selected are particular algorithms known as cellular automata. Further assumptions regarding maximal simplicity of the algorithm lead to two cellular automata only, that in a suitable regime can be described by Weyl’s dierential equations, lying at the basis of the dynamics of relativistic quantum elds. We will nally discuss how the same cellular automaton can give rise to both Fermionic eld dynamics and to Maxwell’s equations, that rule the dynamics of the electromagnetic eld. We will conclude reviewing the discussion of the relativity principle, that must be suitably adapted to the scenario where space-time is not an elementary notion, through the denition of a change of inertial reference frame, and whose formulation leads to the recovery of the symmetry of Minkowski space-time, identied with Poincar´e’s group. Space-time thus emerges as one of the manifestations of physical laws, rather than the background where they occur, and its features are determined by the dynamics of systems, necessarily equipped with dierential equations that express it. In brief, there is no space-time unless an evolution rule requires it.

Conformable derivative applied to experimental Newton's law of cooling

It has been proved that the integer order dierential equation does notrepresent the real behaviour of nature for the Newton's law of cooling.Then, we solve the Newton's cooling law using the conformable deriva-tive, as result we obtain the Kohlrausch stretched exponential function.Due to the free parameter 0 < 1, we can t this function with thegraph of the experimental data set. It is shown that the experimental datacoincide with those theoretical when = 0:77269 and k = 0:018765.

A Clas of A-Stable Runge-Kutta Collocation Methods for the Solution of First Order Ordinary Dierential Equations

This paper presented a class of A-stable Runge-Kutta collocation methods with three free parameters for the solution of rst order ordinary dierential equations. Power series was considered as its basis function, adoption of interpolation and collocation of the approximate solution at some selected grid points to give system of equations was also considered. Gaussian Elimination method was used to solve for the unknown parameters and substituted into the approximate solution to give the continuous method. The three cases considered are the Guass, the Lobatto, and the Radau types. Analysis of the methods was made based on order, zero stability, consistence and convergence. The derived schemes were implemented in the Predictor-Corrector mode. Comparison with existing methods showed that the new developed Schemes compete favorably.

AVERAGING IN MULTIFREQUENCY SYSTEMS WITH DELAY AND LOCAL INTEGRAL CONDITIONS

Multifrequency systems of dierential equations were studied with the help of averaging method in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenko and many other scientists. The complexity of the study of such systems is their inherent resonant phenomena, which consist in the rational complete or almost complete commensurability of frequencies. As a result, the solution of the system of equations averaged over fast variables in the general case may deviate from the solution of the exact problem by the quantity O (1). The approach to the study of such systems, which was based on the estimation of the corresponding oscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works by A.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systems with initial , boundary and integral conditions. For multifrequency systems with an argument delay, the averaging method is substantiated in the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors. In this paper, the averaging method is used to study the solvability of a multifrequency system with an arbitrary nite number of linearly transformed arguments in slow and fast variables and integral conditions for slow and fast variables on parts of the interval [0, L] of the system of equations. An unimproved estimate of the error of the averaging method under the superimposed conditions is obtained, which clearly depends on the small parameter and the number of linearly transformed arguments in fast variables.

A Method to Deal With the Critical Case in Stochastic Population Dynamics

In numerous papers, the behaviour of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the process is persistent in the long run, while if it is negative, the process converges to extinction. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process con-gerves in temporal average to extinction. A number of applications are given, for Stochastic Dierential Equations and Piecewise Deterministic Markov Processes modelling prey-predator, epidemilogical or structured population dynamics.

Generalized Quasilinearization Using Coupled Lower and Upper Solutions for Periodic Boundary Value Problem of an Integro Differential Equation

In this paper we rst develop the method of generalized quasilinearization for initialvalue problem of an integro dierential equation and then use it to develop quasilinearization for the periodic boundary value problem of the integro dierential equation by using the coupled lower and upper solutions of Type-I.

Azela-Ascoli Theorem and Its Applications

In this study we looked at the theorm Azela-Ascoli and its application to functional analysis, ordinary dierential equations and complex analysis. This theorem simplies the checking of compactness for subsets of spaces of continuous functions in much the same way the Heine-Borel theorem does for subsets of Rn

Analytical Study of a System of Dierence Equation

We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. The considered system is given by the following rational difference equations:
 
 where the initial conditions x-2; x-1; x0; y-2; y-1; y0 are arbitrary positive real numbers. Also, we give specic form of the solutions of some special cases of this equation. Some numerical examples are given to verify our theoretical results.

Perturbation in Nonlinear Operator Dynamical Systems

An introduction to nonlinear operator dynamical systems with a brief technical notation will be presented. The operator is designed to include all nonanticipating (causal) delay and functional dierential equations. Condi- tions for the existence and uniqueness of these noanticipating operator dierential equations (NODE) will be investigated. At last the perturbed solutions of these type nonlinear systems and some properties will be presented.

Initial Value Problem for Stochastic Hyprid Hadamard Fractional Differential Equation

In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by 
 
 where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used to obtain the required results.

INEXTENSIBLE FLOWS OF CURVES ACCORDING TO SABBAN FRAME IN GALILEAN SPACE G3

In this paper inextensible ows of curves in 3-dimensional Galilean space is researched. Firstly Sabban frame is dened in 3-dimensional Galilean space, then necessary and sucient conditions for inextensible ows of curves with Sabban frame in 3-dimensional Galilean space are given. Also inextensible curve ow are expressed as a partial dierential equation involving geodesic curvature according to this frame.

A numerical simulation of MHD ow and radiation heat transfer of nano uids through a porous medium with variable surface heat ux and chemical reaction

In this paper, the problem of MHD ow and radiation heat transfer of nanouids against a at plate in porous medium with the eects of variable surface heat ux and rst-order chemicalreaction is investigated numerically. Three dierent types of nanoparticles, namely Cu, Al2O3 andAg are considered by using water as a base uid with Prandtl number Pr = 6:2. The governingpartial dierential equations can be written as a system of nonlinear ordinary dierential equationsover a semi-innite interval using a similarity transformation. A new eective collocation methodis proposed based on exponential Bernstein functions to simulate the solution of the resultingdierential systems. The advantage of this method is that it does not require truncating ortransforming the semi-innite domain of the problem to a nite domain. In addition, this methodreduces the solution of the problem to the solution of a system of algebraic equations. Graphicaland tabular results are presented to investigate the inuence of the solid volume fraction, types ofnanoparticles, radiation and suction/blowing, magnetic eld, permeability, Schmidt number andchemical reaction, on velocity, temperature and concentration proles. The obtained results ofthe current study are in excellent agreement with previous works.

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The main asymptotic property of the continuous stochastic approximation procedure, namely the convergence to the Ornstein-Uhlenbeck process, is considered. The regression function of the procedure depends on a uniformly ergonomic Markov process, which describes the external inuence in the form of switches. To obtain sucient conditions for the convergence of the procedure, we use ergodic properties of the generating Markov process generator, the existence of a stationary distribution of this process, and the existence of the potential for a generating Markov process generator. The stochastic approximation procedure, as a random process, is constructed in the form of a dierential equation in the xed state of a Markov process with a corresponding generator of the equation. Asymptoticity over time is achieved by using a small parameter that normalizes time. This made it possible to obtain a normalized stochastic approximation procedure and its dierential representation. The generator of the obtained dierential equation is used to construct a generator of a two-component Markov process, which consists of a procedure and a switching process. The singular representation of the last generator by a small parameter makes it possible to solve the singular perturbation problem and determine the form of the limited generator. Such form denes the representation of a limited process as the Ornstein-Uhlenbeck diusion process. Note that the convergence to the limited process is weak, which follows from the Koroliuk's theorem. The conditions for the existence of the Lyapunov function for the dynamic procedure, which is averaged by the stationary distribution of the Markov process, are important. Additional conditions on the Lyapunov function make it possible to establish the boundedness of the residual terms of the solution of a singular perturbation. The one-dimensional case of the procedure can be expanded to multidimensional with the corresponding complication of calculations of the components of the limited generator. The work summarizes the studies of Nevelson and Khasminsky in the case of a direct inuence of the Markov process on the regression function.

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Many physical processes are described by the Helmholtz equation. By adding the Dirichlet, Neumann, or Robin boundary conditions, the corresponding problems are obtained. An eective method for solving such boundary problems is the nite dierence method. In this paper, we consider a Dirichlet problem for the Helmholtz equation. A nite dierence approximate scheme of the higher-order was constructed. For rst- and second-order derivatives the fourth-order nite dierence approximations are used. By adding boundary conditions at the grid nodes, we obtain a system of dierence equations (a system of linear algebraic equations) with a symmetric matrix that has a diagonal advantage. Therefore, it is advisable to solve this system by iterative methods. In the paper, the simple iterations method and Seidel method are used. They give an approximate solution with a given accuracy by a small number of iterations. A test problem with a constant wave number and a known exact solution is considered. The dierence equation and the iterative formulas of the simple iterations method and Seidel method for this problem are given. The results of numerical experiments, which conrm the eciency of the method and the theoretical order of convergence, are presented. The calculations are performed under dierent values of the wave number. Iterative processes are compared by the number of iterations and the values of absolute errors.

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In the paper we consider the following Cauchy problem for the systems of the ordinary dierential equations { ψ′′(t) + K(t,ψ(t),ψ′(t)) = S(t), t ∈ (0,T), ψ(0) = ψ0, ψ′(0) = ψ1, (∗) where K and S are some vector-valued functions, ψ0,ψ1 ∈ Rd are some xed vectors. Such systems appear in various classical mechanics problems because they follow from Newton's second law of the dynamics. If we use the Faedo-Galerkin method for the seeking of the weak solution to the initial boundary value problem for the linear or nonlinear hyperbolic equations, then we also use the problems of type (*) for the construction of the approximation functions. In practice, the Cauchy problems of type (*) often are reduced to the following Cauchy problem of the higher dimension: ψ′(t) = θ(t), t ∈ (0,T), θ′(t) = S(t)−K(t,ψ(t),θ(t)), t ∈ (0,T), ψ(0) = ψ0, θ(0) = ψ1. (∗∗) Then the standard Peano/Caratheodory types existence theorems are used. But these are local theorems and they should be complemented by some extension theorems. On the other hand we can seek the global solutions of problems (*) or (**) directly. In the present paper we and the conditions of the global solvability of the Cauchy problem (*). Corresponding results for problem (**) we got in our previous paper. These conditions contain the Caratheodory existence conditions and the Lasalle extension conditions for (*). Finally, we use these facts to proving the existence theorem for the initial boundary value problem for some nonlinear hyperbolic equations with the variable exponent of the nonlinearity.

Nonlinear Partial Dierential Equations Model Related to Oxidation Pond Treatment System: A Case Study of mPHO at Taman Timor Oxidation Pond, Johor Bahru

This study presents a mathematical model examining wastewater pollutant removalthrough an oxidation pond treatment system. This model was developed to describethe reaction between microbe-based product mPHO (comprising Phototrophic bac-teria (PSB)), dissolved oxygen (DO) and pollutant namely chemical oxygen demand(COD). It consists of coupled advection-diusion-reaction equations for the microor-ganism (PSB), DO and pollutant (COD) concentrations, respectively. The couplingof these equations occurred due to the reactions between PSB, DO and COD to pro-duce harmless compounds. Since the model is nonlinear partial dierential equations(PDEs), coupled, and dynamic, computational algorithm with a specic numericalmethod, which is implicit Crank-Nicolson method, was employed to simulate the dy-namical behaviour of the system. Furthermore, numerical results revealed that theproposed model demonstrated high accuracy when compared to the experimental data.Keywords Oxidation pond; nonlinear PDEs; PSB; implicit Crank-Nicolson.