System Of Singular Differential Equations Research Articles

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

On the relations between a singular system of differential equations and a system with delays

In this article, we consider a class of systems of differential equations with multiple delays. We define a transform that reformulates the system with delays into a singular linear system of differential equations whose coefficients are non-square constant matrices, and the number of their columns is greater than the number of their rows. By studying only the singular system, we provide a form of solutions for both systems.

Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane Emden equations

In the present paper, we developed the functional matrix of integration via Bernoulli wavelets and generated a competent numerical scheme to solve the nonlinear system of singular differential equations which is Lane Emden form by Bernoulli wavelets collocation technique (BWCT) with different physical conditions. The system of nonlinear singular models is not smooth to operate as they are singular and nonlinear. This approach obtains the solution for this system by transforming it into a-nonlinear system of algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. These unknown coefficients are calculated using the collocation scheme. The consistency and proficiency of the developed approach are demonstrated via graphs and tables. Attained results confirm that the newly implemented approach is more effective and accurate than other techniques which are available in the literature. All computations have been made using Mathematica software. The convergence of this method is explained in terms of theorems.

Generalized fractional controller for singular systems of differential equations

In this article we consider a class of singular linear systems of first order, and introduce a generalized fractional order feedback controller of Caputo type. The closed loop system in question is a singular system of differential equations having both first, and fractional order derivatives. We provide a comprehensive theory for the existence and uniqueness of solutions, as well as for the stability of the system with inclusion of the fractional order controller. An example of a singular system with a fractional order proportional integral controller, as well as an example on a 3-bus power system with inclusion of a fractional order damping controller, is given to illustrate our theory.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Динамическая термоустойчивость геометрически нерегулярной пологой цилиндрической оболочки под действием периодической, по временной координате, нагрузки

В рамках модели типа Лява рассматривается геометрически нерегулярная изотропная пологая цилиндрическая оболочка (ГНО). За основу берется строгая континуальная модель «оболочка-ребра». Предполагается, что ГНО нагрета до постоянной температуры $\theta_0$, два противоположных края подвергаются воздействию периодической по временной координате тангенциальной нагрузке, амплитуда и частота которой известны ($p(t)=p_0 \cos \vartheta t$). Задача определения динамической неустойчивости (ДН) термоупругой системы сводится к рассмотрению сингулярной системы трех дифференциальных уравнений динамической термоустойчивости ГНО в перемещениях, содержащих слагаемые с тангенциальными усилиями в форме Брайена. Эти усилия, возникающие в оболочке при ее нагреве, предварительно определяются на основе замкнутых решений сингулярной системы дифференциальных уравнений безмоментной термоупругости ГНО. Конкретизированная исходная система уравнений преобразуется к уравнениям Матье, которые записаны в терминах классической атермической теории гладких пластин, содержащих поправки на геометрические параметры - кривизну, относительную высоту подкрепляющих элементов, их число и температуру. Определяются первые три области ДН ГНО. Проводится количественный анализ влияния геометрических параметров упругой системы и температуры на конфигурацию областей ДН и предельного значения коэффициента возбуждения.

Successive iteration technique for singular nonlinear system with four-point boundary conditions

In this paper, the existence of at least one positive solution of the system of singular differential equations with four-point coupled boundary conditions is discussed. A constructive monotonic iterative technique on the equivalent completely continuous nonlinear operator is used to establish the result. This method produces an approximate solution in the form of series which is very helpful in developing a numerical scheme for the positive solution of the system. It is demonstrated through the examples.

A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [0,a] with a singularity of the first kind in the point x=0 is proposed. Using the substitution of the independent variable x=et, we reduce the original initial value problem to the one on the interval (−∞,ln⁡a]. On some finite irregular grid {tn∈(−∞,ln⁡a], n=0,1,...,N,tN=ln⁡a} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t0, new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown.

Participation Factors for Singular Systems of Differential Equations

In this article, we provide a method to measure the participation of system eigenvalues in system states, and vice versa, for a class of singular linear systems of differential equations. This method deals with eigenvalue multiplicities and covers all cases by taking into account both the algebraic and geometric multiplicity of the eigenvalues of the system matrix pencil. A Mobius transform is applied to determine the relative contributions associated with the infinite eigenvalue that appears because of the singularity of the system. The paper is a generalization of the conventional participation analysis, which provides a measure for the coupling between the states and the eigenvalues of systems of ordinary differential equations with distinct eigenvalues. Numerical examples are given including a classical DC circuit and a 2-bus power system dynamic model.

The Möbius transform effect in singular systems of differential equations

The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of differential equations and the solutions of its proper M-systems, i.e., systems that appear after applying the generalized Möbius transform to the pencil of the original singular system (prime system). Firstly, we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of its proper M-systems. Moreover these solutions can be explicitly represented without resorting to any further processes of computations. Finally, numerical examples are given to illustrate our theory.

Study of Existence of Almost Automorphic Solutions to Some Singular Systems of Differential Equations

Study of Existence of Almost Automorphic Solutions to Some Singular Systems of Differential Equations

A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications

The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if A^πAB=0 and BC (I–A^π) =0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by M^d. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABA^π=0 and BA^π=0 respectively. Precisely, for a block matrix M, we give a new representation of M^d under some conditions that AB=0 and DCA^π=0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.

Global smooth solution of a two-dimensional nonlinear singular system of differential equations arising from geostrophics

Consider the Cauchy problems for the following two-dimensional nonlinear singular system of differential equations arising from geostrophics∂∂t[γ(ψ1−ψ2)−△ψ1]+α(−△)ρψ1+β∂ψ1∂x+J(ψ1,γ(ψ1−ψ2)−△ψ1)=0,∂∂t[γδ(ψ2−ψ1)−△ψ2]+α(−△)ρψ2+β∂ψ2∂x+J(ψ2,γδ(ψ2−ψ1)−△ψ2)=0,ψ1(x,y,0)=ψ01(x,y),ψ2(x,y,0)=ψ02(x,y). In this system, α>0, γ>0, δ>0 and ρ>0 are positive constants, β≠0 is a real nonzero constant, the Jacobian determinant is defined by J(p,q)=∂p∂x∂q∂y−∂p∂y∂q∂x.The existence and uniqueness of the global smooth solution of the system of differential equations are very important in applied mathematics and geostrophics, but they have been open for a long time. The singularity generated by the linear parts, the strong couplings of the nonlinear functions and the fractional order of the derivatives make the existence and uniqueness very difficult to study.Very recently, we found that there exist a few special structures in the system. The main purpose of this paper is to couple together the special structures and an unusual method for establishing the uniform energy estimates to overcome the main difficulty to accomplish the existence and uniqueness of the global smooth solution: ψ1∈C∞(R2×R+) and ψ2∈C∞(R2×R+), for all ρ>3/2. The new energy method enables us to make complete use of the special structures of the nonlinear singular system. The results obtained in this paper provide positive solutions to very important open problems and greatly improve many previous results about nonlinear singular systems of differential equations arising from geostrophics.

Essential spectrum of systems of singular differential equations

In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

Systems of singular differential equations with pulse action

Abstract The paper deals with the singular systems of ordinary differential equations with impulsive action under the assumption that the considered systems can be reduced into the central canonical form. An approach which combines the theory of impulsive differential equations and known results from the theory of singular Fredholm boundary value problems is used. Necessary and sufficient conditions for the existence of solutions of the singular boundary value problems with impulsive action are derived. Moreover, an algorithm for the construction of the family of linearly independent solutions is shown. MSC:34A09, 34A37.

Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen–Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λB in which B is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.

On a nonlocal boundary value problem for a system of singular differential equations

This article deals with a system consisting of singular partial differential equations of the first and second order arising in the zero approximation of I. Vekua's hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half-plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress–strain state of the cusped prismatic shell under the action of concentrated forces and moments.

Nonlocal boundary value problem for a system of singular differential equations and applications to cusped prismatic shells

AbstractThe paper deals with a system of singular partial differential equations of the first and second order arising in the zero approximation of I.Vekua's hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half‐plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress‐strain state of the cusped prismatic shell [1,2] under the action of concentrated forces and moments. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point r = 0 r=0 . Because of the singularity at r = 0 r=0 , the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two “shooting” parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.

Generalized quasilinearization for singular system of differential equations

We had earlier employed (A.R. Abd-Ellateef Kamar, The method of generalized quasilinearization for degenerate singular perturbation problem, to appear; A.R. Abd-Ellateef Kamar, Extension of the method of generalized quasilinerization for degenerate singular perturbation problem, to appear) the generalized quasilinearization method for the degenerate singular perturbation problem: y ′=f(t,y,λ),y(0)=y 0, 0=F(t,y,λ) . Here we have employed the generalized quasilinearization method for the following singular system of differential equations: Ay ′= f( t, y), y(0)= y 0, where A is a singular n × n matrix.