• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.287-300
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• 2003

In this paper we shall consider the nonlinear delay differential equation (equation omitted) where m is a positive integer, ${\beta}$(t) and $\delta$(t) are positive periodic functions of period $\omega$. In the nondelay case we shall show that (*) has a unique positive periodic solution (equation omitted), and show that (equation omitted) is a global attractor all other positive solutions. In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about (equation omitted), and establish sufficient conditions for the global attractivity of (equation omitted). Our results extend and improve the well known results in the autonomous case.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권3호
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• pp.189-200
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• 2010

A numerical method is proposed and analyzed to approximate a mathematical model of age-dependent population dynamics with spatial diffusion. The model takes a form of nonlinear and nonlocal system of integro-differential equations. A finite difference method along the characteristic age-time direction is considered and primal mixed finite elements are used in the spatial variable. A priori error estimates are derived for the relevant variables.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.411-423
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• 2010

In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1273-1287
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• 2008

In this paper, the numerical integration method for general singularly perturbed two point boundary value problems with mixed boundary conditions of both left and right end boundary layer is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.743-753
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• 2001

We provide local convergence results in affine form for inexact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first and mth Frechet-derivative (m≥2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Frechet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.

• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.989-1004
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• 2023

In this paper, we are motivated to evaluate and investigate the necessary conditions for the fractional Volterra Fredholm integro-differential equation involving the ς-Hilfer fractional derivative. The given problem is converted into an equivalent fixed point problem by introducing an operator whose fixed points coincide with the solutions to the problem at hand. The existence and uniqueness results for the given problem are derived by applying Krasnoselskii and Banach fixed point theorems respectively. Furthermore, we investigate the convergence of approximated solutions to the same problem using the modified Adomian decomposition method. An example is provided to illustrate our findings.

• Kyungpook Mathematical Journal
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• 제28권2호
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• pp.185-196
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• 1988

We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

• Steel and Composite Structures
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• 제50권2호
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• pp.201-216
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• 2024

This paper is motivated by the lack of studies relating to vibration and nonlinear resonance of fluid-conveying cantilever porous GPLR pipes with fractional viscoelastic model resting on nonlinear foundations. A dynamical model of cantilever porous Graphene Platelet Reinforced (GPLR) pipes conveying fluid and resting on nonlinear foundation is proposed, and the vibration, natural frequencies and primary resonant of such system are explored. The pipe body is considered to be composed of GPLR viscoelastic polymeric pipe with porosity in which Halpin-Tsai scheme in conjunction with fractional viscoelastic model is used to govern the construction relation of the nanocomposite pipe. Three different porosity distributions through the pipe thickness are introduced. The harmonic concentrated force is also applied on pipe and excitation frequency is close to the first natural frequency. The governing equation for transverse motion of the pipe is derived by the Hamilton principle and then discretized by the Galerkin procedure. In order to obtain the frequency-response equation, the differential equation is solved with the assumption of small displacement, damping coefficient, and excitation amplitude by the multiple scale method. A parametric sensitivity analysis is carried out to reveal the influence of different parameters, such as nanocomposite pipe properties, fluid velocity and nonlinear viscoelastic foundation coefficients, on the primary resonance and linear natural frequency. Results indicate that the GPLs weight fraction porosity coefficient, fractional derivative order and the retardation time have substantial influences on the dynamic response of the system.

• International Journal of Control, Automation, and Systems
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• 제2권1호
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• pp.55-67
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• 2004

In this paper, an active vibration control of a translating steel strip in a zinc galvanizing line is investigated. The control objectives in the galvanizing line are to improve the uniformity of the zinc deposit on the strip surfaces and to reduce the zinc consumption. The translating steel strip is modeled as a moving belt equation by using Hamilton’s principle for systems with moving mass. The total mechanical energy of the strip is considered to be a Lyapunov function candidate. A nonlinear boundary control law that assures the exponential stability of the closed loop system is derived. The existence of a closed-loop solution is shown by proving that the closed-loop dynamics is dissipative. Simulation results are provided.

• Steel and Composite Structures
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• 제28권6호
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• pp.671-679
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• 2018

The paper is devoted to the stability analysis of a simply supported five layer sandwich beam. The beam consists of five layers: two metal faces, the metal foam core and two binding layers between faces and the core. The main goal is to elaborate a mathematical and numerical model of this beam. The beam is subjected to an axial compression. The nonlinear hypothesis of deformation of the cross section of the beam is formulated. Based on the Hamilton's principle the system of four stability equations is obtained. This system is approximately solved. Applying the Bubnov-Galerkin's method gives an ordinary differential equation of motion. The equation is then numerically processed. The equilibrium paths for a static and dynamic load are derived and the influence of the binding layers is considered. The main goal of the paper is an analytical description including the influence of binding layers on stability, especially on critical load, static and dynamic paths. Analytical solutions, in particular mathematical model are verified numerically and the results are compared with those obtained in experiments.