Integro-difierential Equation Research Articles

Single-term Walsh series method for solving Volterra's population model

Single-term Walsh series are developed to approximate the solution of the Volterra’s population model. Volterra’s model is a nonlinear integro-difierential equation where the integral term represents the efiect of toxin. Properties of Single-term Walsh series are presented and are utilized to reduce the computation of the Volterra’s population model to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. A comparison is made with existing results.

Exact solutions for some of the fractional integro-differential equations with the nonlocal boundary conditions by using the modification of He’s variational iteration method

AbstractIn this paper, the modiflcation of He’s variational iteration method (MVIM) is developed to solve fractional integro-difierential equations with nonlocal boundary conditions. It is shown that by choosing suitable initial approximation,the exact solution obtains by one iteration. It is illustrated that the propose method is efiective and has high con-vergency rate. Keywords : Fractional integro-difierential equations, nonlocal boundary condition, modiflcation of He’s variational iteration method. 1 Introduction The variational iteration method was flrst proposed by He [7, 8, 9] and has been worked out over a number of yearsby many authors. This method has been shown to efiectively, easily and accurately solve a large class of nonlinearproblems. Generally, one or two iterations lead to high accurate solutions.This method is, in fact, a modiflcation ofthe general Lagrange multiplier method into an iteration method, which is called correction functional. Applicationsof the method have been enlarged due to its °exibility, convenience and e–ciency. The convergence of the methodis systematically discussed by Tatari and Dehghan [17] , Odibat [14]. There are several modiflcations of He’s VIM[6, 7, 8, 9, 10]. In this paper we propose the reliable modiflcation of He’s VIM (MVIM) that was introdused byGhorbani et al. [4] for solving the fractional order integro-difierential equations with nonlocal boundary conditionsby constructing an initial trial-function without unknown parameters, so that one iteration leads to exact solution.Consider fractional order integro-difierential equation of the form

SLOW SCALE MAXWELL-BLOCH EQUATIONS FOR ACTIVE PHOTONIC CRYSTALS

We present a theory to describe the transient and steady state behaviors of the active modes of a photonic crystal with active constituents (active photonic crystal). Using a couple mode model, we showed that the full vectorial Maxwell-Bloch equations describing the physics of light matter interaction in the active photonic crystal can be written as a system of integro-difierential equations. Using the method of moments and the mean value theorem, we showed that the system of integro-difierential equations can be transformed to a set of difierential equations in slow time and slow spatial scales. The slow time (spatial) scale refers to a duration (distance) that is much longer than the optical time period (lattice constant of the photonic crystal). In the steady state, the slow scale equations reduce to a nonlinear matrix eigenvalue problem, from which the nonlinear Bloch modes can be obtained by an iterative method. For cases, where the coupling between the modes are negligible, we describe the transient behavior as an one-dimensional problem in the spatial coordinate, and the steady behaviors are expressed using simple analytical expressions.

Convergence Analysis of Spectral Methods for Integro-Differential Equations with Vanishing Proportional Delays

We describe the application of the spectral method to delay integro-difierential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extensions to equations with more general (nonlinear) vanishing delays are also discussed.

EXISTENCE FOR A NONLINEAR IMPULSIVE FUNCTIONAL INTEGRODIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS IN BANACH SPACES

In this paper, we consider the existence of mild solutions for a certain class of nonlinear impulsive functional evolution integrodifierential equation with nonlocal conditions in Banach spaces. A su-cient condition is established by using Schaefer's flxed point theorem combined with an evolution system. An example is also given to illustrate our result. AMS Mathematics Subject Classiflcation : 34K30, 34G20, 34K05.

EXISTENCE OF SOLUTIONS OF QUASILINEAR INTEGRODIFFERENTIAL EVOLUTION EQUATIONS IN BANACH SPACES

We prove the local existence of classical solutions of quasi- linear integrodifierential equations in Banach spaces. The results are obtained by using fractional powers of operators and the Schauder flxed- point theorem. An example is provided to illustrate the theory.

Perturbation of Singular Chasma Equation

An ionizing beam (electrons, ions, atoms, photons) between plane-parallel plates, either parallel or perpendicular to the plates (like in the multipactor efiect and hf cavities) may create an accumulating ion charge. The resulting steady state of the chasma ('non-quasi-neutral plasma') has a potential ' satisfying a quasi-singular non-linear integro-difierential equation. The investigation of the perturbations reveals problems due to the singular character of the integrand. However, in the case that the beam density (b) and the ion production (P) are constant the potential is given by ' = ikx 2 , where k is the positive root of a cubic equation. Perturbing and linearizing the latter equation yields the change dk for a disturbance db and dP, which may be time dependent provided slow enough that the ions may readapt to the new steady state. Some particular cases are studied. For more involved cases one should return to the singular equation or it may be better to the original set of partial difierential equations, which avoids the singularity problem. However, this makes the investigation much more wide and much more involved. DOI: 10.2529/PIERS060906114445

Elastohydrodynamic response of an ice sheet to forced sub-surface uplift

T he objective of this study is to investigate the dynamic response of an ice sheet under transient subsurface uplift. The ice sheet is modeled as an infinite thin plate undergoing small deflections that is pushed vertically upwards by a rigid flat-ended cylinder ; the water underneath the ice sheet is modeled using incompressible potential flow theory. Ultimately, the deflection of the ice sheet is expressed in terms of an integral of the unknown indentation force. For a given motion of the cylinder, the corresponding indentation force must be determined by solving a Volterra integral equation of the first kind. After some numerical experimentation, it proved advantageous to transform the governing equation into an integro-difierential equation. Once the indentation force is obtained, the other physical quantities of interest — the temporal and spatial distributions of the displacement, moment, shear, the buoyant force, the inertial force of the water underneath the ice sheet and the inertial force of the ice sheet — can be evaluated numerically. The results reveal that the inclusion of the hydrodynamic reaction is of major importance.

Global Existence and Asymptotic Stability for a Nonlinear Integrodifferential Equation Modeling Heat Flow

This paper studies initial value problems that arise from models for one-dimensional heat flow (with finite wave speeds) in materials with memory. Under assumptions that ensure compatibility of the constitutive relations with the second law of thermodynamics, the resulting integrodifierential equation is hyperbolic near equilibrium. The existence of unique, globally (in time) defined, classical solutions to the problems under consideration is established, provided the data are smooth and sufficiently close to equilibrium. Both Dirichlet and Neumann boundary conditions are treated, as well as the problem on the entire real line. Local existence is proved using a contraction-mapping argument which involves estimates for linear hyperbolic partial differential equations with variable coefficients. Global existence is obtained by deriving a priori energy estimates. These estimates are based on inequalities for strongly positive Volterra kernels (including a new inequality that is needed due to the form of the constitutive relations). Furthermore, compatibility with the second law plays an essential role in the proof in order to obtain an existence result under less restrictive assumptions on the data.

Oscillations in a system with material cycling

We study a system of two integrodifierential equations which models the evolution of a biotic species feeding on an abiotic resource. We also consider nutrient recycling with time delay. By Hopf bifurcation theory we prove the existence of stable oscillations for a range of values of the input of nutrients.

Dynamical algebra of invariants of integra-differential equations of quantum theory

The Heisenberg-Weyl dynamical algebra of invariants for the special type of integro-difierential equations of quantum theory is developed.

A Fast Method for the Determination of Emergent Intensities in Radiative Transfer Theory

One of the quantities usually required when solving the equation of radiative transfer is the intensity of radiation emerging from the surface of the medium under consideration. For multi-dimensional situations however, the methods presented to date have been numerical, and these first calculate the so-called source function Sv (r, Ω) as a function of position r, angle Ω and frequency v. This is generally the most difficult part of the exercise since an integro-difierential equation must be solved. The emergent intensity is then determined by solving a relatively simple first order differential equation by any of the well known numerical integration schemes. However, if the emergent intensity is required at a large number of angles, frequencies, and positions on the surface of the medium, and this is usually the case, the amount of computing needed may be considerable.