Integro-differential Differential Equations Research Articles

An epidemiology model involving high-order linear Fredholm integro-differential-difference equations via a novel balancing collocation technique

In this study, the approximations of the higher order linear Fredholm integro-differential-difference equations (IDDEs), with the mixed conditions, have been performed by a new collocation technique based on the balancing polynomials. In particular, an attempt has been made to transform the linear IDDEs and the given boundary conditions into matrix equations which corresponds to a system of linear algebraic equations via the proposed procedure. In addition to that, the solutions obtained by the proposed numerical methodology have been analogized with the exact solutions and the error has been registered to manifest the accuracy of the solutions. Furthermore, the reliability and effectiveness of the proposed scheme have been illustrated by some numerical experiments. In addition to that, the error analysis of the technique has been performed along with the investigation of the error function for the improved approximate solutions for the IDDEs. In particular, the rate of approximation of the balancing polynomials has been deduced and the numerical accuracy of the suggested technique has been demonstrated. Apart from that, the absolute errors have been tabulated and graphical figures have been depicted for the solutions obtained via the proposed technique.

NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD

In this study, Akbari-Ganji’s Method (AGM) was applied to solve Volterra Integro-Differential Difference Equations (VIDDE) using Legendre polynomials as basis functions. Here, a trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. The unknown coefficients are solved for using the new proposed approach, AGM which principally involves the application of the boundary conditions on successive derivatives and integrals of the problem to obtain a system of equations. The system of equation is solved using any appropriate computer software, Maple 18. Some examples were solved and the results compared to the exact solutions.

A spectral collocation matrix method for solving linear Fredholm integro-differential–difference equations

A spectral collocation matrix method for solving linear Fredholm integro-differential–difference equations

Регуляризованная асимптотика решений интегродифференциальных уравнений с частными производными с быстро изменяющимися ядрами

Регуляризованная асимптотика решений интегродифференциальных уравнений с частными производными с быстро изменяющимися ядрами

A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials

In this paper, a matrix method based on the Dickson polynomials and collocation points is introduced for the numerical solution of linear integro-differential–difference equations with variable coefficients under the mixed conditions. In addition, in order to improve the numerical solution, an error analysis technique relating to residual functions is performed. Some linear and nonlinear numerical examples are given to illustrate the accuracy and applicability of the method. Eventually, the obtained results are discussed according to the parameter-α of Dickson polynomials and the residual error estimation.

Hybrid Euler–Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations

The main purpose of this paper is to present a numerical method to solve the linear Fredholm integro-differential difference equations with constant argument under initial-boundary conditions. The proposed method is based on the Euler polynomials and collocation points and reduces the integro-differential difference equation to a system of algebraic equations. For the given method, we develop the error analysis related with residual function. Also, we present illustrative examples to demonstrate the validity and applicability of the technique.

The Approximate Solutions of Fredholm Integrodifferential-Difference Equations with Variable Coefficients via Homotopy Analysis Method

Numerical solutions of linear and nonlinear integrodifferential-difference equations are presented using homotopy analysis method. The aim of the paper is to present an efficient numerical procedure for solving linear and nonlinear integrodifferential-difference equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system.

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.

Polynomial solution of the most general linear Fredholm–Volterra integrodifferential-difference equations by means of Taylor collocation method

In this paper, a Taylor collocation method has been presented for numerically solving the high-order general linear Fredholm–Volterra integrodifferential-difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. Using the Taylor collocation points, the method transforms the equation and conditions into the matrix equations with the unknown Taylor coefficients and hence by finding the Taylor coefficients, the polynomial approach is obtained. Also, to illustrate the accuracy of the method, numerical examples are presented and the results are discussed.

Polynomial solution of the most general linear Fredholm integrodifferential–difference equations by means of Taylor matrix method

In this article, a Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential–difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. Also numerical examples are presented, which illustrate the pertinent features of the method. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.

Kinetics of two phase fuel reflected reactors

In the present work a self-consistent mathematical model for the local dynamics of a quite particular class of fission reactors has been developed and solved. These devices consist of an innermost multiplying region, in which a significant fraction of the fissile fuel is diluted into a liquid phase, while the complementary fuel fraction operates as a standing solid matrix. This unconventional active region is surrounded by a standard peripheral reflector. For cooling purposes, the fluid fraction of the fuel needs to be circulated through external heat exchangers. The pump driven circulation causes the delayed neutron precursors, dissolved inside the fluid phase, to be spatially homogenized in the core volume well before decaying, while a continuous removal of precursor nuclei from the core takes place as a consequence of the outside circulation. Furthermore, the fraction of the extracted precursors still surviving after the solenoidal trip through the heat exchangers is continuously reinserted into the core. A new type of dynamical model is required to account for these unusual technological features. The mathematical structure of the evolution model presented in this paper consists of a system of integro–differential-difference equations, whose solution is derived in closed-form, by means of fully analytical techniques. Many dynamics and safety features of reactors of this type can be clarified a priori, upon inspection of the mathematical properties of the solution of the model. The rigorous time–eigenvalue generating equation can be explicitly established in the present theoretical context, together with the evaluation of any kind of transients. A short survey on the possible fields of application of these reactors is also presented.

Modeling and simulation of vertebrate primary visual system: Basic network

Neuronal activities in the primary visual system in vertebrates were studied by mathematical modeling and simulations. The eight-cell model, called the basic network, is governed by eight coupled nonlinear integro-differential difference equations. Five of the cells in the basic network are from the retina, and the remaining three are from the lateral geniculate nucleus and the visual cortex, forming a functional unit to process the visual information. This functional unit has two local circuitries, the retinal pathway and the thalamocortical pathway, connected by the optic nerve. Each cell in the basic network is characterized by its structure and interaction with other cells. The modeling of the basic network is described. The system equations are derived from the more general neuronal network equations by taking into account the main characteristics of the neurons, the form of the connections, and the manner of the interactions. The experimental determination of the system parameters is briefly discussed. A specific parametric configuration is introduced for a reference framework in the discussions. The simulations are considered under different conditions, as well as the responses of the system to certain rectangular and periodic incoming light intensities.

Initially and eventually dominating nonoscillatory solutions in integro-differential difference equations

For the system T′( t) + p( t) T( t) + q( t) T( t − τ) = ∝ 0 t K( t − μ) T( μ) dμ for t ⩾ 0, T( t) = g( t) for t ∈ [− τ, 0], conditions have been obtained which ensure that a solution of this system is dominated by a nonoscillatory solution in the interval ¦τ, ∞).

Closed Finite Queuing Networks with Time Lags

Consider a closed finite queuing system in which m units move among stations 1, 2, …, N (N ≧ 1). A unit that completes service at station i moves to station j (i, j = 1, 2, …, N) with probability ej∣i (∑j = Nj =1ej∣i=1, i = 1, 2, …, N) where it either commences service immediately if there is a server available, or queues to await service. The time taken for a unit to move from station i to station j is a random variable with distribution function Gi,j(⋯). At each station, the service times are independent and exponentially distributed, with the instantaneous service rate at a station being an arbitrary function of the number of units at that station. The positions of the units enroute between stations, measured in time units, are introduced as supplementary variables yielding a Markov process with states having a combination of discrete and continuous components. The integro-differential difference equations for the state variables are derived, and the steady-state solution is determined. The marginal joint distribution for the number of units at each station is found, and shown to depend, not on the distributions of the time lags, but rather on some linear combination of the means of the time lag distributions. An approximate technique is outlined for determining the absolute probability distribution of the number of units at a station, and some of the limitations of this method are discussed.