Class Of Integro-differential Equations Research Articles

On a new class of integro-differential equations

We consider various initial-value problems for ordinary integro-differential equations of first order that are characterized by convolution-terms, where all factors depend on the solutions of the equations. Applications of such problems are descriptions of certain glass-transition phenomena based on mode-coupling theory, for instance. We will prove results concerning well-posedness of such problems and the asymptotic behaviour of their solutions.

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.

Integral transforms of Hartley–Fourier cosine polyconvolution type

In this paper, after introducing a new polyconvolution for the Hartley–Fourier cosine integral transforms, we consider an integral transformation of this polyconvolution type, namely , where are given functions and is some differential operator. We obtain the necessary and sufficient conditions for the unitary property and the inverse formula of in . A sequence of functions that converges to the original function in norm is defined. We further show that the operator is a bounded operator from to , here and is the conjugate exponent of . Besides showing some nice properties of the Watson and the Plancherel types of the operator , we demonstrate how to use it to solve a class of integro-differential equations and systems of two integro-differential equations in which some other convolutions on are also involved.

Existence and Exponential Stability of Weighted Pseudo Almost Periodic Classical Solutions of Integro-Differential Equations with Analytic Semigroups

In this paper, we investigate a class of integro-differential equations having analytic semigroups given by $$\begin{aligned} x'(t)=Ax(t)+f(t,x(t),Gx(t)). \end{aligned}$$ Our main results concern the existence, uniqueness and globally exponential stability of weighted pseudo-almost periodic classical solutions. An example is given to illustrate our results.

Fixed Points and Stability of a Class of Integrodifferential Equations

We study a class of integrodifferential functional differential equationsx¨+f(t,x,x˙)x˙+∑j=1N∫t-rj(t)taj(t,s)gj(s,x(s))ds=0with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable.

A mathematical model for value estimation with public information and herding

This paper deals with a class of integro-differential equations modeling the dynamics of a market where agents estimate the value of a given traded good. Two basic mechanisms are assumed to concur in value estimation: interactions between agents and sources of public information and herding phenomena. A general well-posedness result is established for the initial value problem linked to the model and the asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic different economic scenarios. Analytical results are illustrated by means of numerical simulations and lead us to conclude that, in spite of its oversimplified nature, this model is able to reproduce some emerging behaviors proper of the system under consideration. In particular, consistently with experimental evidence, the obtained results suggest that if agents are highly confident in the product, imitative and scarcely rational behaviors may lead to an over-exponential rise of the value estimated by the market, paving the way to the formation of economic bubbles.

On some classes of integro-differential equations on the half-line and related operator functions

On some classes of integro-differential equations on the half-line and related operator functions

Evolutionary branching patterns in predator-prey structured populations

Predator-prey ecosystems represent, among others, a natural context where evolutionary branching patterns may arise. Moving from this observation, the paper deals with a class of integro-differential equations modeling the dynamics of two populations structured by a continuous phenotypic trait and related by predation. Predators and preys proliferate through asexual reproduction, compete for resources and undergo phenotypic changes. A positive parameter $\varepsilon$ is introduced to model the average size of such changes. The asymptotic behavior of the solution of the mathematical problem linked to the model is studied in the limit $\varepsilon \rightarrow 0$ (i.e., in the limit of small phenotypic changes). Analytical results are illustrated and extended by means of numerical simulations with the aim of showing how the present class of equations can mimic the formation of evolutionary branching patterns. All simulations highlight a chase-escape dynamics, where the preys try to evade predation while predators mimic, with a certain delay, the phenotypic profile of the preys.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A CLASS OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED BOUNDARY CONDITIONS

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A CLASS OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED BOUNDARY CONDITIONS

The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability

We study discretization in classes of integro-differential equations $\begin{gathered} u'(t) + \int_0^t {(\lambda _1 a_1 (t - \tau ) + \lambda _2 a_2 (t - \tau ) + \cdots + \lambda _n a_n (t - \tau ))} u(\tau )d\tau = 0,t > 0, \hfill \\ u(0) = 1,\lambda _j \geqslant 1,j = 1,2,...,n, \hfill \\ \end{gathered} $ , where the functions a j (t), 1 ⩽ j ⩽ n, are completely monotonic on (0,∞) and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted l 1(ρ; 0,∞) norm, where ρ is a given weight function. Applications to the weighted l 1 stability of the numerical solutions of a related equation in Hilbert space are given.

Dynamical behaviors of recurrently connected neural networks and linearly coupled networks with discontinuous right-hand sides

The aim of this paper is to provide a systematic review on the framework to analyze dynamics in recurrently connected neural networks with discontinuous right-hand sides with a focus on the authors’ works in the past three years. The concept of the Filippov solution is employed to define the solution of the neural network systems by transforming them to differential inclusions. The theory of viability provides a tool to study the existence and uniqueness of the solution and the Lyapunov function (functional) approach is used to investigate the global stability and synchronization. More precisely, we prove that the diagonal-dominant conditions guarantee the existence, uniqueness, and stability of a general class of integro-differential equations with (almost) periodic self-inhibitions, interconnection weights, inputs, and delays. This model is rather general and includes the well-known Hopfield neural networks, Cohen-Grossberg neural networks, and cellular neural networks as special cases. We extend the absolute stability analysis of gradient-like neural network model by relaxing the analytic constraints so that they can be employed to solve optimization problem with non-smooth cost functions. Furthermore, we study the global synchronization problem of a class of linearly coupled neural network with discontinuous right-hand sides.

A splitting positive definite mixed finite element method for two classes of integro-differential equations

In this article, we establish a new mixed finite element procedure to solve the second-order hyperbolic and pseudo-hyperbolic integro-differential equations, in which the mixed element system is symmetric positive definite without requiring the LBB consistency condition. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(Ω) norm for u and in H( div;Ω) norm for the flux σ. Numerical experiments are given to verify the theoretical results.

An efficient numerical method to solve a special cass of integro-differential equations relating to the Levy models

The paper presents a new efficient numerical method for solving a special class of integro-differential equations with numerical derivatives. These problems arise in applications used to calculate special functionals of the Levy processes. The method is based on the Wiener-Hopf approximated factorization and numerical inversion of the Laplace transform.

A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions

In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: \[\begin{aligned}D_{t}^{q}x(t)=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&\;t\in [0,T],\;n\in Z^{+},\\qx(0)=g(x)+x_{0}.\end{aligned}\] Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: , , , where is a constant and . An approach which combines topological method of T. Wazewski and Schauder's fixed point theorem is used.

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

New algorithm for the numerical solution of the integro-differential equation with an integral boundary condition

In this paper, a sequence of approximate solution converging uniformly to the exact solution for a class of integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics and other field of physics and mathematical chemistry is obtained by using an iterative method. Its exact solution is represented in the form of series in the reproducing kernel space. The n-term approximation u n (x) is proved to converge to the exact solution u(x). Moreover, the first derivative of u n (x) is also convergent to the first derivative of u(x).

Homogenization of a Class of Integro-Differential Equations with Lévy Operators

The periodic homogenization of the integro-differential equation (PIDE) with the Lévy operator with the alpha-stable density, is studied in this paper. The formal asymptotic expansion method is employed to derive the cell problem, the ergodic problem for the Lévy operator without the second-order uniformly elliptic term. The effective equation is then obtained by using the result of the ergodic problem. Finally, the formal argument is justified rigorously by the perturbed test function method.

Solvability of a class of integro-differential equations of first order with variable coefficients

In the present paper, we consider a class of linear integro-differential equations of first order with a stochastic kernel and with variable coefficients on the semiaxis. These equations have important applications in physical kinetics. By combining special factorization methods with methods involving integral Fredholm equations of the second kind, we can construct solutions of such equations in the Sobolev space W 1 1 (ℝ+). In certain singular cases, we can also describe the structure of the obtained solutions.

New Integral Inequalities for Iterated Integrals with Applications

Some new nonlinear retarded integral inequalities of Gronwall type are established. These inequalities can be used as basic tools in the study of certain classes of integrodifferential equations.