Hyperbolic Integro-differential Equations Research Articles

Finite element approximation and analysis of damped viscoelastic hyperbolic integrodifferential equations with [formula omitted] kernel

In this paper, we propose a finite element method for solving viscoelastic hyperbolic integrodifferential equations with L1 kernel, involving a nonlocal and nonlinear damped coefficient. Firstly, we discuss and deduce that the L1 kernel is of positive type in two cases. Subsequently, based on the continuous Galerkin technique and the energy argument, we prove the global existence and uniqueness of the discrete solution. Finally, using the basic properties of the positive-type kernel, we derive the error estimates of the finite element solutions.

Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

On the null controllability of integer order integro-differential equations

Abstract This paper is addressed to the study of the null controllability for integer order integro-differential equations. Unlike the known results for partial differential equations, we need to consider the equation involving a $\beta -$power of the Laplace operator $(-\varDelta )^\beta $ and an integral term. The key point is to construct a suitable state space of the controlled system at the final time. We first discuss a class of hyperbolic integro-differential equation. We prove that the controlled system is null controllable by an Ingham-type estimate. Also, the controllability time is given. On the other hand, by reduction to absurdity, we deduce that the null controllability property fails for a class of parabolic integro-differential equation with $\beta \in \mathbb{N}^+$.

A necessary optimality condition of the Pontryagin maximum principle type in one problem of optimal control of a system with distributed parameters

We consider a variable-structure optimal control problem described in different domains by a hyperbolic integro-differential equation and a Volterra integral equation, respectively. The quality functional is terminal. A formula for the increment of the quality criterion is constructed and an analogue of L.S. Pontryagin's maximum principle is proved by investigating on special McShane-type variations.

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

Numerical Analysis of Alternating Direction Implicit Orthogonal Spline Collocation Scheme for the Hyperbolic Integrodifferential Equation with a Weakly Singular Kernel

This paper studies an alternating direction implicit orthogonal spline collocation (ADIOSC) technique for calculating the numerical solution of the hyperbolic integrodifferential problem with a weakly singular kernel in the two-dimensional domain. The integral term is approximated with the help of the second-order fractional quadrature formula introduced by Lubich. The stability and convergence analysis of the proposed strategy are proven in L2-norm. Numerical results highlight the high accuracy and efficiency of the proposed strategy and clarify the theoretical prediction.

The problem of determining the memory of an environment with weak horizontal heterogeneity

The problem of determining the convolutional kernel $k(t,x)$, $t>0$, $x \in {\Bbb R}$, included in a hyperbolic integro-differential equation of the second order, is investigated in a domain bounded by a variable $z$ and having weakly horizontal heterogeneity. It is assumed that this kernel weakly depends on the variable $x$ and decomposes into a power series by degrees of a small parameter $\varepsilon$. A method for finding the first two coefficients $k_{0}(t)$, $k_{1}(t)$ of this expansion is constructed according to the given first two moments in the variable $x$ of the solution of the direct problem at $z=0$.

Finite element approximation of fractional hyperbolic integro-differential equation

<abstract><p>In this article, we propose a Galerkin finite element method for numerically solving a type of fractional hyperbolic integro-differential equation, which can be considered as the generalization of the classical hyperbolic Volterra integro-differential equation. Along with Galerkin finite element method in spatial direction, we apply a second order symmetric difference method in time. Next we discuss the regularity analysis of the weak solution and convergence analysis of the semi-discrete scheme. Then we further study the stability analysis and the error estimation of the fully discrete problems, according to the properties of fractional Ritz-Volterra projection, Ritz projection and so on. Numerical examples with comparisons among the proposed schemes verify our theoretical analyses.</p></abstract>

A fully discrete two-grid finite element method for nonlinear hyperbolic integro-differential equation

Two-grid fully discrete finite element approximations of the solution for a nonlinear hyperbolic integro-differential equation are considered and analyzed in this paper. The H1-norm error estimate is derived, which shows that the optimal convergence order can be obtained when the coarse-grid of size H and the fine-grid of size h satisfy h=O(H2). Besides reducing the storage and saving a large amount of time, two-grid methods also keep the accuracy of convergence in calculations. Numerical examples are given to support our theoretical results and demonstrate the efficiency of the methods.

Existence of asymptotically almost periodic solutions for some second-order hyperbolic integrodifferential equations

In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type in a separable Hilbert space H. To illustrate our abstract results, the existence of asymptotically almost periodic mild solutions to the well-known Kirchoff plate equation is studied.

Inverse problem for a second-order hyperbolic integro-differential equation with variable coefficients for lower derivatives

Inverse problem for a second-order hyperbolic integro-differential equation with variable coefficients for lower derivatives

Краевая задача для системы псевдогиперболических уравнений четвертого порядка с нелокальным условием

We consider a boundary value problem for a system of the fourth order pseudo-hyperbolic equations with nonlocal condition on a rectangular domain. By introducing a new unknown function, the considered problem is reduced to an equivalent nonlocal problem with integral condition for a system of hyperbolic integro-differential equations of the second order. We propose an algorithm for finding an approximate solution to the equivalent problem, and its convergence is proved basing on the functional parametrization method. Sufficient conditions of the unique existence of the classical solution to the boundary value problem for the system of the fourth order pseudo-hyperbolic equations with nonlocal condition are established in the terms of initial data.

Analytical and numerical solutions of a class of nonlinear integro-differential equations with [formula omitted] kernels

We study the finite element approximations to a general class of nonlinear and nonlocal hyperbolic integro-differential equations with L1 convolution kernels. The continuous time Galerkin procedures are defined and global existence of a unique discrete solution is derived. Moreover, optimal error estimates are shown in the L∞(H01(Ω))-norms. For the completely discrete scheme, linearized backward Euler method is defined and error estimates in l∞(H01(Ω))-norm are proved. Several numerical experiments are reported to confirm our theoretical findings.

Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids.

Plasmids are autonomously replicating genetic elements in bacteria. At cell division, plasmids are distributed among the two daughter cells. This gene transfer from one generation to the next is called vertical gene transfer. We study the dynamics of a bacterial population carrying plasmids and are in particular interested in the long-time distribution of plasmids. Starting with a model for a bacterial population structured by the discrete number of plasmids, we proceed to the continuum limit in order to derive a continuous model. The model incorporates plasmid reproduction, division and death of bacteria, and distribution of plasmids at cell division. It is a hyperbolic integro-differential equation and a so-called growth-fragmentation-death model. As we are interested in the long-time distribution of plasmids we study the associated eigenproblem and show existence of eigensolutions. The stability of this solution is studied by analyzing the spectrum of the integro-differential operator given by the eigenproblem. By relating the spectrum with the spectrum of an integral operator we find a simple real dominating eigenvalue with a non-negative corresponding eigenfunction. Moreover, we describe an iterative method for the numerical construction of the eigenfunction.

Output regulation boundary control of first-order coupled linear MIMO hyperbolic PIDE systems

ABSTRACTThe work addresses the output regulation problem for coupled linear multiple input multiple output (MIMO) hyperbolic partial integro-differential equation systems with disturbances affecting the systems through the space and boundary input. The exosystems are extended to generate ramp signals and general family of polynomial signals. The system decomposition is applied through the state transformation and yields a decoupled equivalent system. Based on the decoupled form, the backstepping transformation is applied and then in the new coordinate, the full state and output-feedback regulators are designed, respectively. For the state feedback regulator, the corresponding regulator equation is obtained and its solvability conditions are provided to facilitate the regulator design and feasibility. The design of observer-based regulator is based on the decoupling of the observer error system into a PDE subsystem and an ODE subsystem so that the backstepping approach achieves stabilisation by eigenvalue assignment leading to design of observer stabilizing gains.

Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.

Inverse Problem for an Integro-Differential Equation of Acoustics

We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t > 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.

Analysis of integro-differential equations modeling the vertical decomposition of soil organic matter

In this paper, a family of first-order hyperbolic integro-differential equations introduced to model the decomposition of organic matter (OM) are studied. These original equations depend on an extra variable named “quality”. We prove that these equations admit solutions in particular Banach spaces ensuring the continuity and the N N -order closure of equations ( N ∈ N ∗ N\in \mathbb {N}^* ) according to “quality”. We first give a result of existence, uniqueness and smoothness in a general framework. Then, this result is applied to specific transport equations. Finally, a numerical illustration of solutions properties is given by using an implicit-explicit finite difference scheme.

A Galerkin method for two-dimensional hyperbolic integro-differential equation with purely integral conditions

The present paper is devoted to the solution for two-dimensional hyperbolic integro-differential equations subject to purely integral conditions. First, it is demonstrated the existence and uniqueness of the solution under certain conditions. For the numerical approach, a Galerkin method based on least squares is proposed. The numerical examples illustrate the technique and show the efficiency of the proposed method.

Dust grain coagulation modelling : From discrete to continuous

Abstract In molecular clouds, stars are formed from a mixture of gas, plasma and dust particles. The dynamics of this formation is still actively investigated and a study of dust coagulation can help to shed light on this process. Starting from a pre-existing discrete coagulation model, this work aims to mathematically explore its properties and its suitability for numerical validation. The crucial step is in our reinterpretation from its original discrete to a well-defined continuous form, which results in the well-known Smoluchowski coagulation equation. This opens up the possibility of exploiting previous results in order to prove the existence and uniqueness of a mass conserving solution for the evolution of dust grain size distribution. Ultimately, to allow for a more flexible numerical implementation, the problem is rewritten as a non-linear hyperbolic integro-differential equation and solved using a finite volume discretisation. It is demonstrated that there is an exact numerical agreement with the initial discrete model, with improved accuracy. This is of interest for further work on dynamically coupled gas with dust simulations.