Integro-differential Equations Of Hyperbolic Type Research Articles

Inverse problem for non-homogeneous integro-differential equation of hyperbolic type

An inverse problem is considered, which consists in finding a solution and a one-dimensional kernel of the integral term of an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given intego-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral k(t), of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. And also the theorem of conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronoullo’s inequality is used.

Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time t as well as how the absolute error in each example is calculated.

Initial-Boundary Value Problem for a Multidimensional Integro-Differential Equation of Hyperbolic Type and a Difference Method for its Numerical Implementation

Initial-Boundary Value Problem for a Multidimensional Integro-Differential Equation of Hyperbolic Type and a Difference Method for its Numerical Implementation

Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations

A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

General decay properties of abstract linear viscoelasticity

We consider a linear Volterra integro-differential equation of hyperbolic type, which can be viewed as an abstract version of the equation $$\begin{aligned} \partial _{tt} u(t)- \varDelta u(t) +\displaystyle \int \limits _0^t\mu (s)\varDelta u(t-s)\mathrm{d}s=0 \end{aligned}$$describing the motion of linearly viscoelastic solids. We establish some decay results for the associated energy, under assumptions that do not involve differential inequalities for the convolution kernel $$\mu $$.

A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation

An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is presented such that it can be used for adaptive strategies based on dual weighted residual methods. A posteriori error estimates based on weighted global projections and local projections are also proved.

Decadimento uniforme per equazioni integro-differenziali lineari di Volterra

This talk is devoted to some recent results concerning the exponential and the polynomial decays of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation describing the motion of a linearly viscoelastic solid occupying a (bounded) volume at rest. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel. A similar analysis is carried on in the whole N-dimensional real space, although both the polynomial and the exponential decay of the memory kernel lead to a polynomial decay of the energy, with a rate influenced by the space dimension N. These results are contained in two joint papers with Monica Conti and Vittorino Pata (Politecnico di Milano).

Wave equations with memory: The minimal state approach

Recently, in M. Conti et al. (2010) [6] and M. Fabrizio et al. (2010) [12], a new theoretical scheme has been developed in order to study equations with memory, the so-called minimal state approach. The aim of this work is to provide the technical body needed to study the asymptotic behavior of semilinear integro-differential equations of hyperbolic type in the novel framework.

Integro-differential equations of hyperbolic type with positive definite kernels

The main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in Hilbert spaces. For kernels that decay exponentially at infinity and possess strongly positive definite primitives, the exponential stability of weak solutions is obtained in the energy norm. It is also shown that this theory applies to several examples of kernels with possibly variable sign, and to a problem in nonlinear viscoelasticity.

Semigroup Analysis of Structured Parasite Populations

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

On nonlinear integro-differential equations of hyperbolic type

This paper considers unified initial-boundary value problems for integro-differential equations of hyperbolic type. For such problems where the forcing functions satisfy certain monotonicity conditions, monotone sequences converging to coupled minimal–maximal solutions may be constructed from a single pair of coupled lower–upper solutions. The coupled minimal–maximal solutions coincide in a unique solution provided the problem satisfies certain one-sided Lipschitz conditions.

UNIFORM DECAY PROPERTIES OF LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation [Formula: see text] describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ.

Semilinear Integrodifferential Equations of Hyperbolic Type: Existence in the Large

The main object of this work is the existence of global solutions to semilinear integrodifferential equations in Hilbert spaces for absolutely continuous convolution kernels, where the nonlinear term is the gradient of a functional having superlinear growth at infinity. An application to the theory of viscoelasticity concludes the paper providing motivations for the abstract theory.

Some problems for multidimensional integro-differential equations of hyperbolic type

We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations of the hyperbolic type encountered in biology.

Weak solution to hyperbolic Stefan problems with memory

A model for Stefan problems in materials with memory is considered. This model is mainly characterized by a nonlinear Volterra integrodifferential equation of hyperbolic type. Colli and Grasselli proved the uniqueness of a weak solution under the natural assumptions on data and the existence of a strong solution for smoother data. Taking advantage of these two results and assuming just the hypotheses ensuring uniqueness, the existence of a weak solution is here shown.

Well-posedness of the weak formulation for the phase-field model with memory

A phase-field model based on the Gurtin-Pipkin heat flux law is considered. This model consists in a Volterra integrodifferential equation of hyperbolic type coupled with a nonlinear parabolic equation. The system is then associated with a set of initial and Neumann boundary conditions. The resulting problem was already studied by the authors who proved existence and uniqueness of a smooth solution. A~careful and detailed investigation on weak solutions is the goal of this paper, going from the aspects of the approximation to the proof of continuous dependence estimates. In addition, a sufficient condition for the boundedness of the phase variable is given.

Nonautonomous integro-differential equations of hyperbolic type

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

A quasilinear integrodifferential equation of hyperbolic type

A quasilinear integrodifferential equation of hyperbolic type

On the Speed of Propagation of Solutions of Integrodifferential Equations

Linear Volterra integrodifferential equations of hyperbolic type from the electromagnetic theory of dielectrics, heat conduction theory of materials with memory and from viscoelasticity theory are examined. For a particular class of kernels solutions of initial value problems for these equations can all be split into a part propagating outwards and a part whose support remains within the support of the forcing term. The speed of the outward propagating part of each equation is exhibited.

Existence theorems for optimization problems concerning hyperbolic partial differential equations

Existence theorems are given for optimal control problems with partial integro-differential equations of hyperbolic type. A variety of conditions are given depending on whether (i) the cost functional is in the Lagrange form or Mayer form, (ii) the controls vary in a fixed compact set or variable but closed sets, (iii) the required solutions are generalized or usual, (iv) the defining equations are nonlinear or linear in the state variable or linear in the control, (v) the trajectories belong to the usual Sobolev classW p 2 ,p>1, or the special Sobolev classW p * , 1⩽p⩽∞. Several examples are given to illustrate the above cases. Necessary conditions for some of these problems have been discussed in an earlier paper.