Hyperbolic Integro-differential Equations Research Articles

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.

Analysis of a viscosity model for concentrated polymers

The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier–Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic–hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier–Stokes system, includes dependence on the shear rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We prove the existence of global-in-time, large-data weak solutions under fairly general hypotheses.

Symmetric finite volume element approximations of second order linear hyperbolic integro-differential equations

In this paper, based on barycenter dual meshes, we develop one semi-discrete and two full discrete symmetric finite volume element schemes for second order linear hyperbolic integro-differential equations. The optimal order error estimates in L2 and H1-norms are derived for the semi-discrete scheme. Numerical experiments confirm the performance of the symmetric schemes, and further show that the L2-norm convergence rate of the full discrete backward Euler and Crank–Nicolson schemes to be O(h2+τ) and O(h2+τ2), respectively.

A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations

An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz–Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

An Approximation Method for Solving Volterra Integrodifferential Equations with a Weighted Integral Condition

We apply the reproducing kernel Hilbert space (RKHS) method for getting analytical and approximate solutions for second-order hyperbolic integrodifferential equations with a weighted integral condition. The analytical solution is represented in the form of series; thus, then-terms approximate solutions are obtained. The results of the numerical examples are compared with the exact solutions to illustrate the accuracy and the effectivity of this method.

Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis

In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results.

Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.

Continuous Galerkin finite element methods for hyperbolic integro-differential equations

A hyperbolic integro-differential equation is considered, as a model problem, where the convolution kernel is assumed to be either smooth or no worse than weakly singular. Well-posedness of the problem is studied in the context of semigroup of linear operators, and regularity of any order is proved for smooth kernels. A continuous space-time finite element method of order one is formulated for the problem. Stability of the discrete dual problem is proved, that is used to obtain optimal order a priori estimates via duality arguments. The theory is illustrated by an example.

Controllability for systems governed by semilinear evolution inclusions without compactness

In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.

Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions . Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity , under the appropriate assumptions on data. • Well-posedness of a hyperbolic integro-differential equation is studied. • The model arises in fractional viscoelasticity with two Mittag-Leffler type kernels. • Homogeneous Dirichlet and nonhomogeneous Neumann boundary conditions are considered. • Existence, uniqueness and regularity of the solution are proved by Galerkin's method. • We extend the method to prove regularity of any order for models with smooth kernels.

Nonlocal semilinear evolution equations without strong compactness: theory and applications

Abstract A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integro-differential equation arising in age-structure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation. MSC:34G25, 34B10, 34B15, 47H04, 28B20, 34H05.

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.

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

AbstractIn this article, we investigate the L∞(L2) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

SUPERCONVERGENCE AND EXTRAPOLATION OF QUASI-WILSON NONCONFORMING ELEMENT METHOD FOR HYPERBOLIC INTEGRO-DIFFERENTIAL EQUATION

In this paper,the nonconforming quasi-Wilson element method is applied to approximate the hyperbolic integro-differential equation with semi-discrete scheme.This element has a special character that the consistency error can reach to order O(h~2)/O(h~3) in broken H~1-norm(one/two order higher than its interpolaion error) when exact solution u belongs to H~3(Ω)/H~4(Ω).Thus,the superclose property and global superconvergence result with order O(h~2)are obtained by combining the known high accuracy analysis of bilinear element and post-processing technique,which are the same as that of bilinear element in the existing literatures.Furthermore,the accuracy of extrapolation solution with third order is presented by constructing a new extrapolation scheme.

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.

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.

Weak solvability of a hyperbolic integro-differential equation with integral condition

Weak solvability of a hyperbolic integro-differential equation with integral condition

Two-dimensional reaction-diffusion equations with memory

In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonin-creasing summable memory kernel κ. This equation models several phenomena arising from many different areas. After rescaling κ by a relaxation time e > 0, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that e is small enough, namely, when the equation is sufficiently close to the standard reaction-diffusion equation formally obtained by replacing κ with the Dirac mass at 0. Then, we provide an estimate of the difference between e-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit e → 0. In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.