Class Of Integro-differential Equations Research Articles

Relaxed Trajectories of Integrodifferential Equations and Optimal Control on Banach Space

Relaxation optimal control for a class of integrodifferential equation associated with fractional power operator is investigated. By Gronwall lemma with singularity and delay the existence of mild solutions of integrodifferential equations is proved. Introducing finitely additive measure control, we convexificate original control systems and obtain corresponding relaxed control systems. Approximation results and equivalent relationship between control differential equations and integrodifferential inclusion are given. Existence of relaxed optimal control is verified and one relaxed optimal control is just an original one under generalized Cesari conditions.

A quasilinearization method for a class of integro-differential equations with mixed nonlinearities

We develop an extended form of the quasilinearization method for an initial value problem involving a nonlinear integro-differential equation with mixed nonlinearities, occurring in distributed-infective and distributed-contact models. In fact, a monotone sequence of iterates converging uniformly and quadratically to a solution of the problem is obtained. Some interesting observations are presented.

On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations

We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R2, for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit limt→∞u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.

On a functional equation related to the intermediate long wave equation

We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin–Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain.

Integrodifferential equations with analytic semigroups

In this paper we study a class of integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness, regularity and continuation of solutions to these integrodifferential equations.

New Versions of Spline Methods for a Class of Integro-Differential Equations

New Versions of Spline Methods for a Class of Integro-Differential Equations

A numerical solution technique for a class of integro-differential equations in elastodynamic crack propagation problems

A strategy for the numerical solution of a type of integro-differential equation arising from the analysis of dynamic crack propagation problems is presented. Specifically, the problem of interest is that of a mode III crack dynamically propagating in a homogeneous linear elastic medium while the crack tip consists of a nonlinear rate dependent cohesive (or failure) zone. The mode of propagation is general, that is, not restricted to steady state or other special regimes. Furthermore, the presented solution technique is in no way restricted to mode III problems.

Existence of solutions of nonlinear integrodifferential equation with nonlocal condition

In this paper we prove the existence and uniqueness of local and global solutions of a nonlocal Cauchy problem for a class of integrodifferential equation. The method of semigroups and the contraction mapping principle are used to establish the results.

Lyapunov Stability of a Class of Operator Integro-differential Equations with Applications to Viscoelasticity

The Lyapunov stability is analysed for a class of integro-differential equations with unbounded operator coefficients. These equations arise in the study of non-conservative stability problems for viscoelastic thin-walled elements of structures. Some sufficient stability conditions are derived by using the direct Lyapunov method. These conditions are formulated for arbitrary kernels of the Volterra integral operator in terms of norms of the operator coefficients. Employing these conditions the supersonic flutter of a viscoelastic panel is studied and explicit expressions for the critical gas velocity are derived. Dependence of the critical flow velocity on the material characteristics and compressive load is analysed numerically.

Cauchy problem for a class of integrodifferential equations on the semiaxes with a difference kernel

The equation is considered together with initial conditions . The looked for solution should belong to L 2(0,∞) as well as its derivatives up to the mth order. The question under consideration is: what conditions should fulfill parameters to ensure solvability of the Cauchy problem. Under certain assumptions this problem is reduced to the one of finding some term, in the Lorent series of a particular function.

Asymptotic behavior in a class of integrodifferential equations with diffusion

Asymptotic behavior in a class of integrodifferential equations with diffusion

Painleve analysis and singularity confinement: the ultimate conjecture

The authors propose a new integrability criterion based on the combination of two powerful integrability detectors: the 'Painleve method' (for continuous systems) and the singularity confinement (for discrete systems). With this new criterion they can treat a large variety of systems that combine discrete and continuous characteristics: differential-difference systems (ordinary and partial), differential-delay problems and even a class of integro-differential equations.

ON THE UNIQUENESS OF SOLUTIONS OF THE SINGULAR PROBLEM FOR CERTAIN CLASS OF INTEGRO-DIFFERENTIAL EQUATIONS

Article ON THE UNIQUENESS OF SOLUTIONS OF THE SINGULAR PROBLEM FOR CERTAIN CLASS OF INTEGRO-DIFFERENTIAL EQUATIONS was published on October 1, 1992 in the journal Demonstratio Mathematica (volume 25, issue 4).

Asymptotic Behavior and Oscillation of Classes of Integrodifferential Equations

Asymptotic Behavior and Oscillation of Classes of Integrodifferential Equations

Hamiltonian treatment of quasi-linear systems via a Bateman Lagrangian

A Hamiltonian treatment for Bateman Lagrangians is presented for a class of integro-differential equations. This includes a restricted Navier-Stokes equation and the Korteweg-de Vries equation.

Generic Hopf bifurcation in a class of integro-differential equations

Generic Hopf bifurcation in a class of integro-differential equations

On the oscillatory properties of the solutions of a class of integro‐differential equations of neutral type

In the present paper the oscillatory properties of the solutions of the equation urn:x-wiley:01611712:media:ijmm163126:ijmm163126-math-0001 are investigated where n ≥ 1, L is an operator of the difference type, It ⊂ ℝ, K : DK → ℝ, DK⫅ℝ3, x : [αx, ∞] → ℝ. Under natural conditions imposed on L, It and K it is proved that for n even all ultimately nonzero solutions oscillate and for n odd they either oscillate or tend to zero as t → ∞.

Mutual finite-element universal matrices in skin effect problems

Mutual finite-element universal matrices are defined for the numerical solution of a certain class of integrodifferential equations. The term 'mutual' comes from the fact that these matrices are relative to a couple of finite elements. By using these matrices an extension of the classical per-element assembly procedure of the global algebraic system is easily implemented. An example of application to the skin effect analysis of shielded multiconductor transmission lines is given. The proposed approach seems very promising because it allows easy inclusion of integrodifferential formulation into general-purpose N-dimensional finite-element codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Uniform $L^1$ behavior in classes of integro-differential equations with convex kernels

Uniform $L^1$ behavior in classes of integro-differential equations with convex kernels

Uniform $L^1 $ Behavior in Classes of Integrodifferential Equations with Completely Monotonic Kernels

We find conditions on a family $\mathfrak{a}$ of completely monotone, locally integrable, nonconstant functions a which enable us to write the solution $u(t;a)$ of $u'(t) + \int_0^t {a(t - s)u(s)ds = 0} $, $u(0) = 1$, as $u(t;a) = - \int_{ - \varepsilon }^0 {e^{\sigma t} } d\mu (\sigma ;a) + u_1 (t;a)$, where $\mu ( \cdot ;a)$ is a finite nonnegative measure on $[ - \varepsilon ,0]$ and $|u_1 (t;a)| \leqslant Qe^{ - \varepsilon t} $ with Q, $\varepsilon $ positive constants independent of $a \in \mathfrak{a}$. This formula is then utilized to give conditions on the collection $\mathfrak{a}$ which ensure that $\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)| \to 0(t \to \infty )$ and $\int_0^\infty {\rho (t)\sup _{a \in \mathfrak{a}} |u(t;a)|dt < \infty } $ , where $\rho $ is a given weight function. These results can be combined with a resolvent formula to investigate the asymptotic behavior as $t \to \infty $ of solutions of certain integrodifferential equations in Hilbert space.