Order Integro-differential Equations Research Articles

The Lyusternik‒Sobolev lemma and the specific asymptotic stability of solutions of linear homogeneous Volterra-type integro-differential equations of order 3

Sufficient conditions for the asymptotic stability on a half-line of the solutions of a linear homogeneous Volterra-type integro-differential equation of order 3 in the case where the solutions of the corresponding linear homogeneous differential equation are asymptotically unstable are determined. A new method is proposed, and an illustrative example is constructed.

Bayesian sequential testing for Lévy processes with diffusion and jump components

We study the Bayesian problem of sequential testing of two simple hypotheses about the Lévy-Khintchine triplet of a Lévy process, having diffusion component, represented by a Brownian motion with drift, and jump component of finite variation. The method of proof consists of reducing the original optimal stopping problem to a free-boundary problem. We show it is characterized by a second order integro-differential equation, that the unknown value function solves on the continuation region, and by the smooth fit principle, which holds at the unknown boundary points. Several examples are presented.

Periodic solutions to nonlocal MEMS equations

Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices.

The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces

By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).

Viscosity Solutions of Integro-Differential Equations for Nonruin Probabilities

We consider a model of an insurance company investing its reserve into a risky asset whose price follows a geometric Levy process. We show that the nonruin probability is a viscosity solution of a second order integro-differential equation and prove a uniqueness theorem for the latter.

Viscosity solutions for second order integro-differential equations without monotonicity condition: the probabilistic approach

In this paper, we establish a new existence and uniqueness result of a continuous viscosity solution for integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver which is classically assumed in the literature of viscosity solutions of equation with non-local terms. Our method strongly relies on the link between IPDEs and backward stochastic differential equations with jumps for which we already know that the solution exists and is unique for general drivers. In the second part of the paper, we deal with the IPDE with obstacle and we obtain similar results.

On solvability of a class of nonlinear second order integro-differential equations in the space W 1 2 (ℝ+)

In this paper we study an initial-boundary value problem for a second order nonlinear integro-differential equation with Hammerstein type noncompact operator on the positive semi-axis. This problem has a direct application in the nonlocal problems of physical kinetics. We prove the existence of a nonnegative (nontrivial) solution of this equation in the space W12 (ℝ+). Some special examples of these equations are also discussed, which represent independent theoretical and applied interest.

Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

In this article, we study the concept of Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. We establish the results with Lipschitz condition and without Lipschitz condition on the forcing term. An interesting example is presented to illustrate the main findings. The results proven are new and complement the existing ones.

Complete Volterra Integrodifferential Equations of the Second Order Unsolved with Respect to the Higher Derivative

We prove the theorem on solvability of the Cauchy problem for the complete Volterra integrodifferential linear equations of the second order in a Hilbert space. For a complete equation, we consider three main classes of equations depending on the ordering of operator coefficients.

The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations

The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations

Convergence of the Generalized Homotopy Perturbation Method for Solving Fractional Order Integro-Differential Equations

In this paper,the homtopy perturbation method (HPM) was applied to obtain the approximate solutions of the fractional order integro-differential equations . The fractional order derivatives and fractional order integral are described in the Caputo and Riemann-Liouville sense respectively. We can easily obtain the solution from convergent the infinite series of HPM . A theorem for convergence and error estimates of the HPM for solving fractional order integro-differential equations was given. Moreover, numerical results show that our theoretical analysis are accurate and the HPM can be considered as a powerful method for solving fractional order integro-diffrential equations.

Convergence of the Generalized Homotopy Perturbation Method for Solving Fractional Order Integro-Differential Equations

In this paper,the homtopy perturbation method (HPM) was applied to obtain the approximate solutions of the fractional order integro-differential equations . The fractional order derivatives and fractional order integral are described in the Caputo and Riemann-Liouville sense respectively. We can easily obtain the solution from convergent the infinite series of HPM . A theorem for convergence and error estimates of the HPM for solving fractional order integro-differential equations was given. Moreover, numerical results show that our theoretical analysis are accurate and the HPM can be considered as a powerful method for solving fractional order integro-diffrential equations.

Jacobi spectral Galerkin methods for fractional integro-differential equations

We propose general spectral and pseudo-spectral Jacobi---Galerkin methods for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide rigorous error analysis for spectral and pseudo-spectral Jacobi---Galerkin methods, which show that the errors of the approximate solution decay exponentially in $$L^\infty $$L? norm and weighted $$L^2$$L2-norm. The numerical examples are given to illustrate the theoretical results.

On the numerical solution of fractional-order Fredholm-Volterra integro-differential and multi-fractional order integro-differential equations

In this work, we have applied the homotopy perturbation method (HPM) and variational iteration method (VIM) for solving fractional Fredholm and Volterra integrodifferential equation. Further, we have shown that multi-fractional order integrodifferential equation is a special case of fractional Fredholm and Volterra integrodifferential equation. For this, the fractional derivatives are described in the Caputo sense. Some numerical experiments are solved by using the aforesaid methods wherein the high accuracy and efficiency of both methods are observed.

Solution of System of Higher Order Integro-Differential Equations by Perturbed Variational Iteration Method

Variational Iteration Method (VIM) is a numerical method for solving a wide class of non-linear problems, first envisioned by the Chinese Mathematician He (1998). In this paper, higher order integro differential equations are reduced to a system of integral equations. The reduced system is then perturbed by using Chebyshev Polynomial and solved by Variational Iteration method. The results obtained for some illustrative examples showed that the perturbed variational iteration method is efficient and reliable. Examples are given to illustrate the efficiency and implementation of the method.

Asymptotic behavior for second order stochastic evolution equations with memory

In this paper, a general second order integro-differential evolution equation with memory driven by multiplicative noise is considered. We prove the existence of global mild solution and asymptotic stability of the zero solution using Lyapunov function techniques. Moreover, we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations.

Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examples are given to illustrate the theoretical results.

Some new dynamic Opial type inequalities and applications for second order integro-differential dynamic equations on time scales

In this paper, we give some new Opial type inequalities on time scales, at the same time we study the similar problems as disfocal and disconjugacy properties of solutions to second order integro-differential dynamic equations on time scales.

Stability Conditions of Second Order Integrodifferential Equations with Variable Delay

We investigate integrodifferential functional differential equationsẍ+f(t,x,ẋ)ẋ+∫t-r(t)t‍a(t,s)g(x(s))ds=0with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.

Moving least square method for treating nonlinear fourth order integro-differential equations

In this paper, we introduce a numerical scheme based on the moving least squares (MLS) method with utilizing collocation points for solving linear and nonlinear fourth order integro-differential equations. The main advantage of the presented method is that it requires no domain elements for interpolation or approximation. Some cases of the mentioned equations are solved as examples to illustrate the ability and reliability of the method. The results reveal that the method is very effective and convenient.