Numerical verification of solutions for nonlinear hyperbolic equations

We consider a numerical method to verify the solutions for nonlinear hyperbolic problems with guaranteed error bounds in the one-space dimensional case. We present verification procedures and show some numerical examples.     

Sample path approximation for stochastic integro-differential equations

We consider the problem of mean square sample path convergence for a specialized stochastic difference scheme defined for stochastic integro-differential equations under the assumption of pathwise uniqueness property of solutions. We apply the method of weak convergence and the technique of Skorohod representation in the space of continuous trajectories     

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.     

Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.      

Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C¹ finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's fixed point theorem for a continuous map on L^p-space in a computer. We present detailed verification procedures and give some numerical examples.     

Numerical solutions of systems of nonlinear integro-differential equations by Homotopy-perturbation method

In this study, the numerical solutions of a system of two nonlinear integro-differential equations, which describes biological species living together, are derived employing the well-known Homotopy-perturbation method. The approximate solutions are in excellent agreement with those obtained by the Adomian decomposition method. Furthermore, we present an analytical approximate solution for a more general form of the system of nonlinear integro-differential equations. The numerical result indicates that the proposed method is straightforward to implement, efficient and accurate for solving nonlinear integro-differential equations.     

Numerical methods for singular nonlinear integro-differential equations

For the numerical integration of singular nonlinear integro-differential equations we consider fractional linear multistep methods. We prove convergence of these methods and discuss their stability (as an extension of A-stability for stiff differential equations). Numerical experiments with the Basset equation are included.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

Lipschitz regularity of solutions for mixed integro-differential equations

We establish new Hölder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii–Lions?s method. We thus extend the Hölder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local–nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.     

Asymptotics of solutions of matrix integro-differential equations

We obtain an asymptotic expansion of the solution of a system of first-order integro-differential equations with the influence of the roots of the characteristic equation taken into account. A similar expansion is established for a system of Volterra integral equations.     

Interpolation correction for collocation solutions of Fredholm integro-differential equations

In this paper we discuss the collocation method for a large class of Fredholm linear integro-differential equations. It will be shown that, when a certain higher order interpolation operation is added to the collocation solution of this equation, the new approximations will, under suitable assumptions, admit a multiterm error expansion in even powers of the step-size . Based on this expansion, ideal multilevel correction results of this collocation solution are obtained.

     

Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations

In this paper, we study the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of class r for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.     

Asymptotically almost periodic solutions for abstract neutral integro-differential equations

We study the existence of asymptotically almost periodic classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations which arise in the study of heat conduction in fading memory material is considered.     

Existence and approximation of solutions of forced duffing type integro-differential equations with three-point nonlinear boundary conditions

A generalized quasilinearization technique is developed to obtain an analytic approximation of the solutions of the forced Duffing type integro-differential equation with nonlinear three-point boundary conditions. Monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of the problem are presented.