Class Of Linear Integro-differential Equations Research Articles

Hölder estimates for viscosity solutions of nonlocal equations with variable-order fractional Laplace term

We consider a class of linear integro-differential equations with variable-order fractional Laplacian. Under sharp assumptions on the variable-order α(x,y), we prove the local Hölder continuity of bounded viscosity solutions. In particular, for the continuous right-hand side f, we show that weak solutions are viscosity solutions. Furthermore, we prove a comparison principle for viscosity solutions when the function f is strictly away from zero.

Cubic spline quasi-interpolation operator to numerically solve integro-differential equations with weakly singular kernels

In this paper we use cubic spline quasi-interpolant operator to numerically address a class of linear integro-differential equations with weakly singular kernel. As stated in Pedas and Tamme (2006), the exact solution of this equation lacks the desired level of smoothness and belongs to a particular function space. Then, in the first part of this paper, we analyze the approximation properties of the cubic spline quasi-interpolant operator in this particular space. Subsequently, we use these results in the analysis of the quasi-collocation method used to solve an integro-differential equation with weakly singular kernel. Also, some numerical tests are provided to confirm the theoretical results.

Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation

We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in [6].

STABILITY OF INTEGRO DIFFERENTIAL EQUATIONS WITH IMPULSES

Sufficient conditions have been derived for the asymptotic stability of the trivial solutions of a class of linear integrodifferential equations under impulsive per-turbations. The sufficient conditions are formulated in terms of the parameters of the equations such that in the absence of the impulsive effects, these conditions are reduced to those of the non-impulsive equations. doi: http://dx.doi.org/10.22342/jims.13.1.71.17-26

Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is polynomial, then the exact solution can be obtained. If the exact solution is not a polynomial, then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver. In addition, the method is presented with error and stability analysis.

Uniqueness of traveling waves for nonlocal lattice equations

We establish the uniqueness (up to translation) of traveling waves for a nonlocal lattice equation with time delay. Our approach is based on exact a priori asymptotics of the wave profiles. This we accomplish by developing a structure theorem of entire solutions to a class of linear integro-differential equations.

Approximate solution of a class of linear integro-differential equations by Taylor expansion method

In this paper, a simple and effective Taylor expansion method is presented for solving a class of linear integro-differential equations including those of Fredholm and of Volterra types. By means of the nth-order Taylor expansion of an unknown function at an arbitrary point, a linear integro-differential equation can be converted approximately to a system of linear equations for the unknown function itself and its first n derivatives under initial conditions. The nth-order approximate solution is exact for a polynomial of degree equal to or less than n. Some examples are given to illustrate the accuracy of this method.

Solvability of a class of integro-differential equations of first order with variable coefficients

In the present paper, we consider a class of linear integro-differential equations of first order with a stochastic kernel and with variable coefficients on the semiaxis. These equations have important applications in physical kinetics. By combining special factorization methods with methods involving integral Fredholm equations of the second kind, we can construct solutions of such equations in the Sobolev space W 1 1 (ℝ+). In certain singular cases, we can also describe the structure of the obtained solutions.

A Method for the Approximate Solution of Linear Integro-Differential Equations

Approximate methods for solving a class of linear integro-differential equations of the Fredholm type are discussed. The integro-differential equations are reduced to Fredholm integral equations and Nyström’s method is used to establish numerical procedures. A convergence theorem is given for a simple integro-differential equation. Extensions of the method to more complicated cases are indicated.

On the existence and uniqueness of solution of the Cauchy problem for a class of linear integro-differential equations

Some problems in the theory of viscoelasticity may be described by means of integro-differential equations. In the paper a class of initial-value problems is considered which includes these physical examples, covering also their analogues - equations of the second order in time coordinate. The theory is restricted to the equations only, possessing in the same term both the highest spatial and the highest derivatives. The weak solution is defined on the base of variational principles, introduced in a previous article, and its existence, uniqueness and continuous dependence on the given data is proved, using the theory of integral Volterra's equations in Banach spaces.

On a class of linear integro-differential equations

This paper is a sequel to a previous one(1) with the same title which dealt with the general solution of equations of the typeWe consider here the more general equationwhere g(x) is a given function. We are interested particularly in the existence and uniqueness of solutions of the latter equation and show how these are related to the closure and completeness properties of sets of functions {xneωnx} derived from the kernels kr(y).