Class Of Singular Integro-differential Equations Research Articles

On Approximate Solution of One Class of Singular Integro-Differential Equations

A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized versions of the methods of moments and subdomains are proposed and substantiated. Optimality of the methods in order of accuracy is established.

On Approximate Solution of One Class of Singular Integro-Differential Equations

On Approximate Solution of One Class of Singular Integro-Differential Equations

Collocation Methods for a Class of Singular Integro-Differential Equations

Collocation Methods for a Class of Singular Integro-Differential Equations

On a New Class of Singular Integro-differential Equations

In this paper for a new class of model and non-model partial integro-differential equations with singularity in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions. Such type of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of analytic functions. In the process of our research the new types of singular integro-differential operators are introduced and main property of entered operators are learned. It is shown that the solution of studied equation is equivalent to the solution of system of two equations with respect to x and y, one of which is integral equation and the other is integro-differential equation. Further, non-model integro-differential equations are studied by regularization method. This regularization method for non-model equation is based on selecting and analysis of a model part of the equation and reduced to the solution of two second kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence of a non-model part in the equation does not affect to the general structure of the solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important. In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a Cauchy type problems are investigated.

Justification of Galerkin and collocations methods for one class of singular integro-differential equations on interval

Justification of Galerkin and collocations methods for one class of singular integro-differential equations on interval

A new B-spline collocation method for singular integro-differential equations of higher orders

Two numerical schemes based on B-spline collocation method are presented for approximate solution of the systems of singular integro-differential equations of higher orders appearing in fracture problems with Steigmann–Ogden surface energy. It is shown that the resulting systems of linear equations are well-conditioned and the numerical examples are in good agreement with known solutions. Although this paper is concerned with the numerical solution to the specific equations, the proposed method can be easily generalized to a large class of singular integro-differential equations.

On asymptotic convergence of polynomial collocation method for one class of singular integro-differential equations

On asymptotic convergence of polynomial collocation method for one class of singular integro-differential equations

A polynomial collocation method for singular integro-differential equations in weighted spaces

A polynomial collocation method is proposed for the numerical solution of a class of singular integro-differential equations of Cauchy type; the collocation points are chosen to be the Chebyshev nodes. Function spaces are defined and theorems concerning the boundedness of certain operators are developed. Convergence of the numerical method is demonstrated in weighted uniform normed spaces of continuous functions; convergence rates are then determined in accordance with the smoothness of the functions characterizing the problem. Numerical examples are provided which go some way to confirming these estimates.

An approximate solution of one class of singular integro-differential equations

The problem of definition of mechanical field in a homogeneous plate supported by finite inhomogeneous inclusion is considered. The contact between the plate and inclusion is realized by a thin glue layer. The problem is reduced to the boundary value problem for singular integro-differential equations. Asymptotic analysis is carried out. Using the method of orthogonal polynomials, the problem is reduced to the solution of an infinite system of linear algebraic equations. The obtained system is investigated for regularity.

Numerical Minimum Time Control to a Class of Singular Integro-Differential Equations

Numerical Minimum Time Control to a Class of Singular Integro-Differential Equations

Correct formulation and a spline method for solving a class of singular integro-differential equations

Conditions for the unique solvability of a class of singular integro-differential equations with Hilbertian kernels are examined. A spline method for solving such equations is justified, and numerical results are presented.

Projection methods for the solution of one class of singular integrodifferential equations

In this paper we propose a computational scheme of the general projection method for the solution of singular integrodifferential equations in the theory of stream lines and thermal conductivity. We theoretically substantiate this scheme from the standpoint of the theory of approximate methods of functional analysis. We consider particular cases of the general projection method, namely, the method of moments and the collocation method.

Justification of the Galerkin method for one class of singular integro-differential equations on an interval

Here we present a justification of a Galerkin method for solving the linear singular integro-differential equations with Cauchy kernel on an interval. The proofs of the convergence and the error estimations are given.

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions

On well-posedness of integro-differential equations in weighted $L^2$-spaces

In this paper we consider the problem of constructing a well-posed state space model for a class of singular integro-differential equations of neutral type. The work is motivated by the need to develop a framework for the analysis of numerical methods for designing control laws for aeroelastic systems. Semigroup theory is used to establish existence and well-posedness results for initial data in weighted $L^{2}-$spaces. It is shown that these spaces lead naturally to the dissipativeness of the basic dynamic operator. The dissipativeness of the solution generator combined with the Hilbert space structure of these weighted spaces make this choice of a state space more suitable for use in the design of computational methods for control than previously used product spaces.

Numerical Methods for a Class of Singular Integro-Differential Equations Based on Semigroup Approximation

In this paper computational methods for singular integro-differential equations of neutral type are discussed. The study is based on a well-posed state-space formulation and uses semigroup theoretical framework to analyze the convergence of semi- and fully discrete numerical schemes. Feasibility of the fully discrete scheme is demonstrated by applying it to Abel–Volterra equations of the first kind and to a singular neutral functional differential equation.