Methods For Singular Integro-differential Equations Research Articles

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.

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.

A reliable mixed method for singular integro-differential equations of non-integer order

It is our goal in this article to apply a method which is based on the assumption that combines two methods of conjugating collocation and multiple shooting method. The proposed method can be used to find the numerical solution of singular fractional integro-differential boundary value problems (SFIBVPs) Dϑ y(t) + η ∫0t (t−s)ς−1 y(s) ds = g(t), 1 < ϑ ≤ 2, 0 < ς< 1, η ∈ ℝ, where Dϑ denotes the Caputo derivative of order ϑ. Meanwhile, in a separate section the existence and uniqueness of this method is also discussed. Two examples are presented to illustrate the application and further understanding of the methods.

Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval

In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation.

Convergence of the collocation methods for singular integro-differential equations in Lebesgue spaces

In this article, the numerical schemes of collocation methods for an approximate solution of singular integro- differential equations with kernels of Cauchy type are explained. The equations are defined on arbitrary smooth closed contours. The theoretical background of collocations methods in Lebesgue spaces is obtained.

A fast algorithm for Prandtl's integro-differential equation

Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given.

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.

Spline collocation for singular integro-differential equations over (0.1)

This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.