This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

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Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

The present paper is devoted to the application of local polynomial integro-differential splines to the solution of integral equations, in particular, to the solution of the integral equations of Fredholm of the second kind. To solve the Fredholm equation of the second kind, we apply local polynomial integro-differential splines of the second and third order of approximation. To calculate the integral in the formulae of a piecewise quadratic integro-differential spline and piecewise linear integro-differential spline, we propose the corresponding quadrature formula. The results of the numerical experiments are given.

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On the solution of the Fredholm equation of the second kind

One of the important tasks of interpolation is the good selection of interpolation nodes. As is well known, the roots of the Chebyshev polynomial are optimal ones for interpolation with algebraic polynomials on the interval [-1,1]. Nevertheless, there are still some difficulties in constructing the grid of nodes when the number of points tend to infinity. Here we offer the formula for constructing the interpolation nodes for a rapidly increasing function or decreasing function. The formula takes into account the local behavior of the function on the previous three grid nodes, and it is based on the interpolation by local quadratic polynomial splines. Particular attention is paid to the interpolation of functions on radial-ring grids.

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About Adaptive Grids Construction

Interpolation by local splines in some cases gives a better result than other splines or interpolation by classical interpolation polynomials. Integro-differential splines are one of the types of local splines that use, in addition to the values of the functions at the nodes of the grid, integrals over grid intervals. To construct an approximation on a finite interval, in order to improve the approximation quality, we use left or right integrodifferential splines near the ends of this interval. At some distance from the ends, besides the left or right splines, we can also use the middle integro-differential splines. Sometimes it is not necessary to calculate the approximation of the function at intermediate points in [𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑗𝑗+1]. Instead of calculating the approximation in many points in [𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑗𝑗+1] it is sufficient to estimate only the upper and lower boundaries of the variety of the approximation on this interval. The paper discusses the estimation of the boundaries of approximation of functions using left, right and middle trigonometrical integro-differential splines of the third order of approximation. The process of constructing the basic splines is discussed. The approximation theorems are given. Unimprovable constants in the approximation inequalities are given. Numerical examples of construction of the approximations and interval estimation are given.

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Interval Estimation using Integro-Differential Splines of the Third Order of Approximation

In many cases, approximation by local interpolating splines is preferable to approximation by interpolating polynomials or interpolating by other types of splines. In the case of integro-differential splines we use the values of integrals over net intervals. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. In this paper we present the estimation of approximation and the algorithm for constructing an interval extension of approximation when values of function in nodes, values of its first derivative in nodes, and values of its integrals over net intervals are given. The algorithm of approximation is based on the method of approximating functions using integro-differential splines. For constructing this interval extension, we use techniques from interval analysis. Numerical examples are given.

Interval Estimation of Polynomial Splines of the Fifth Order

This paper deals with cases when the values of derivatives of a function are given at grid nodes or the values of integrals of a function over grid intervals are known. Polynomial and trigonometric integrodifferential splines for computing the value of a function from given values of its nodal derivatives and/or from its integrals over grid intervals are constructed. Error estimates are obtained, and numerical results are presented.

Application of integrodifferential splines to solving an interpolation problem

The present article is one of a number of articles in which the authors study the properties of integro-differential splines. The paper deals with the construction of integro-differential polynomial and nonpolynomial splines of the fifth order. The order of approximation with integrodifferential polynomial and nonpolynomial splines of the fifth order are given. We use the tensor product of polynomial and non-polynomial splines constructed in this paper for the approximation of functions of two variables. The results of these numerical experiments are given.

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On approximations by polynomial and nonpolynomial integro-differential splines

Construction of meansquare approximation with integro-differential splines of the fifth order and first level is considered. The results of numerical experiments for computation acceleration with the algorithm parallelization are presented.

Construction of meansquare approximation with integro-differential splines of fifth order and first level

This paper deals with the construction of integro-differential polynomial splines of the fifth order on a uniform grid of nodes. It is supposed that values of function in nodes and the values of integrals over intervals are known. The properties of the left, the right and the middle integro-differential polynomial splines are investigated. The approximation with these splines is constructed on every grid interval separately. The results of numerical approximation by the left, the right, and the middle integro-differential splines show that the middle splines are preferable. Errors of approximation of the left, the right and the middle integro-differential polynomial splines of one variable of the fifth order are given. The approximation of functions of two variables is constructed using the tensor product. Numerical examples are presented.

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A Comparison of Approximations with Left, Right and Middle Integro-Differential Polynomial Splines of the Fifth Order

This paper considers the numerical solution of the Fredholm integral equation of the second kind using local polynomial splines of the fifth order of approximation and the fourth order of approximation (cubic splines). The basis splines in these cases occupy five and four adjacent grid intervals respectively. Different local spline approximations of the fifth (or fourth) order of approximation are used at the beginning of the integration interval, in the middle of the integration interval, and at the end of the integration interval. The construction of the calculation schemes for solving the Fredholm equation of the second kind with these splines is considered. The results of the numerical experiments on the approximation of functions and on the solution of the Fredholm integral equations are presented. The results of the solution of the integral equation which uses the polynomial splines of the fifth order of approximation are compared with ones obtained with cubic splines and with the application of the Simpson’s method. Note that in order to achieve a given error using the approximation with quadratic splines, a denser grid of nodes is required than when using the approximation with the cubic splines or splines of the fifth order of approximation.

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I. G. Burova

Papers

Application of integrodifferential splines to solving an interpolation problem

On approximations by polynomial and nonpolynomial integro-differential splines

Construction of meansquare approximation with integro-differential splines of fifth order and first level

A Comparison of Approximations with Left, Right and Middle Integro-Differential Polynomial Splines of the Fifth Order

Fredholm Integral Equation and Splines of the Fifth Order of Approximation