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 this paper, a cubic integral smoothing spline with roughness penalty for restoring a function by integrals is described. A mathematical method for building such a spline is described in detail. The method is based on cubic integral spline with a penalty function, which minimizes the sum of squares of the difference between the observed integrals of the unknown function and the integrals of the spline being constructed, plus an additional penalty for the nonlinearity (roughness) of the spline. This method has a matrix form, and this paper shows in detail how to fill in each matrix. The parameter \( \alpha \) governs the desired smoothness of the restored function. Spline knots can be chosen independently of observations, and a weight can be defined for each observation for more control over the resulting spline shape. An implementation in the R language as function int_spline is given. The function int_spline is easy to use, with all arguments completely described and corresponding examples given. An example of the application of the method in rare event analysis and forecasting is given.

Algorithm 1023: Restoration of Function by Integrals with Cubic Integral Smoothing Spline in R

Currently there are a number of papers in which certain types of splines are used to solve the Fredholm equation. Now much attention is paid to the application of a new type of spline, the so-called integro-differential spline to the solution of various problems. In this paper we consider the solution of the Fredholm equation using polynomial integro-differential splines of the third order approximation. To calculate the integral in the formula of a quadratic integro-differential spline, we propose the corresponding quadrature formula. The results of numerical experiments are given.

On the solution of the fredholm equation with the use of quadratic integro-differential splines

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

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. 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. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

/pdf/integro-differential-polynomial-and-trigonometrical-splines-3jtokwjmwy.pdf

Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulae

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. 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. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

O. V. Rodnikova

Papers

Application of integrodifferential splines to solving an interpolation problem

Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulae

Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulas