Basma Magdy Ahmed Mohamed

Emil Shoukralla

01/01/2020

In this paper, the Lagrange functions of Lagrange interpolation are expanded into Maclaurin polynomials to improve the performance of an improved formula of the Barycentric Lagrange interpolation with uniformly spaced nodes and was used for solving Volterra integral equations of the second kind. For the implementation of this technique, the given data function, the kernel and the unknown function are approximated by the given improved formula to get interpolated polynomials of the same degree. Furthermore, the interpolated unknown function is represented by four matrices and is substituted twice into both sides of the considered integral equation, while the kernel is represented by five matrices. This enforcement provided the possibility to reduce the solution of the Volterra equation into an equivalent algebraic linear system in a matrix form. To show the efficiency of this method, four examples are solved. It turns out, that the obtained approximate solutions were equal to the exact ones. Moreover, it is noticed that a smaller number of nodes are applied if the given function and the kernel were algebraic functions and the upper bound of the integration domain variable was canceled. For a non–algebraic given function and kernel, the exact solutions were obtained by increasing the number of nodes and taking the upper bound of the integration domain to be equal to one, which ensures the accuracy and authenticity of the presented method.

Basma Magdy Ahmed Mohamed

Emil Shoukralla

01/12/2019

This paper presents a new computational method for solving linear Volterra integral equations of the second kind. Three techniques are used to establish the method; the first technique is based on re-describing the Barycentric Lagrange interpolation in a new formula that reduces the round-off error resulting from the high degree interpolant polynomials; the second technique is based on expanding the Lagrange Barycentric functions into Maclaurin polynomials and expressing them via a monomial basis that facilitates calculations and reduces the procedure's steps. In the third technique, the equidistance Chebyshev interpolation nodes have been chosen so that the bad behavior of the solution near the endpoints of the integration domain is treated. Moreover, the method reduces the solution to the solution of an equivalent matrix equation that can be easily solved by using the undermined coefficients method. The obtained results of the five illustrated examples show that if the unknown function is algebraic, the numerical solutions are found in explicit mathematical form equal to the exact solutions, regardless of the properties of the given function or the kernel. If the unknown function is non-algebraic, the numerical solutions are strongly converging to the exact solutions rather close to the endpoint of the integration domain which ensures the accuracy, efficiency, and authenticity of the presented method.

Basma Magdy Ahmed Mohamed

E S Shoukralla.

01/09/2019

A new method is established for solving Volterra integral equations of the second kind using Lagrange interpolation through the Vandermonde approach. The goal is to minimize the interpolation errors of the high-degree polynomials on equidistance interpolations by redefining the original Lagrange functions in terms of the monomial basis. Accordingly, the complexity of the calculations is significantly reduced, and time is saved. To achieve this, the given data and the unknown functions are interpolated using Lagrange polynomials of the same degree via the Vandermonde matrix. Moreover, the interpolant unknown function is substituted twice into both sides of the integral equation so that the solution is reduced to an equivalent matrix equation without any need to apply collocation points. The error norm estimation is proved to be equal to zero. It was found that the obtained Vandermonde numerical solutions were equal to the exact ones, the calculation time was remarkably reduced, the round-off error was significantly reduced and the problematics due to the high-degree interpolating polynomial was completely faded regardless of whether the given functions were analytical or not. Thus, interpolation via the Vandermonde matrix ensures the accuracy, efficiency and authenticity of the presented method.

Basma Magdy Ahmed Mohamed

E S Shoukralla , H Elgohary

01/09/2019

An improved version of Barycentric Lagrange interpolation with uniformly spaced interpolation nodes is established and applied to solve Volterra integral equations of the second kind. The given data function and the unknown functions are transformed into two separate interpolants of the same degree, while the kernel is interpolated twice. The presented technique provides the possibility to reduce the solution of the Volterra equation into an equivalent algebraic linear system in matrix form without any need to apply collocation points. Convergence in the mean of the solution is proved and the error norm estimation is found to be equal to zero. Moreover, the improved Barycentric numerical solutions converge to the exact ones, which ensures the accuracy, efficiency, and authenticity of the presented method.