Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind
02 Mar 2021-WSEAS Transactions on Mathematics archive (World Scientific and Engineering Academy and Society (WSEAS))-Vol. 20, pp 9-23
TL;DR: In this paper, the application of polynomial and non-polynomial splines to the solution of nonlinear Volterra integral equations is discussed. And the results of the numerical experiments are presented.
Abstract: The present paper is devoted to the application of local polynomial and non-polynomial
interpolation splines of the third order of approximation for the numerical solution of the Volterra integral
equation of the second kind. Computational schemes based on the use of the splines include the ability to
calculate the integrals over the kernel multiplied by the basis function which are present in the computational
methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra
integral equations is also discussed. The results of the numerical experiments are presented.
Citations
More filters
04 Oct 2021
TL;DR: In this article, the existence and uniqueness of the solution of the above-mentioned equations are investigated, and some stable methods with the degree p < 8 are constructed to solve some problems, and obtained results are compared with other known methods.
Abstract: The mathematical model for many problems is arising in different industries of natural science, basically formulated using differential, integral and integro-differential equations. The investigation of these equations is conducted with the help of numerical integration theory. It is commonly known that a class of problems can be solved by applying numerical integration. The construction of the quadrature formula has a direct relation with the computation of definite integrals. The theory of definite integrals is used in geometry, physics, mechanics and in other related subjects of science. In this work, the existence and uniqueness of the solution of above-mentioned equations are investigated. By this way, the domain has been defined in which the solution of these problems is equivalent. All proposed four problems can be solved using one and the same methods. We define some domains in which the solution of one of these problems is also the solution of the other problems. Some stable methods with the degree p<=8 are constructed to solve some problems, and obtained results are compared with other known methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. On the intersection of multistep and hybrid methods have been constructed multistep methods and have been proved that these methods are more exact than others. And also has been shown that, hybrid methods constructed here are more exact than Gauss methods. Noted that constructed here hybrid methods preserves the properties of the Gauss method.
17 citations
TL;DR: This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines, and the main focus is given to the use of splines of the fourth order of approximation.
Abstract: This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.
2 citations
31 Aug 2021
TL;DR: In this article, the Sinc function is used to deal with the external value problem of Volterra Integro differential equation, which reduces the error of the external boundary value problem.
Abstract: In the process of traditional methods, the error rate of external boundary value problem is always at a high level, which seriously affects the subsequent calculation and cannot meet the requirements of current Volterra products. To solve this problem, Volterra's preprocessing method for the external boundary value problem of Integro differential equations is studied in this paper. The Sinc function is used to deal with the external value problem of Volterra Integro differential equation, which reduces the error of the external value problem and reduces the error of the external value problem. In order to prove the existence of the solution of the differential equation, when the existence of the solution can be proved, the differential equation is transformed into a Volterra integral equation, the Taylor expansion equation is used, the symplectic function is used to deal with the external value problem of homogeneous boundary conditions, and the uniform effective numerical solution of the external value problem of the equation is obtained by homogeneous transformation according to the non-homogeneous boundary conditions.
1 citations
References
More filters
01 Jan 1962
33 citations
TL;DR: A new and efficient method for solving three-dimensional Volterra–Fredholm integral equations of the second kind, first kind and even singular type and three-variable Bernstein polynomials and their properties is presented.
27 citations
TL;DR: A numerical scheme for approximating the solutions of nonlinear system of fractional-order Volterra–Fredholm integral–differential equations (VFIDEs) based on the orthogonal functions defined over 0, 1 combined with their operational matrices of integration and fractional -order differentiation is proposed.
22 citations
"Application Local Polynomial and No..." refers methods in this paper
...In study [1] a numerical scheme for approximating the solutions of the nonlinear system of fractional-order VolterraFredholm integral differential equations was proposed....
[...]
...The authors of papers [1]-[10] devoted a lot of attention to the modification of the known numerical methods and the construction of new numerical methods for solving integral equations....
[...]
22 citations
"Application Local Polynomial and No..." refers methods in this paper
...The proposed numerical methods extend the set of known numerical methods for solving integral equations [18]....
[...]
...[18] Zdeněk Kopal, Numerical Analysis with emphasis on the applications of numerical techniques to problems of infinitesimal calculus in single variable, Wiley, New York, 1955, 556 p....
[...]
TL;DR: The important contributions are to provide convergence order theorems of the scheme using spline function theories and propose a new scheme with high convergence order for solving the approximate solutions to oscillation and non-oscillation of exact solutions.
18 citations