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Open accessJournal ArticleDOI: 10.37394/23206.2021.20.2

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
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.

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Topics: Volterra integral equation (69%), Polynomial (61%), Interpolation (52%) ... read more
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6 results found



Open accessJournal ArticleDOI: 10.46300/9106.2021.15.136
31 Aug 2021-
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.

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Topics: Boundary value problem (68%), Volterra integral equation (67%), Differential equation (62%) ... read more

1 Citations


Open accessJournal ArticleDOI: 10.46300/9106.2021.15.167
04 Oct 2021-
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.

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Topics: Numerical integration (56%), Gauss–Seidel method (55%), Gauss (53%) ... read more

1 Citations



Open accessDOI: 10.37394/232011.2021.16.27
Sandip Saha1, Vikash Kumar1, Apurba DasInstitutions (1)
22 Nov 2021-

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16 results found




Journal ArticleDOI: 10.1016/J.AMC.2018.07.021
Abstract: A new and efficient method is presented for solving three-dimensional Volterra–Fredholm integral equations of the second kind (3D-VFIEK2), first kind (3D-VFIEK1) and even singular type of these equations. Here, we discuss three-variable Bernstein polynomials and their properties. This method has several advantages in reducing computational burden with good degree of accuracy. Furthermore, we obtain an error bound for this method. Finally, this method is applied to five examples to illustrate the accuracy and implementation of the method and this method is compared to already present methods. Numerical results show that the new method provides more efficient results in comparison with other methods.

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21 Citations


Journal ArticleDOI: 10.1016/J.CAM.2018.06.008
Jiaquan Xie1, Mingxu Yi2Institutions (2)
Abstract: In this study, a numerical scheme for approximating the solutions of nonlinear system of fractional-order Volterra–Fredholm integral–differential equations (VFIDEs) has been proposed. This method is based on the orthogonal functions defined over 0 , 1 combined with their operational matrices of integration and fractional-order differentiation. The main characteristic behind this approach is that it reduces such problems to a linear system of algebraic equations. In addition the error analysis of the system is investigated in detail. Lastly, several numerical examples are presented to test the effectiveness and feasibility of the given method.

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Topics: Nonlinear system (67%), Linear system (58%), Algebraic equation (57%) ... read more

14 Citations


Journal ArticleDOI: 10.1016/J.CAM.2018.05.035
Neda Negarchi1, Kazem Nouri1Institutions (1)
Abstract: This paper presents a computational technique based on a special family of the Muntz–Legendre polynomials to solve a class of Volterra–Fredholm integral equations. The relationship between the Jacobi polynomials and Muntz–Legendre polynomials, in a particular state, are expressed. The proposed method reduces the integral equation into algebraic equations via the Chebyshev–Gauss–Lobatto points, so that the system matrix coefficients are obtained by the least squares approximation method. The useful properties of the Jacobi polynomials are exploited to analyze the approximation error. The performance and accuracy of our method are examined with some illustrative examples.

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Topics: Jacobi polynomials (74%), Collocation method (61%), Integral equation (58%) ... read more

12 Citations