scholarly journals Multi-Wavelets Galerkin Method for Solving the System of Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1369
Author(s):  
Hoang Viet Long ◽  
Haifa Bin Jebreen ◽  
Stefania Tomasiello
Keyword(s):  
Wavelet Transform ◽  
Galerkin Method ◽  
Integral Equations ◽  
Operational Matrix ◽  
Computational Effort ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Linear Type ◽  
Numerical Tests ◽  
Operational Matrix Of Integration

In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order O(2−Jr), where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method.

Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itô-Volterra integral equations

2019 ◽  
Vol 15 (3) ◽  
pp. 575-598 ◽  
Author(s):  
Farshid Mirzaee ◽  
Nasrin Samadyar
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Stochastic Operational Matrix ◽  
Bernoulli Wavelet ◽  
Operational Matrix Of Integration

Purpose The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically. Design/methodology/approach For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time. Findings By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated. Originality/value Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

scholarly journals Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

2020 ◽  
pp. 395-410
Author(s):  
S. C. Shiralashetti ◽  
Lata Lamani
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Effective Strategy ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Stochastic Operational Matrix ◽  
Nonlinear Algebraic Equations ◽  
Operational Matrix Of Integration

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.

Stochastic operational matrix of Chebyshev wavelets for solving multi-dimensional stochastic Itô–Volterra integral equations

2019 ◽  
Vol 17 (03) ◽  
pp. 1950007 ◽  
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
General Procedure ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
System Of Integral Equations ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

In this paper, the numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations have been obtained by second kind Chebyshev wavelets. The second kind Chebyshev wavelets are orthonormal and have compact support on [Formula: see text]. The block pulse functions and their relations to second kind Chebyshev wavelets are employed to derive a general procedure for forming stochastic operational matrix of second kind Chebyshev wavelets. The system of integral equations has been reduced to a system of nonlinear algebraic equations and solved for obtaining the numerical solutions. Convergence and error analysis of the proposed method are also discussed. Furthermore, some examples have been discussed to establish the accuracy and efficiency of the proposed scheme.

scholarly journals Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 200
Author(s):  
Ji-Huan He ◽  
Mahmoud H. Taha ◽  
Mohamed A. Ramadan ◽  
Galal M. Moatimid
Keyword(s):  
Linear System ◽  
Computer Program ◽  
Integral Equations ◽  
Operational Matrix ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

scholarly journals Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2105
Author(s):  
Elham Hashemizadeh ◽  
Mohammad Ali Ebadi ◽  
Samad Noeiaghdam
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Group Symmetry ◽  
Algebraic Equations ◽  
Weakly Singular Kernels ◽  
Euler’S Method ◽  
Genocchi Polynomials ◽  
Weakly Singular ◽  
Singular Kernels

In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority of the proposed method.

scholarly journals NUMERICAL SOLUTION OF MIXED LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY MODIFIED BLOCK PULSE FUNCTIONS

2021 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Linear System ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration ◽  
Modified Block

A numerical method based on modified block pulse functions is proposed for solving the mixed linear Volterra-Fredholm integral equations. We obtain an integration operational matrix of modified block pulse functions on interval [0,T). A modified block pulse functions and their operational matrix of integration, the mixed linear Volterra-Fredholm integral equations can be reduced to a linear system of algebraic equations. The rate of convergence is O(h) and error analysis of the proposed method are discussed. Some examples are provided to show that the proposed method have a good degree of accuracy.

scholarly journals An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Galerkin Method ◽  
Operational Matrix ◽  
Computational Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Dispersion Parameters ◽  
Wavelet Galerkin Method ◽  
Operational Matrix Of Integration ◽  
Nonlinear Physical Phenomena ◽  
The Galerkin Method

Abstract In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

scholarly journals An operational Haar wavelet method for solving fractional Volterra integral equations

2011 ◽  
Vol 21 (3) ◽  
pp. 535-547 ◽  
Author(s):  
Habibollah Saeedi ◽  
Nasibeh Mollahasani ◽  
Mahmoud Moghadam ◽  
Gennady Chuev
Keyword(s):  
Integral Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Global Error Bound ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Abel Integral Equations

An operational Haar wavelet method for solving fractional Volterra integral equationsA Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations

2020 ◽  
Vol 27 (1) ◽  
pp. 81-95 ◽  
Author(s):  
Mohammad Hossein Heydari ◽  
Mohammad Reza Hooshmandasl ◽  
Carlo Cattani
Keyword(s):  
Nonlinear Systems ◽  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
Computational Method ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
Stochastic Operational Matrix ◽  
A New Technique

AbstractIn this paper, a new computational method based on the Chebyshev wavelets (CWs) is proposed for solving nonlinear stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix (SOM) for the CWs is obtained. By using these basis functions and their SOM, such problems can be transformed into nonlinear systems of algebraic equations which can be simply solved. Moreover, a new technique for computation of nonlinear terms in such problems is presented. Further error analysis of the proposed method is also investigated and the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient.

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