scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Changqing Yang
Keyword(s):  
Integral Equations ◽  
Chebyshev Polynomials ◽  
Mathematical Physics ◽  
Fredholm Integral Equations ◽  
Chebyshev Series ◽  
Fredholm Type ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Hybrid Function ◽  
Nonlinear Fredholm Integral Equations

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm-type equations, which have many applications in mathematical physics, are then considered. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Chebyshev series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Jianhua Hou ◽  
Beibo Qin ◽  
Changqing Yang
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Taylor Series ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Hybrid Function ◽  
Nonlinear Fredholm Integral Equations

A numerical method to solve nonlinear Fredholm integral equations of second kind is presented in this work. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Taylor series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of algebraic equations. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

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 Iterative Method for a System of Nonlinear Fredholm Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Keyan Wang ◽  
Qisheng Wang
Keyword(s):  
Integral Equations ◽  
Rates Of Convergence ◽  
Fredholm Integral Equations ◽  
Iteration Algorithm ◽  
Nonlinear Integral Equations ◽  
Approximation Solution ◽  
Fredholm Type ◽  
Integration Rule ◽  
Nonlinear Fredholm Integral Equations ◽  
The Fixed Point Theorem

In this paper, the iteration method is proposed to solve a class of system of Fredholm-type nonlinear integral equations. First, the existence and uniqueness of solution are theoretically proven by the fixed-point theorem. Second, the approximation solution method is given by using the appropriate integration rule. The error analysis for the approximated solution with the exact solution is discussed for infinity-norm, and the rates of convergence are obtained. Furthermore, an iteration algorithm is constructed, and the convergence of the proposed numerical method is rigorously derived. Finally, some numerical examples are given to illustrate the theoretical results.

scholarly journals Polynomial Least Squares Method for the Solution of Nonlinear Volterra-Fredholm Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bogdan Căruntu ◽  
Constantin Bota
Keyword(s):  
Least Squares ◽  
Integral Equations ◽  
Least Squares Method ◽  
Fredholm Integral Equations ◽  
Nonlinear Integral Equations ◽  
Straightforward Manner ◽  
Fredholm Type ◽  
Numerical Examples ◽  
Polynomial Solutions ◽  
Approximate Polynomial

The present paper presents the application of the polynomial least squares method to nonlinear integral equations of the mixed Volterra-Fredholm type. For this type of equations, accurate approximate polynomial solutions are obtained in a straightforward manner and numerical examples are given to illustrate the validity and the applicability of the method. A comparison with previous results is also presented and it emphasizes the accuracy of the method.

scholarly journals Numerical solution of two dimensional nonlinear fuzzy Fredholm integral equations of second kind using hybrid of block-pulse functions and Bernstein polynomials

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4923-4935 ◽  
Author(s):  
Vahid Mahaleh ◽  
Reza Ezzati
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Approximation Method ◽  
Numerical Stability ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Successive Approximation Method ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Block Pulse Functions

In this paper, first, we introduce a successive approximation method in terms of a combination of Bernstein polynomials and block-pulse functions. The proposed method is given for solving two dimensional nonlinear fuzzy Fredholm integral equations of the second kind. Then, we present the convergence of the proposed method. Also we investigate the numerical stability of the method with respect to the choice of the first iteration. Finally, two numerical examples are presented to show the accuracy of the method.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

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