scholarly journals Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3623-3635 ◽  
Author(s):  
Haman Azodi ◽  
Mohammad Yaghouti
Keyword(s):  
Differential Equations ◽  
Numerical Procedure ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Weakly Singular ◽  
The Matrix ◽  
Singular Kernels

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

scholarly journals Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Mikhail Bulatov ◽  
Pedro Lima ◽  
Ewa Weinmüller
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Main Stream ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
The Third ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Dimensional Integral

AbstractWe consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

A New Bernoulli Wavelet Method for Numerical Solutions of Nonlinear Weakly Singular Volterra Integro-Differential Equations

2017 ◽  
Vol 14 (03) ◽  
pp. 1750022 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Present Method ◽  
Numerical Solutions ◽  
Bernoulli Polynomials ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Weakly Singular ◽  
Nonlinear Algebraic Equations ◽  
Bernoulli Wavelet

In this paper, Bernoulli wavelet method has been developed to solve nonlinear weakly singular Volterra integro-differential equations. Bernoulli wavelets are generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton’s method. Convergence analysis of the present method has been discussed in this paper. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method.

scholarly journals Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Guodong Shi ◽  
Yanlei Gong ◽  
Mingxu Yi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Tabular Form ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Singular Kernels

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

Sinc-Galerkin Technique for the Numerical Solution of Fractional Volterra–Fredholm Integro-Differential Equations with Weakly Singular Kernels

2016 ◽  
Vol 17 (6) ◽  
pp. 315-323 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Galerkin Method ◽  
Function Approximation ◽  
Present Method ◽  
Sinc Function ◽  
Algebraic Equations ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

AbstractThe sinc-Galerkin method is developed to approximate the solution of fractional Volterra–Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the sinc function approximation. Usually, this type of integral equations is very difficult to solve analytically as well as numerically. The present method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by sinc-Galerkin method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method.

EULER AND TAYLOR POLYNOMIALS METHOD FOR SOLVING VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS WITH NONLINEAR TERMS

2021 ◽  
Vol 21 (2) ◽  
pp. 395-406
Author(s):  
DENİZ ELMACI ◽  
NURCAN BAYKUŞ SAVAŞANERİL ◽  
FADİME DAL ◽  
MEHMET SEZER
Keyword(s):  
Differential Equations ◽  
Matrix Equation ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Mean Value Theorem ◽  
Euler Polynomials ◽  
Algebraic Equations ◽  
Volterra Type ◽  
Collocation Points ◽  
The Matrix

In this study, the first order nonlinear Volterra type integro-differential equations are used in order to identify approximate solutions concerning Euler polynomials of a matrix method based on collocation points. This method converts the mentioned nonlinear integro-differential equation into the matrix equation with the utilization of Euler polynomials along with collocation points. The matrix equation is a system of nonlinear algebraic equations with the unknown Euler coefficients. Additionally, this approach provides analytic solutions, if the exact solutions are polynomials. Furthermore, some illustrative examples are presented with the aid of an error estimation by using the Mean-Value Theorem and residual functions. The obtained results show that the developed method is efficient and simple enough to be applied. And also, convergence of the solutions of the problems were examined. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB.

scholarly journals Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudospectral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
N. H. Sweilam ◽  
M. M. Khader ◽  
W. Y. Kota
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Upper Bound ◽  
Unknown Function ◽  
Approximate Formula ◽  
Analytical Study ◽  
Pseudospectral Method ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Collocation Points

A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshev coefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency and the accuracy of the present work.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

scholarly journals A numerical scheme for problems in fractional calculus

2018 ◽  
Vol 20 ◽  
pp. 02001
Author(s):  
M. Razzaghi
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Boundary Value ◽  
Bernoulli Polynomials ◽  
Algebraic Equations ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Fractional Differential

In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Sign in / Sign up

Export Citation Format

Share Document