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.

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.

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.

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 RETRACTED ARTICLE: Existence and uniqueness of solutions for the Schrödinger integrable boundary value problem

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Jianjie Wang ◽  
Ali Mai ◽  
Hong Wang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Linear Maps ◽  
Retracted Article

Abstract This paper is mainly devoted to the study of one kind of nonlinear Schrödinger differential equations. Under the integrable boundary value condition, the existence and uniqueness of the solutions of this equation are discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.

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