scholarly journals A Numerical Method for Solving Singular Integral Algebraic Equations with Weakly Singular Kernels

2021 ◽  
Vol 10 (3) ◽  
Keyword(s):  
Numerical Method ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

scholarly journals On multi-singular integral equations involving n weakly singular kernels

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1323-1333 ◽  
Author(s):  
Sales Nabavi ◽  
O. Baghani
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Singular Integral ◽  
Applied Mathematics ◽  
Uniform Space ◽  
Singular Integral Equations ◽  
Restrictive Assumption ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

We deal with some sources of Banach spaces which are closely related to an important issue in applied mathematics i.e. the problem of existence and uniqueness of the solution for the very applicable weakly singular integral equations. In the classical mode, the uniform space (C[a,b], ||.||?) is usually applied to the related discussion. Here, we apply some new types of Banach spaces, in order to extend the area of problems we could discuss. We consider a very general type of singular integral equations involving n weakly singular kernels, for an arbitrary natural number n, without any restrictive assumption of differentiability or even continuity on engaged functions. We show that in appropriate conditions the following multi-singular integral equation of weakly singular type has got exactly a solution in a defined Banach space x(t) = ?p,i=1 ?i/?(^?i) ?t,0 fi(s,x(s)) (tn-tn-1)1-?i,n...(t1-s)1-?i,1 dt + ?(t). In particular we consider the famous fractional Langevin equation and by the method we could extend the region of variations of parameter ?+ ? from interval [0,1) in the earlier works to interval [0,2).

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.

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.

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