Reproducing kernel method for solving Fredholm integro-differential equations with weakly singularity

Du, Hongliang; Zhao, Guoliang; Zhao, Chunyan

doi:10.1016/j.cam.2013.04.006

Cited by 29 publications

(15 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years the reproducing kernel methods have been used to solve singular integral equations types, linear integro-differential difference equations and the kind of singular integral equations whith difference kernel [17,18,19,20,21,25]. This method can be applied to solve singular integral equations since it is well known that singular integral operators on L 2 space are bounded linear operators, see [22].…”

Section: Introductionmentioning

confidence: 98%

A new efficient method for cases of the singular integral equation of the first kind

Lotfi

Mahdiani

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution u(x) is represented in the form of a series in the reproducing kernel space, afterwards the n-term approximate solution u n (x) is obtained and it is proved to converge to the exact solution u(x). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.

show abstract

Section: Introductionmentioning

confidence: 98%

A new efficient method for cases of the singular integral equation of the first kind

Lotfi

Mahdiani

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Because of the limitations of several traditional analytical methods for solving these equations, some new numerical methods have been proposed. For instance, Du et al [7] proposed the reproducing kernel method for solving Fredholm integro-differential equations (FIDEs) with weakly singularity. Lifanov et al [12] also suggested some new numerical methods are for solving singular integral equations.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations

Rza̧dkowski

Tohidi

2019

Filomat

View full text Add to dashboard Cite

In the present paper we use the generalized Euler-Maclaurin summation formula to study the convergence and to solve weakly singular Fredholm and Volterra integral equations. Since these equations have different nature, the proposed convergence analysis for each equation has a different structure. Moreover, as an application of this summation formula, we consider the numerical solution of the fractional ordinary differential equations (FODEs) by transforming FODEs into the associated weakly singular Volterra integral equations of the first kind. Some numerical illustrations are designed to depict the accuracy and versatility of the idea.

show abstract

“…Alvandi and Paripour solved nonlinear Abel's integral equations with weakly singular kernel in the reproducing kernel space and removed the singularity of the equation considered [13]. The same authors presented a simple and efficient method to solve linear Volterra integro-differential equations [14], nonlinear Volterra-Fredholm integro-differential equations [15], and see [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel method for a class of weakly singular Fredholm integral equations

Alvandi

Paripour

2018

Journal of Taibah University for Science

View full text Add to dashboard Cite

Numerical methods for solving integral equations have been the focus of much research, including reproducing kernel methods. We present a new algorithm to solve weakly singular Fredholm integral equations (WSFIEs). The advantage of this method is that it is possible to pick any point in the interval of integration and also the approximate solution. The advantage of this method is used to remove singularity and reproducing kernel functions are used as a basis. The convergence of approximation solution to the exact solution is also proved. Some examples are displayed to demonstrate that the method is accurate and efficient for WSFIEs.

show abstract

Reproducing kernel method for solving Fredholm integro-differential equations with weakly singularity

Cited by 29 publications

References 8 publications

A new efficient method for cases of the singular integral equation of the first kind

A new efficient method for cases of the singular integral equation of the first kind

Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations

Reproducing kernel method for a class of weakly singular Fredholm integral equations

Contact Info

Product

Resources

About