Convergence analysis of spectral collocation methods for a class of weakly singular Volterra integral equations

Ma, Xiaohua; Huang, Chengming; Xin, Niu

doi:10.1016/j.amc.2014.10.100

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…In addition, with some modifications in our novel and accurate method, noncompact Volterra‐Fredholm integral equations can be considered in details both numerically and theoretically 16,34 :

u (x) - {true\int}_{a}^{x} \frac{t^{μ - 1}}{x^{μ}} k_{1} (x, t) u (t) d t - {true\int}_{a}^{b} \frac{t^{μ - 1}}{x^{μ}} k_{2} (x, t) u (t) d t = f (x), x \in [a, b] .

…”

Section: Discussionmentioning

confidence: 99%

“…Numerical results in Section 4 verify our theoretical findings. Moreover, in the last Section, some interesting models including linear Volterra integro‐differential equations of the first order 14 and the second order, 15 pantograph type delay differential equations, 32 nonlinear fractional integro‐differential equations, 33 noncompact Volterra integral equations, 16,34 integral algebraic equations, 35 and Volterra integro‐differential algebraic equations 36 are introduced as future works that can be solved by our proposed least‐squares based reproducing kernel method.…”

Section: Introductionmentioning

confidence: 99%

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A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

Jing

Tohidi

et al. 2021

Math Methods in App Sciences

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Based on the least‐squares method, we proposed a new algorithm to obtain the solution of the second kind of regular and weakly singular Volterra‐Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in detail. Numerical experiments verify the theoretical findings. Meanwhile, this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non‐smooth solutions are also considered, and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR.

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“…In addition, with some modifications in our novel and accurate method, noncompact Volterra‐Fredholm integral equations can be considered in details both numerically and theoretically 16,34 :

u (x) - {true\int}_{a}^{x} \frac{t^{μ - 1}}{x^{μ}} k_{1} (x, t) u (t) d t - {true\int}_{a}^{b} \frac{t^{μ - 1}}{x^{μ}} k_{2} (x, t) u (t) d t = f (x), x \in [a, b] .

…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

Jing

Tohidi

et al. 2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Moreover, the discrete Galerkin method, for solving FIDEs with weakly singular kernels, was provided by Pedas and Tamme [17]. However, although many other attempts have been considered for solving weakly singular integral equations, but only a few explore both numerical discussions and theoretical analysis such as [14,15,23].…”

Section: Introductionmentioning

confidence: 99%

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

Rza̧dkowski

Tohidi

2019

Filomat

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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.

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Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels

Tang

Tohidi

2020

Comp. Appl. Math.

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Convergence analysis of spectral collocation methods for a class of weakly singular Volterra integral equations

Cited by 6 publications

References 20 publications

A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

A new least‐squares‐based reproducing kernel method for solving regular and weakly singular Volterra‐Fredholm integral equations with smooth and nonsmooth solutions

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

Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels

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