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

Abstract

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.

A new algorithm based on least-squares method for fractional integro-differential equations

“…Cui and Lin did some pioneering works for operator equations by RKM in [9]- [11]. Then RKM has been chosen as an alternative technique to solve different types of differential equations and integral equations [12]- [17]. Recently, Li 0 E-mail address: xumq9@mail2.sysu.edu.cn in [18] presented an RK collocation method for nonlocal fractional BVPs.…”

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

“…Approximate methods using block-pulse functions [52,61,67], Bernstein polynomials [59], Haar wavelets [4], hyperbolic basis functions [84] and Hosoya polynomials [85] have also been presented. Other successful methods that have been reported are the reproducing kernel methods [14,78,88], Tau methods [50,55], modified hat functions method [56], optimal control method [28], scaling function interpolation wavelet method [39], hybrid contractive mapping and parameter continuation method [72], sinusoidal basis functions and a neural network approach [74]. Finally, existence and uniqueness results are considered in [6][7][8][9]11,20,83,91].…”