Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Babolian, E.; Bazm, Sohrab; Lima, Pedro M.

doi:10.1016/j.cnsns.2010.05.029

Cited by 70 publications

(40 citation statements)

References 17 publications

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“…In Table 10, we compare the maximum absolute error and measured CPU time(s) for each of the present method, block-pulse functions method [45] and rationalized Haar functions method [44]. This fact is obvious from Table 10 that the results obtained by the present method is better than that obtained in [44] and [45]. f (x, y) = x + y − 1 12…”

Section: Example 4 Consider the Following Two-dimensional Nonlinear Fmentioning

confidence: 54%

Application of two-dimensional hat functions for solving space-time integral equations

Mirzaee

Hadadiyan

2015

J. Appl. Math. Comput.

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In this paper, we introduce two-dimensional hat functions and derive operational matrix of integration of these functions. Then, we utilize them to solve some classes of integral equations. The method is based upon expanding functions as their truncated hat functions. Also, an error analysis is provided under several mild conditions. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the method.

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Section: Example 4 Consider the Following Two-dimensional Nonlinear Fmentioning

confidence: 54%

Application of two-dimensional hat functions for solving space-time integral equations

Mirzaee

Hadadiyan

2015

J. Appl. Math. Comput.

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“…where the coefficients {c 1 , … , c N } are determined by solving the nonlinear system that is obtained by replacing the expansion (21) with u(x, t) and pick distinct the node points in the integral equation (20). The iteration methods, such as Newton's method for solving these types of cumbersome nonlinear systems, require to compute several definite integrals at each step of the iteration, and so they are usually sensitive to the selection of initial guess.…”

Section: Solution Of Integral Equationsmentioning

confidence: 99%

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Assari

Asadi‐Mehregan

2018

Int J Numerical Modelling

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The main purpose of this article is to present a numerical method for solving two-dimensional Fredholm-Hammerstein integral equations of the second kind with weakly singular kernels. The scheme utilizes locally supported (inverse) multiquadric functions constructed on scattered points as a basis in the discrete collocation method. The local (inverse) multiquadrics estimate a function in any dimensions via a small set of data instead of all points in the solution domain. The proposed method uses a special accurate quadrature formula based on the nonuniform Gauss-Legendre integration rule for approximating singular integrals appeared in the scheme. In comparison with the globally supported (inverse) multiquadric for the numerical solution of integral equations, the proposed method is stable and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is examined over several two-dimensional Hammerstein integral equations and obtained results confirm the theoretical error estimates. KEYWORDS discrete collocation method, Hammerstein integral equation, local (inverse) multiquadric, meshless method, weakly singular kernel Int J Numer Model. 2019;32:e2488.wileyonlinelibrary.com/journal/jnm

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“…Therefore, in recent years, several numerical approaches have been proposed. For instance, a reduced differential transform method, a method using radial basis functions, a numerical method based on Haar wavelet, 2‐dimensional orthogonal triangular functions, and the 2‐dimensional rationalized Haar can be used to approximate the solution, in approximation method based on a polynomial system . There are furthermore a spectral meshless radial point interpolation method, a method based on developing the 2‐dimensional differential transform, a discrete Galerkin and iterated Galerkin method, an integral mean‐value method, a piecewise interpolating polynomial technique (a 1‐, 2‐, and 3‐dimensional modification of the hat functions method, respectively), a 2‐dimensional triangular orthogonal functions method, a numerical scheme based on the moving least squares method, an operational matrix and 2‐dimensional block pulse functions methods, a method using Legendre polynomials, an iterative numerical method of successive approximations, and He variational method …”

Section: Introductionmentioning

confidence: 99%

Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

Berenguer

Gámez

2018

Math Methods in App Sciences

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Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Cited by 70 publications

References 17 publications

Application of two-dimensional hat functions for solving space-time integral equations

Application of two-dimensional hat functions for solving space-time integral equations

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

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