Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

Bazm, Sohrab; Babolian, E.

doi:10.1016/j.cnsns.2011.08.017

Cited by 32 publications

(8 citation statements)

References 9 publications

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“…The literature on the numerical solution methods of such equations is fairly extensive [2][3][4][5][6][7][8][9]. But the analysis of computational methods for two-dimensional integral equations seem to have been discussed in only a few papers [10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

Mirzaee

Hadadiyan

2015

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 99%

Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

Mirzaee

Hadadiyan

2015

Applied Mathematics and Computation

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“…The discrete collocation‐type method, Walsh‐Hybrid functions, the collocation method and positive definite functions, the discrete Legendre spectral method, the modified iterated projection method, the Adomian decomposition method, and wavelet method() have adapted to solve one‐dimensional Hammerstein integral equations. The iterated discrete Galerkin method, the iterated collocation method, the Galerkin method with spline functions as basis, the Nystrom method, two‐dimensional rationalized Haar functions, the two‐dimensional differential transform method, the degenerate method, fast collocation methods, piecewise polynomial projection methods, and the Gauss product quadrature rules have been applied to obtain the approximate solution of two‐dimensional Hammerstein integral equations of the second kind.…”

Section: Introductionmentioning

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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“…Integral equations of various types and kinds are playing an important role in branches of mathematical physics (Bai, 2013), mathematical engineering (Assari et al, 2013) and contact problems in the theory of elasticity (Heydari et al, 2013;Li and HuaZou, 2013;Aleksandrov and Covalenko, 1986). Therefore, many different methods are established and used to solve the linear and nonlinear integral equation analytically and numerically (Abdou, 2002;Diogo and Lima, 2008; Anastassiou George and Ali, 2009;Bazm and Babolian, 2012;Biazar et al, 2003;Yu¨zbasi, 2014;Yu¨zbasi et al, 2011;Toutounian and Tohidi, 2013).…”

Section: Introductionmentioning

confidence: 99%

Solution of mixed integral equation in position and time using spectral relationships

Abdou

Basseem

2017

Journal of the Association of Arab Universities for Basic and A

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In this article, the existence of a unique solution of Fredholm-Volterra integral equation of the second kind is guaranteed. The Fredholm integral term is assumed in position with bad kernel, while the Volterra integral term is considered in time with continuous kernel. Under certain conditions and new discussions, the bad kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main theorem of spectral relationships of Fredholm integral equation of the first kind with logarithmic kernel multiplying by a smooth kernel is stated and used to obtain numerically the Fredholm-Volterra integral equation of the second kind. Finally, numerical results are obtained and the error, in each case, is computed.

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Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

Cited by 32 publications

References 9 publications

Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

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

Solution of mixed integral equation in position and time using spectral relationships

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