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Laeli Dastjerdi, H.; Nili Ahmadabadi, M.

The numerical solution of nonlinear two-dimensional Volterra-Fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis. (English) Zbl 1411.65167

Appl. Math. Comput. 293, 545-554 (2017).
Summary: In this paper, we present a numerical method for solving two-dimensional nonlinear Volterra-Fredholm integral equations of the second kind. The method approximates the solution by the discrete collocation method based on radial basis functions (RBFs) constructed on a set of disordered data. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Numerical examples which compare the proposed method with 2D-TFs method [E. Babolian et al., Comput. Math. Appl. 60, No. 6, 1711–1722 (2010; Zbl 1202.65168)] approve its supremacy in terms of accuracy and computational cost. Using various RBFs we have concluded that MQ-RBF is the best choice for the proposed method.

Cited in 9 Documents

MSC:

65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45D05 Volterra integral equations

Keywords:

two-dimensional problems; Volterra-Fredholm integral equations; radial basis functions; numerical method

Citations:

Zbl 1202.65168
Cite Review PDF
Full Text: DOI

References:

[1] Abdou, M. A.; Badr, A. A.; Soliman, M. B., On a method for solving a two-dimensional nonlinear integral equation of the second kind, J. Comput. Appl. Math., 235, 12, 3589-3598 (2011) · Zbl 1221.65325
[2] Hosseini, S. A.; Shahmorad, S.; Tari, A., Existence of an \(l^p\)-solution for two dimensional integral equations of the Hammerstein type, Bull. Iranian Math. Soc., 40, 4, 851-862 (2014) · Zbl 1337.45001
[3] Wendland, H., Scattered Data Approximation (2005), Cambridge University Press · Zbl 1075.65021
[4] Babolian, E.; Maleknejad, K.; Roodaki, M.; Almasieh, H., Two-dimensional triangular functions and their applications to nonlinear 2d Volterra-Fredholm integral equations, Comput. Math. Appl., 60, 6, 1711-1722 (2010) · Zbl 1202.65168
[5] Vainikko, G. M., A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations, USSR Comput. Math. Math. Phys., 7, 4, 1-41 (1967) · Zbl 0213.16201
[6] Hardy, R. L., Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-1988, Comput. Math. Appl., 19, 8-9, 163-208 (1990) · Zbl 0692.65003
[7] Kansa, E. J., Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I. surface approximations and partial derivative estimates, Comput. Math. Appl., 19, 8-9, 127-145 (1990) · Zbl 0692.76003
[8] Kansa, E. J., Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. II. solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19, 8-9, 147-161 (1990) · Zbl 0850.76048
[9] Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for Burgers equation, Appl. Math. Comput., 95, 37-50 (1998) · Zbl 0943.65101
[10] Hon, Y. C.; Cheung, K. F.; Mao, X. Z.; Kansa, E. J., Multiquadric solution for shallow water equations, ASCE J. Hydraul. Eng., 125, 524-533 (1999)
[11] Zerroukat, M.; Power, H.; Chen, C. S., A numerical method for heat transfer problem using collocation and radial basis functions, Int. J. Numer. Meth. Eng., 42, 1263-1278 (1992) · Zbl 0907.65095
[12] O’Regan, D., Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192, 705-726 (1995) · Zbl 0851.45003
[13] Tricomi, F. G., Integral equations (1982), Dover: Dover New york
[14] Atkinson, K., The Numerical Solution of Integral Equations of the Second Kind (1997), Cambridge University Press: Cambridge University Press New York · Zbl 0899.65077
[15] Brunner, H.; Ningning, Y., Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Numer. Math., 101, 1, 1-27 (2005) · Zbl 1076.65057
[16] Han, G.; Wang, J., Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equation, Appl. Math. Comput., 112, 49-61 (2000) · Zbl 1023.65140
[17] Atkinson, K. E., The numerical evaluation of fixed points for completely continuous operators, SIAM J. Numer. Anal., 10, 799-807 (1973) · Zbl 0237.65040
[18] Han, G.; Wang, J., Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations, J. Comput. Appl. Math., 134, 259-268 (2001) · Zbl 0989.65150
[19] Das, P.; Nelakanti, G.; Long, G., Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations, J. Comput. Appl. Math., 278, 293-305 (2015) · Zbl 1304.65266
[20] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions II, Math. Comp., 54, 211-230 (1990) · Zbl 0859.41004
[21] Hosseini, S. A.; Shahmorad, S.; Talati, F., A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations, Numer. Algor., 68, 511-529 (2015) · Zbl 1317.65252
[22] Dastjerdi, H. L.; Ghaini, F. M., Numerical solution of Volterra-Fredholm integral equations by moving least square method and Chebyshev polynomials, Appl. Math. Model., 36, 3283-3288 (2012) · Zbl 1252.65212
[23] Assari, P.; Adibi, H.; Dehghan, M., A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis, J. Comput. Appl. Math., 239, 72-92 (2013) · Zbl 1255.65233
[24] Mirzaei, D.; Dehghan, M., A meshless based method for solution of integral equations, Appl. Numer. Math., 60, 245-262 (2010) · Zbl 1202.65174
[25] Yang, X.-J.; Baleanu, D.; Lazarevi, M. P.; Caji, M. S., Fractal boundary value problems for integral and differential equations with local fractional operators, Therm. Sci., 19, 959-966 (2015)
[26] Ma, X.-J.; Srivastava, H. M.; Baleanu, D.; Yang, X.-J., A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations, Math. Probl. Eng., 2013, 1-6 (2013) · Zbl 1296.65193
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