Abstract
This paper investigates a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The scheme utilizes the shape functions of the moving least squares (MLS) approximation constructed on scattered points as a basis in the discrete collocation method. The MLS methodology is an effective technique for approximating unknown functions which involves a locally weighted least square polynomial fitting. The proposed method is meshless, since it does not need any background mesh or cell structures and so it is independent of the geometry of the domain. The scheme reduces the solution of two-dimensional nonlinear integral equations to the solution of nonlinear systems of algebraic equations. The error analysis of the proposed method is provided. The efficiency and accuracy of the new technique are illustrated by several numerical examples.
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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, 3589–3598 (2011)
Alipanah, A., Esmaeili, S.: Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function. J. Comput. Appl. Math. 235, 5342–5347 (2011)
Armentano, M.G.: Error estimates in Sobolev spaces for moving least square approximations. SIAM J. Numer. Anal. 39, 38–51 (2001)
Armentano, M.G., Duran, R.G.: Error estimates for moving least square approximations. Appl. Numer. Math. 37, 397–416 (2001)
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)
Assari, P., Adibi, H., Dehghan, M.: A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl. Math. Model. 37, 9269–9294 (2013)
Atkinson, K.E.: The Numerical Evaluation of Fixed Points for Completely Continuous Operators. SIAM J. Numer. Anal. 10, 799–807 (1973)
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Atkinson, K.E., Potra, F.A.: Projection and iterated projection methods for nonlinear integral equations. SIAM J. Numer. Anal. 24, 1352–1373 (1987)
Atkinson, K.E., Flores, J.: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13, 195–213 (1993)
Babolian, E., Bazm, S., Lima, P.: Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions. Commun. Nonlinear Sci. Numer. Simul. 16, 1164–1175 (2011)
Babuska, I., Banerjee, U., Osborn, J.: Survey of meshless and generalized finite element methods: a unied approach. Acta Numer. 12, 1–125 (2003)
Bazm, S., Babolian, E.: Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules. Commun. Nonlinear Sci. Numer. Simul. 17, 1215–1223 (2012)
Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)
Boersma, J., Danicki, E.: On the solution of an integral equation arising in potential problems for circular and elliptic disks. SIAM J. Appl. Math. 53, 931–941 (1993)
Bremer, J., Rokhlin, V., Sammis, I.: Universal quadratures for boundary integral equations on two-dimensional domains with corners. J. Comput. Phys. 229, 8259–8280 (2010)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)
Carutasu, V.: Numerical solution of two-dimensional nonlinear Fredholm integral equations of the second kind by spline functions. Gen. Math. 9, 31–48 (2001)
Dehghan, M., Mirzaei, D.: Numerical solution to the unsteady two-dimensional Schrodinger equation using meshless local boundary integral equation method. Int. J. Numer. Methods Eng. 76, 501–520 (2008)
Dehghan, M., Salehi, R.: The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math. 236, 2367–2377 (2012)
Dobner, H.J.: Bounds for the solution of hyperbolic problems. Computing 38, 209–218 (1987)
Fang, W., Wang, Y., Xu, Y.: An implementation of fast wavelet Galerkin methods for integral equations of the second kind. J. Sci. Comput. 20, 277–302 (2004)
Fasshauer, G.E.: Meshfree methods. In: Rieth, M., Schommers, W. (eds.) Handbook of Theoretical and Computational Nanotechnology, vol. 27, pp. 33-97. American Scientific Publishers (2006)
Farengo, R., Lee, Y.C., Guzdar, P.N.: An electromagnetic integral equation: application to microtearing modes. Phys. Fluids 26, 3515–3523 (1983)
Graham, I.G.: Collocation methods for two dimensional weakly singular integral equations. Aust. Math. Soc. Ser. B 22, 456–473 (1981)
Han, G., Wang, J.: Extrapolation method of iterated collocation solution for two-dimensional non-linear Volterra integral equation. Appl. Math. Comput. 112, 49–61 (2000)
Han, G., Wang, J.: Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations. J. Comput. Appl. Math. 134, 259–268 (2001)
Han, G., Wang, J.: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations. J. Comput. Appl. Math. 139, 49–63 (2002)
Hanson, R.L., Phillips, J.L.: Numerical solution of two-dimensional integral equations using linear elements source. SIAM J. Numer. Anal. 15, 113–121 (1978)
Hansen, P.C., Jensen, T.K.: Large-scale methods in image deblurring. Lect. Notes. Comput. Sci. 4699, 24–35 (2007)
Kaneko, H., Xu, Y.: Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math. Comput. 62, 739–753 (1994)
Kress, B.: Linear Integral Equations. Springer, Berlin (1989)
Kumar, S.: A discrete collocation-type method for Hammerstein equations. SIAM J. Numer. Anal. 25, 328–341 (1988)
Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37, 141–158 (1981)
Li, X.F., Rong, E.Q.: Solution of a class of two-dimensional integral equations. J. Comput. Appl. Math. 145, 335–343 (2002)
Lin, Q., Sloan, I.H., Xie, R.: Extrapolation of the iterated collocation method for integral equations of the second kind. SIAM J. Numer. Anal. 27, 1535–1541 (1990)
Manzhirov, A.V.: On a method of solving two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology. J. Appl. Math. Mech. 49, 777–782 (1985)
Mirkin, M.V., Bard, A.J.: Multidimensional integral equations: a new approach to solving microelectrode diffusion problems. J. Electroad. Chem. 323, 29–51 (1992)
Mirzaei, D., Dehghan, M.: A meshless based method for solution of integral equations. Appl. Numer. Math. 60, 245–262 (2010)
Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32, 983–1000 (2012)
Mukherjee, Y.X., Mukherjee, S.: The boundary node method for potential problems. Int. J. Numer. Methods Eng. 40, 797–815 (1997)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, Texts in Applied Mathematics, 2nd edn. Springer, New York (2007)
Radlow, J.: A two-dimensional singular integral equation of diffraction theory. Bull. Amer. Math. Soc. 70, 596–599 (1964)
Rajan, D., Chaudhuri, S.: Simultaneous estimation of super-resolved scene and depth map from low resolution defocused observations. IEEE. T. Pattern. Anal. Mach. Intell. 25, 1102–1117 (2003)
Salehi, R., Dehghan, M.: A moving least square reproducing polynomial meshless method. Appl. Numer. Math. 69, 34–58 (2013)
Salehi, R., Dehghan, M.: A generalized moving least square reproducing kernel method. J. Comput. Appl. Math. 249, 120–132 (2013)
Shepard, D.: A two-dimensional interpolation function for irregularly spaced points. In: Proceedings 23rd National Conference ACM, pp. 517–524. ACM Press, New York (1968)
Sladek, J., Sladek, V., Atluri, S.N.: Local boundary integral equation (LBIE) method for solving problem of elasticity with nonhomogeneous material properties. Comput. Mech. 24, 456–462 (2000)
Tari, A., Rahimi, M.Y., Shahmorad, S., Talati, F.: Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method. J. Comput. Appl. Math. 228, 70–76 (2009)
Weiss, R.: On the approximation of fixed points of nonlinear compact operators. SIAM J. Numer. Anal 11, 550–553 (1974)
Wazwaz, A.M.: Linear and Nonlinear Integral Equations: Methods and Applications. Publisher: Higher Education Press and Springer Verlag (2011)
Wendland, H.: Scattered Data Approximation. Cambridge University Press (2005)
Wendland, H.: Local polynomial reproduction, and moving least squares approximation. IMA J. Numer. Anal. 21, 285–300 (2001)
Zuppa, C.: Error estimates for moving least square approximations. Bull. Braz. Math. Soc, New Series 34, 231–249 (2001)
Zuppa, C.: Good quality point sets error estimates for moving least square approximations. Appl. Numer. Math. 47, 575–585 (2003)
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Assari, P., Adibi, H. & Dehghan, M. A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains. Numer Algor 67, 423–455 (2014). https://doi.org/10.1007/s11075-013-9800-1
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DOI: https://doi.org/10.1007/s11075-013-9800-1
Keywords
- Nonlinear integral equation
- Two-dimensional integral equation
- Moving least squares (MLS) approximation
- Meshless method
- Non-rectangular domain
- Error analysis