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The numerical solution of the non-linear integro-differential equations based on the meshless method. (English) Zbl 1243.65154
The paper is concerned with numerical solution of nonlinear integro-differential equations based on a moving least squares method. The material is presented as follows: the authors introduce the moving least squares method and then show how it can be applied to problems of Fredholm type. They provide a detailed error analysis. This is repeated for the Volterra-type problem. The paper concludes with some numerical examples that demonstrate the efficacy of the approach.

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
45G10 Other nonlinear integral equations
47G20 Integro-differential operators
45B05 Fredholm integral equations
45J05 Integro-ordinary differential equations
Full Text: DOI
[1] Akyüz, A.; Sezer, M., A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form, Int. J. comput. math., 84, 527-539, (2007) · Zbl 1118.65129
[2] Bülbül, B.; Sezer, M., Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients, Int. J. comput. math., 88, 533-544, (2011) · Zbl 1211.65131
[3] Streltsov, I.P., Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations, Comput. phys. comm., 126, 178-181, (2000) · Zbl 0963.65143
[4] Maleknejad, K.; Tavassoli Kajani, M., Solving second kind integral equation by Galerkin methods with hybrid Legendre and block – pulse functions, Appl. math. comput., 145, 623-629, (2003) · Zbl 1101.65323
[5] Bildik, N.; Konuralp, A.; Yalcı̧nbaş, S., Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations, Comput. math. appl., 59, 1909-1917, (2010) · Zbl 1189.65307
[6] Wang, S.H.; He, J.H., Variational iteration method for solving integro-differential equations, Phys. lett. A, 367, 188-191, (2007) · Zbl 1209.65152
[7] Ramos, J.I., Iterative and non-iterative methods for non-linear Volterra integro-differential equations, Appl. math. comput., 214, 287-296, (2009) · Zbl 1202.65179
[8] Feldstein, A.; Sopka, J.R., Numerical methods for nonlinear Volterra integro-differential equations, SIAM J. numer. anal., 11, 826-846, (1974) · Zbl 0291.65029
[9] Tavassoli Kajani, M.; Ghasemi, M.; Babolian, E., Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Appl. math. comput., 180, 569-574, (2006) · Zbl 1102.65137
[10] Atkinson, K., The numerical solution of integral equations of the second kind, (1997), Cambridge University Press
[11] Wazwaz, A.M., A first course in integral equations, (1997), World Scientific Publishing
[12] Lancaster, P.; Salkauskas, K., Surface generated by moving least squares methods, Math. comp., 37, 41-58, (1981) · Zbl 0469.41005
[13] Levin, D., The approximation power of moving least-squares, Math. comp., 67, 17-31, (1998)
[14] Breitkopf, P.; Naceur, H.; Rassineux, A.; Villon, P., Moving least squares response surface approximation: formulation and metal forming applications, Comput. struct., 83, 11-28, (2005)
[15] Bucher, C.; Macke, M.; Most, T., Approximate response functions in structural reliability analysis, ()
[16] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the FEM: diffuse approximations and diffuse elements, Comput. mech., 10, 7-18, (1992) · Zbl 0764.65068
[17] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Internat. J. numer. methods engrg., 37, 29-56, (1994) · Zbl 0796.73077
[18] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 17-27, (1998) · Zbl 0932.76067
[19] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. comput. simul., 79, 700-715, (2008) · Zbl 1155.65379
[20] Dehghan, M.; Mirzaei, D., Meshless local petrov – galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. numer. math., 59, 1043-1058, (2009) · Zbl 1159.76034
[21] Dehghan, M.; Ghesmati, A., Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation, Eng. anal. bound. elem., 34, 324-336, (2010) · Zbl 1244.65147
[22] Dehghan, M.; Salehi, R., A boundary-only meshless method for numerical solution of the eikonal equation, Comput. mech., 47, 283-294, (2011) · Zbl 1398.65324
[23] Dehghan, M.; Salehi, R., The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas, Comput. phys. comm., 182, 2540-2549, (2011) · Zbl 1263.76047
[24] Tatari, M.; Dehghan, M., On the solution of the non-local parabolic partial differential equations via radial basis functions, Appl. math. model., 33, 1729-1738, (2009) · Zbl 1168.65403
[25] Mirzaei, D.; Dehghan, M., A meshless based method for solution of integral equations, Appl. numer. math., 60, 245-262, (2010) · Zbl 1202.65174
[26] Shepard, D., A two-dimensional interpolation function for irregularly spaced points, (), 517-524
[27] McLain, D.H., Drawing contours from arbitary data points, Computing, 17, 318-324, (1974)
[28] McLain, D.H., Two dimensional interpolation from random data, Computing, 19, 178-181, (1976) · Zbl 0321.65009
[29] Franke, R.; Nielson, G., Smooth interpolation of large sets of scattered data, Internat. J. numer. methods engrg., 15, 1691-1704, (1980) · Zbl 0444.65011
[30] Armentano, M.G.; Durán, R.G., Error estimates for moving least square approximations, Appl. numer. math., 37, 397-416, (2001) · Zbl 0984.65096
[31] Armentano, M.G., Error estimates in Sobolev spaces for moving least square approximations, SIAM J. numer. anal., 39, 38-51, (2001) · Zbl 1001.65014
[32] Zuppa, C., Error estimates for moving least square approximations, Bull. braz. math. soc. (N.S.), 34, 231-249, (2003) · Zbl 1056.41007
[33] Li, X.; Zhu, J., A Galerkin boundary node method and its convergence analysis, J. comput. appl. math., 230, 314-328, (2009) · Zbl 1189.65291
[34] Zuppa, C., Good quality point sets and error estimates for moving least square approximations, Appl. numer. math., 47, 575-585, (2003) · Zbl 1040.65034
[35] Babuška, I.; Banerjee, U.; Osborn, J.E.; Zhang, Q., Effect of numerical integration on meshless methods, Comput. methods appl. mech. engrg., 198, 2886-2897, (2009) · Zbl 1229.65204
[36] Kambo, N.S., Error of the newton – cotes and gauss – legendre quadrature formulas, Math. comp., 24, 261-269, (1970) · Zbl 0264.65018
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