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A MESHLESS LOCAL GALERKIN INTEGRAL EQUATION METHOD FOR SOLVING A TYPE OF DARBOUX PROBLEMS BASED ON RADIAL BASIS FUNCTIONS

Part of: Integral equations, integral transforms

Published online by Cambridge University Press:  09 November 2021

P. ASSARI Open the ORCID record for P. ASSARI [Opens in a new window] ,
F. ASADI-MEHREGAN Open the ORCID record for F. ASADI-MEHREGAN [Opens in a new window]  and
M. DEHGHAN Open the ORCID record for M. DEHGHAN [Opens in a new window]
P. ASSARI*
Affiliation:
Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan65178, Iran; e-mail: f.asadi@sci.basu.ac.ir.
F. ASADI-MEHREGAN
Affiliation:
Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan65178, Iran; e-mail: f.asadi@sci.basu.ac.ir.
M. DEHGHAN
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave, Tehran15914, Iran; e-mail: mdehghan@aut.ac.ir.
*
e-mail: passari@basu.ac.ir
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Abstract

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions.

Keywords

Darboux equationVolterra integral equationlocal radial basis functionsdiscrete Galerkin methodmeshless methoderror analysis

MSC classification

Primary: 65R20: Integral equations
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 4 , October 2021 , pp. 469 - 492
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Assari, P., Adibi, H. and 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 (2013) 72–92; doi:10.1016/j.cam.2012.09.010.CrossRefGoogle Scholar
Assari, P., Adibi, H. and Dehghan, M., “The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis”, Appl. Numer. Math. 81 (2014) 76–93; doi:10.1016/j.apnum.2014.02.013.CrossRefGoogle Scholar
Assari, P., Asadi-Mehregan, F. and Dehghan, M., “On the numerical solution of Fredholm integral equations utilizing the local radial basis function method”, Int. J. Comput. Math. 96 (2018) 1416–1443; doi:10.1080/00207160.2018.1500693.CrossRefGoogle Scholar
Assari, P. and Dehghan, M., “The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision”, Eng. Comput. 33 (2017) 853–870; doi:10.1007/s00366-017-0502-5.CrossRefGoogle Scholar
Assari, P. and Dehghan, M., “A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions”, Appl. Math. Comput. 315 (2017) 424–444; doi:10.1016/j.amc.2017.07.073.Google Scholar
Atkinson, K. E., The numerical solution of integral equations of the second kind (Cambridge University Press, Cambridge, 1997); doi:10.1017/CBO9780511626340.CrossRefGoogle Scholar
Atkinson, K. E. and Bogomolny, A., “The discrete Galerkin method for integral equations”, Math. Comp. 48 (1987) 595–616; doi:10.1090/S0025-5718-1987-0878693-6.CrossRefGoogle Scholar
Babolian, E., Bazm, S. and Lima, P., “Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1164–1175; doi:10.1016/j.cnsns.2010.05.029.CrossRefGoogle Scholar
Berenguer, M. I. and Gamez, D., “A computational method for solving a class of two dimensional Volterra integral equations”, J. Comput. Appl. Math. 318 (2017) 403–410; doi:10.1016/j.cam.2016.05.028.CrossRefGoogle Scholar
Brunner, H. and Kauthen, J. P., “The numerical solution of two-dimensional Volterra integral equations by collocation and iterated collocation”, IMA J. Numer. Anal. 9 (1989) 47–59; doi:10.1093/imanum/9.1.47.CrossRefGoogle Scholar
Buhmann, M. D., Radial basis functions: theory and implementation (Cambridge University Press, Cambridge, 2003); doi:10.1017/CBO9780511543241.CrossRefGoogle Scholar
Chandhini, G., Prashanthi, K. S. and Vijesh, V. A., “A radial basis function method for fractional Darboux problems”, Eng. Anal. Bound. Elem. 86 (2018) 1–18; doi:10.1016/j.enganabound.2017.10.001.Google Scholar
Czlapinski, T., “Difference methods for the Darboux problem for functional partial differential equations”, Ann. Polon. Math. 71 (1999) 171–193; doi:10.4064/ap-71-2-171-193.CrossRefGoogle Scholar
Day, J. T., “A Runge–Kutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations”, Comput. J. 9 (1966) 81–83; doi:10.1093/comjnl/9.1.81.CrossRefGoogle Scholar
Dehghan, M. and Abbaszadeh, M., “The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations”, Alexandria Eng. J. 57 (2018) 1137–1156; doi:10.1016/j.aej.2017.02.024.CrossRefGoogle Scholar
Dehghan, M. and Nikpour, A., “Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method”, Appl. Math. Model. 37 (2013) 8578–8599; doi:10.1016/j.apm.2013.03.054.CrossRefGoogle Scholar
Dobner, H., “Bounds for the solution of hyperbolic problems”, Computing 38 (1987) 209–218; doi:10.1007/BF02240096doi:10.1007/BF02240096.CrossRefGoogle Scholar
Evans, D. J. and Sanugi, B., “Numerical solution of the Goursat problem by a nonlinear trapezoidal formula”, Appl. Math. Lett. 1 (1988) 221–223; doi:10.1016/0893-9659(88)90080-8.CrossRefGoogle Scholar
Fang, W., Wang, Y. and Xu, Y., “An implementation of fast wavelet Galerkin methods for integral equations of the second kind”, J. Sci. Comput. 20 (2004) 277–302; doi:10.1023/B:JOMP.0000008723.85496.ce.CrossRefGoogle Scholar
Fasshauer, G. E., “Meshfree methods”, in: Handbook of theoretical and computational nanotechnology (eds. Rieth, M. and Schommers, W.), (American Scientific Publishers, Valencia, CA, 2005).Google Scholar
Fasshauer, G. E. and Zhang, J. G., “On choosing ‘optimal’ shape parameters for RBF approximation”, Numer. Algorithms 45 (2007) 345–368; doi:10.1007/s11075-007-9072-8.CrossRefGoogle Scholar
Gou, K. and Sun, B., “Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions”, Appl. Math. Comput. 217 (2011) 8765–8777; doi:10.1016/j.amc.2011.03.129.Google Scholar
Gourlay, A. R., “A note on trapezoidal methods for the solution of initial value problems”, Math. Comput. 24 (1970) 629–933; doi:10.1090/S0025-5718-1970-0275680-3.CrossRefGoogle Scholar
Jain, M. K. and Sharma, K. D., “Cubature method for the numerical solution of the characteristic initial value problem  ${u}_{\mathrm{xy}} = f(x,y,u,{u}_x,{u}_y)$ ”, J. Austral. Math. Soc. 8 (1968) 355–368; doi:10.1017/S1446788700005425.CrossRefGoogle Scholar
Kaneko, H. and Xu, Y., “Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind”, Math. Comp. 62 (1994) 739–753; doi:10.1090/S0025-5718-1994-1218345-X.CrossRefGoogle Scholar
Kosec, G. and Sarler, B., “Solution of a low Prandtl number natural convection benchmark by a local meshless method”, Int. J. Numer. Methods Heat Fluid Flow 23 (2013) 189–204; doi:10.1108/09615531311289187.CrossRefGoogle Scholar
Lee, C. K., Liu, X. and Fan, S. C., “Local multiquadric approximation for solving boundary value problems”, Comput. Mech. 30 (2003) 396–409; doi:10.1007/s00466-003-0416-5.CrossRefGoogle Scholar
Lees, M., “The Goursat problem”, J. Soc. Ind. Appl. Math. 8 (1960) 518–530; doi:10.1137/0108036.CrossRefGoogle Scholar
Lether, F. G., “On the construction of Gauss–Legendre quadrature rules”, J. Comput. Appl. Math. 4 (1978) 47–52; doi:10.1016/0771-050X(78)90019-0.CrossRefGoogle Scholar
Li, X., “Meshless Galerkin algorithms for boundary integral equations with moving least square approximations”, Appl. Numer. Math. 61 (2011) 1237–1256; doi:10.1016/j.apnum.2011.08.003.CrossRefGoogle Scholar
Li, X. and Dong, H., “Error analysis of the meshless finite point method”, Appl. Math. Comput. 382 (2020) 125326; doi:10.1016/j.amc.2020.125326.Google Scholar
Li, X. and Li, S., “A complex variable boundary point interpolation method for the nonlinear Signorini problem”, Comput. Math. Appl. 79 (2020) 3297–3309; doi:10.1016/j.camwa.2020.01.030.CrossRefGoogle Scholar
Li, X. and Zhu, J., “A Galerkin boundary node method for biharmonic problems”, Eng. Anal. Bound. Elem. 33 (2009) 858–865; doi:10.1016/j.enganabound.2008.11.002.CrossRefGoogle Scholar
Li, X. and Zhu, J., “A meshless Galerkin method for Stokes problems using boundary integral equations”, Comput. Methods Appl. Mech. Engrg. 198 (2009) 2874–2885; doi:10.1016/j.cma.2009.04.009.CrossRefGoogle Scholar
Mavric, B. and Sarler, B., “Local radial basis function collocation method for linear thermoelasticity in two dimensions”, Int. J. Numer. Methods Heat Fluid Flow 25 (2015) 1488–1510; doi:10.1108/HFF-11-2014-0359.CrossRefGoogle Scholar
Mirzaei, D. and Dehghan, M., “A meshless based method for solution of integral equations”, Appl. Numer. Math. 60 (2010) 245–262; doi:10.1016/j.apnum.2009.12.003.CrossRefGoogle Scholar
Mramor, K., Vertnik, R. and Sarler, B., “Simulation of natural convection influenced by magnetic field with explicit local radial basis function collocation method”, CMES Comput. Model. Eng. Sci. 92 (2013) 327–352; doi:10.3970/cmes.2013.092.327.Google Scholar
Nemati, S., Lima, P. M. and Ordokhani, Y., “Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials”, J. Comput. Appl. Math. 242 (2013) 53–69; doi:10.1016/j.cam.2012.10.021.CrossRefGoogle Scholar
Parand, K. and Rad, J. A., “Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions”, Appl. Math. Comput. 218 (2012) 5292–5309; doi:10.1016/j.amc.2011.11.013.Google Scholar
Quarteroni, A., Sacco, R. and Saleri, F., Numerical analysis for electromagnetic integral equations (Artech House, Boston, MA, 2008).Google Scholar
Sarler, B. and Vertnik, R., “Meshfree explicit local radial basis function collocation method for diffusion problems”, Comput. Math. Appl. 51 (2006) 1269–1282; doi:10.1016/j.camwa.2006.04.013doi:10.1016/j.camwa.2006.04.013.CrossRefGoogle Scholar
Sun, J., Yi, H. and Tan, H., “Local radial basis function meshless scheme for vector radiative transfer in participating media with randomly oriented axisymmetric particles”, Appl. Opt. 55 (2016) 1232–1240; doi:10.1364/AO.55.001232.CrossRefGoogle ScholarPubMed
Tari, A., Rahimi, M. Y., Shahmorad, S. and Talati, F., “Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method”, J. Comput. Appl. Math. 228 (2009) 70–76; doi:10.1016/j.cam.2008.08.038.CrossRefGoogle Scholar
Vertnik, R. and Sarler, B., “Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems”, Int. J. Numer. Methods Heat Fluid Flow 16 (2006) 617–640; doi:10.1108/09615530610669148.CrossRefGoogle Scholar
Wazwaz, A. M., “On the numerical solution of the Goursat problem”, Appl. Math. Comput. 59 (1983) 89–95; doi:10.1016/0096-3003(93)90036-E.Google Scholar
Wendland, H., Scattered data approximation (Cambridge University Press, Cambridge, 2005); doi:10.1017/CBO9780511617539.Google Scholar
Yun, D. F. and Hon, Y. C., “Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems”, Eng. Anal. Bound. Elem. 67 (2016) 63–80; doi:10.1016/j.enganabound.2016.03.003.CrossRefGoogle Scholar
Zhang, S., Lin, Y. and Rao, M., “Numerical solutions for second-kind Volterra integral equations by Galerkin methods”, Appl. Math. 45 (2000) 19–39; doi:10.1023/A:1022284616125.CrossRefGoogle Scholar
Zhang, T. and Li, X., “Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems”, Appl. Math. Comput. 380 (2020) 125306; doi:10.1016/j.amc.2020.125306.Google Scholar
Zhang, T. and Li, X., “Variational multiscale interpolating element-free Galerkin method for the nonlinear Darcy–Forchheimer model”, Comput. Math. Appl. 79 (2020) 363–377; doi:10.1016/j.camwa.2019.07.003.CrossRefGoogle Scholar