A numerical solution of mixed Volterra Fredholm integral equations of Urysohn type on non-rectangular regions using meshless methods

Subject Areas : History and biography

M.  Nili Ahmadabadi ¹ (Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran)
H.  Laeli Dastjerdi ² (Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran)

Received: 2015-10-02 Accepted : 2016-03-11 Published : 2015-12-01

Keywords: collocation method, Mixed Volterra-Fredholm integral equations, Radial basis functions, Meshless method, Numerical treatment,

Abstract :

In this paper, we propose a new numerical method for solution of Urysohn two dimensional mixed Volterra-Fredholm integral equations of the second kind on a non-rectangular domain. The method approximates the solution by the discrete collocation method based on inverse multiquadric radialbasis functions (RBFs) constructed on a set of disordered data. The method is a meshless method, because it is independent of the geometry of the domain and it does not require any background interpolation or approximation cells. The error analysisof the method is provided. Numerical results are presented, which confirm the theoretical prediction of the convergence behavior of the proposed method.

References:

[1] H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005.

[2] W.R. Madych, S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comput. 54 (189) (1990) 211-230.

[3] W.R. Madych, S.A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory 70 (1992) 94-114.

[4] P.J. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math. 56 (1989) 409-424.

[5] H. Brunner, P.J. van der Houwen, The numerical solution of Volterra equations, CWI Monographs, vol. 3, North-Holland, Amsterdam, 1986.

[6] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal. 27 (4) (1990) 987-1000.

[7] J.P. Kauthen, Continuous time collocation method for Volterra-Fredholm integral equations, Numer. Math. 56 (1989) 409-424.

[8] K. Maleknejad, M. Hadizadeh, A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl. 37 (1999) 1-8.

[9] A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput. 127 (2002) 405-414.

[10] E. Banifatemi, M. Razzaghi, S. Yousefi, Two-dimensional Legendre Wavelets Method for the mixed VolterraFredholm integral equations, J. Vibr. Control. 13 (2007) 1667-1675.

[11] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 1905-1915.

[12] H.R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4 (1977) 337-351.

[13] G. Han, L. Zhang, Asymptotic expansion for the trapezoidal Nystrom method of linear VolterraFredholm equations, J. Comput. Appl. Math. 51 (1994) 339-348.

[14] M. Hadizadeh, M. Asgari, An effective numerical approximation for the linear class of mixed integral equations, Appl. Math. Comput. 167 (2005) 1090-1100.

[15] E. Babolian, A.J. Shaerlar, Two dimensional block pulse functions and application to solve Volterra-Fredholm integral equations with Galerkin method, Int. J. Contemp. Math. Sci. 6 (2011) 763-770.

[16] H. Laeli Dastjerdi, F. M. Maalek Ghaini, M. Hadizadeh, A meshless approximate solution of mixed VolterraFredholm integral equations, Int. J. Comput. Math. 90 (2013) 527-538.

[17] R.L. Hardy, Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968- 1988, Comput. Math. Appl. 19 (8-9) (1990) 163-208.

[18] E.J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluiddynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (8-9) (1990) 127-145.

[19] E.J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluiddynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (8-9) (1990) 147-161.

[20] A. Cardone, E. Messina, E. Russo, A fast iterative method for discretized Volterra-Fredholm integral equations, J. Comput. Appl. Math. 189 (2006) 568-579.

[21] Y.C. Hon, X.Z. Mao, An efficient numerical scheme for Burgers equation, Appl. Math. Comput. 95 (1998) 37-50.

[22] Y.C. Hon, K.F. Cheung, X.Z. Mao, E.J. Kansa, Multiquadric solution for shallow water equations, ASCE J. Hydraul. Eng. 125 (1999) 524-533.

[23] M. Zerroukat, H. Power, C.S. Chen, A numerical method for heat transfer problem using collocation and radial basis functions, Int. J. Numer. Meth. Eng. 42 (1992) 1263-1278.

[24] K. E. Atkinson, F. A. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal. 24 (1987) 1352-1373.

[25] K. Atkinson, J. Flores, The discrete collocation method for nonlinear integral equations, IMA J. Numer. Anal. 13 (1993) 195-213.

[26] G.E. Fasshauer, Meshfree methods. In: Rieth, M., Schommers, W. (eds.) Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers, 27 (2006) 33-97.