An Efficient Operational Matrix Method for Solving a Class of Two-Dimensional Singular Volterra Integral Equations

Year 2018, Volume: 1 Issue: 3, 192 - 201, 30.12.2018

Somayeh Nemati

https://doi.org/10.33187/jmsm.423059

Abstract

In this paper, we consider a spectral method to solve a class of two-dimensional singular Volterra integral equations using some basic concepts of fractional calculus. This method uses a modification of hat functions for finding a numerical solution of the considered equation. Some properties of the modification of hat functions are presented. The main contribution of this work is to introduce the fractional order operational matrix of integration for the considered basis functions. Making use of the Riemann-Liouville fractional integral operator helps us to reduce the main problem to a system of linear algebraic equations which can be solved easily. After that, error analysis of the method is discussed. Finally, numerical examples are included to confirm the accuracy and applicability of the suggested method.

Keywords

Two-dimensional singular Volterra integral equations, Riemann-Liouville integral operator, Modification of hat functions, Operational matrix, Two-dimensional singular Volterra integral equations

References

• [1] R. Estrada, R.P. Kanwal, Singular Integral Equations, Birkh¨auser Basel, 2000.
• [2] N.I. Ioakimidis, On the quadrature methods for the numerical solution of singular integral equations, J. Comput. Appl. Math., 8 (1982), 81–86.
• [3] A. Gerasoulis, R.P. Srivastav, A method for the numerical solution of singular integral equations with a principal value integral, Int. J. Eng. Sci., 19 (1981), 1293–1298.
• [4] R. P. Srivastav, Numerical solution of sigular integral equations using Gauss-type formulae II: quadrature and collocation on Chebyshev nodes, IMA J. Numer. Anal., 3 (1983), 305–318.
• [5] G. Monegato, L. Scuderi, High order methods for weakly singular integral equations with nonsmooth input functions, Math. Comput., 67 (1998), 1493–1515.
• [6] G. C. Sih (Ed.), Methods of analysis and solutions of crack problems, Chapter 7, 368-425, Springer Netherlands, 1973.
• [7] B. N. Mandal, A. Chakrabarti, Applied Singular Integral Equations, CRC Press, 2011.
• [8] P.K. Kythe, P. Puri, Computational Methods for Linear Integral Equations, Birkhauser Basel, 2002.
• [9] M.L. Dow, D. Elliott, The numerical solution of singular integral equations over (-1;1), SIAM J. Numer. Anal., 16 (1979), 115–134.
• [10] D. Berthold, P. Junghanns, New error bounds for the quadrature method for the solution of Cauchy singular integral equations, SIAM J. Numer. Anal., 30 (1993), 1351–1372.
• [11] R. P. Srivastav, F. Zhang, Solving Cauchy singular integral equations by using general quadrature-collocation nodes, Comput. Math. Appl., 21 (1991), 59–71.
• [12] V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (1999), 763–775.
• [13] P. Assari, H. Adibi, M. Dehghan, 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.
• [14] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161–208.
• [15] R. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210.
• [16] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
• [18] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [19] A. Loverro, Fractional Calculus: History, Definitions and Applications for Engineer, USA: Department of Aerospace and Mechanical Engineering, University of Notre Dame, 2004.
• [20] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
• [21] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
• [22] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2011.
• [23] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2012.
• [24] H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, C. M. Khalique, Application of Legendre wavelets for solving fractional differential equations, Comput. Math. Appl., 62 (2011), 1038–1045.
• [25] F. Mirzaee, E. Hadadiyan, Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions, Appl. Math. Comput., 250 (2015), 805–816.
• [26] F. Mirzaee, E. Hadadiyan, Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions, J. Comput. Appl. Math., 302 (2016), 272–284.
• [27] F. Mirzaee, E. Hadadiyan, Numerical solution of Volterra-Fredholm integral equations via modification of hat functions, Appl. Math. Comput., 280 (2016), 110–123.
• [28] F. Mirzaee, E. Hadadiyan, Solving system of linear Stratonovich Volterra integral equations via modification of hat functions, Appl. Math. Comput., 293 (2017), 254–264.
• [29] A. Lotfi, M. Dehghan, S.A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl., 62 (2011), 1055–1067.
• [30] A. H. Bhrawy, A. S. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett., 26 (2013), 25–31.
• [31] Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276–2285.
• [32] M. P. Tripathi, V. K. Baranwal, R. K. Pandey, O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1327–1340.
• [33] A. N. Vityuk, A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil., 7 (2004), 318–325.
• [34] S. Nemati, P. M. Lima, Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput., 327 (2018), 79–92.
• [35] S. Nemati, P. M. Lima, Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math., 242 (2013), 53–69.