Numerical Scheme Based on Operational Matrices for Integro-Differential Equations

  • Authors

    • C. Singh
    • A. K. Singh
    • J. K. Sahoo
    2018-12-19
    https://doi.org/10.14419/ijet.v7i4.41.24298
  • Legendre wavelets, Bernstein polynomials, Operational matrices, Singular voltera integro-differential equations.

  • Abstract

    An effective numerical tool based on wavelets and orthogonal polynomials are presented for the solution of a class of system of singular Voltera integro-differential equations (SSVIDEs). We also presented the convergence analysis for the derivative of the approximation in terms of Legendre wavelets. In this paper, we propose a numerical wavelet and polynomial methods for solving SSVIDEs of second kind. The method is based on the operational and almost operational matrix of integration based on wavelet and orthogonal polynomials. We use the concept of operation matrix of integration to convert the main problem into linear system of algebraic equations. Some numerical examples along with error evaluation are given to illustrate the accuracy and efficiency of the proposed method. The advantage of the proposed technique is computationally most simple, low cost of setting the algebraic equations without using artificial smoothing factors.

     

     

  • References

    1. [1] Amari S., Dynamics of pattern formation in lateral inhibition type neural fields, Biological Cybernetics 1977; 27: 77-87.

      [2] Gutkin, B., Pinto, D. and Ermentrout B. Mathematical neuroscience: from neurons to circuits to systems, J. Physiol. Paris 2003; 97: 209-219.

      [3] Vineet K. Singh, Om P. Singh, and Rajesh K. Pandey, Almost Bernstein Operational Matrix Method for Solving System of Volterra Integral Equations of Convolution Type, Nonlinear Science Letters A, 2010; 1: 201-206.

      [4] O. P Singh, V. K. Singh, and R. K. Pandey, A new stable algorithm for Abel inversion using Bernstein polynomials, International Journal of Nonlinear Sciences and Numerical Simulation 2009; 10: 891-896.

      [5] K. Maleknejad and F. Mirzaee, Solving linear integro-differential equations system by using rationalized Haar functions method, Int. J. Computer math.2003: 1-9.

      [6] K. Maleknejad and M. Tavassoli Kajani, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 2004 ; 159: 603-612.

      [7] K. Maleknejad and M. Shahrezaee, Using Runge-Kutta method for numerical solution of the system of Volterra integral equation, Appl. Math. Comput.2004; 149: 399-410.

      [8] K. Maleknejad and A. Salimi Shamloo, Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Appl. Math. Comput. 2008; 195: 500-505.

      [9] Pinto, D. and Ermentrout, B., Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses, SIAM J. Appl. Math 2001; 62 :206-225

      [10] Pinto, D. and Ermentrout, B. Spatially structured activity in synaptically coupled neuronal networks: II. lateral inhibition and standing pulses, SIAM J. Appl. Math 2001; 62: 226-243.

      [11] Wilson, H. R. and Cowan, J. D., Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. j. 1972; 12: 1-24.

      [12] Wilson, H. R. and Cowan, J. D., A mathematical theory of the functional dynamics of control and thalamic nervous tissue, Kybernetik 1973; 13: 55-80.

      [13] A.K. Singh, V. K. Singh, and O. P. Singh, The Betnstein operational matrix of integration. Appl. Math. Sci.2009; 3: 2427-2436.

      [14] O. P. Singh, V. K. Singh, and R. K. Pandey, A stable numerical inversion of Abel’s integral equation using almost Betnstein operational matrix, J. Quant. Spectrosc. Radiat. Transfer 2010; 111: 245-252.

      [15] Vineet K. Singh, and E. B. Postnikov, Operational matrix approach for solution of integro-differential equations arising in theory of anomalous relaxation processes in vicinity of singular point, Appl. Math. Modell 2013; 37: 6609-6616.

      [16] S.A. Yousefi, Numerical solution of Abel’s integral equation by using Legendre wavelets, Appl. Math. Comput.2006;175:574–580.

      [17] M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci.2011; 32: 495–502.

      [18] Richard V. Kadison, Zhe Liu, The Heisenberg Relation - Mathematical, Symmetry, Integrability and Geometry: Methods and applications, 2014; 10: 009-40.

  • Downloads

  • How to Cite

    Singh, C., K. Singh, A., & K. Sahoo, J. (2018). Numerical Scheme Based on Operational Matrices for Integro-Differential Equations. International Journal of Engineering & Technology, 7(4.41), 50-54. https://doi.org/10.14419/ijet.v7i4.41.24298

    Received date: 2018-12-18

    Accepted date: 2018-12-18

    Published date: 2018-12-19