Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.
References
[1]
Davis, H.T. (1962) Introduction to Nonlinear Differential and Integral Equations. Dover, Publications, New York.
[2]
Jerri, A. (1999) Introduction to Integral Equations with Applications. Wiley, New York.
[3]
Linz, P. (1985) Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970852
[4]
Miller, R.K. (1967) Nonlinear Volterra Integral Equations. W. A. Benjamin, Menlo Park.
[5]
Wazwaz, A.M. (1997) A First Course in Integral Equations. World Scientific, Singapore City. http://dx.doi.org/10.1142/3444
[6]
Wazwaz, A.M. (2009) Partial Differential Equations and Solitary Waves Theory. Higher Education, Beijing, and Springer, Berlin. http://dx.doi.org/10.1007/978-3-642-00251-9
[7]
Wazwaz, A.M. (2011) Linear and Nonlinear Integral Equations: Methods and Applications. Higher Education, Beijing, and Springer, Berlin. http://dx.doi.org/10.1007/978-3-642-21449-3
[8]
Duan, J.S. and Rach, R. (2011) A New Modification of the Adomian Decomposition Method for Solving Boundary Value Problems for Higher Order Nonlinear Differential Equations. Applied Mathematics and Computation, 218, 4090-4118. http://dx.doi.org/10.1016/j.amc.2011.09.037
[9]
Daftardar-Gejji, V. and Jafari, H. (2006) An Iterative Method for Solving Nonlinear Functional Equations. Journal of Mathematical Analysis and Application, 316, 753-763. http://dx.doi.org/10.1016/j.jmaa.2005.05.009
[10]
Maleknejad, K. and Najafi, E. (2011) Numerical Solution of Nonlinear Volterra Integral Equations Using the Idea of Quasilinearization. Communications in Nonlinear Science and Numerical Simulation, 16, 93-100. http://dx.doi.org/10.1016/j.cnsns.2010.04.002
[11]
Brunner, H. (2004) Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511543234
[12]
Brunner, H., Pedas, A. and Vainikko, G. (2001) A Spline Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels. BIT Numerical Mathematics, 41, 891-900. http://dx.doi.org/10.1023/A:1021920724315
[13]
Brunner, H., Pedas, A. and Vainikko, G. (2001) Piecewise Polynomial Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels. SIAM Journal on Numerical Analysis, 39, 957-982. http://dx.doi.org/10.1137/S0036142900376560
[14]
Te Riele, H.J.J. (1982) Collocation Methods for Weakly Singular Second-Kind Volterra Integral Equations with Non-Smooth Solution. IMA Journal of Numerical Analysis, 2, 437-449. http://dx.doi.org/10.1093/imanum/2.4.437
[15]
Atkinson, K.E., Han, W. and Stewart, D. (2009) Numerical Solution of Ordinary Differential Equations. John Wiley & Sons, Hoboken. http://dx.doi.org/10.1002/9781118164495
[16]
Fazeli, S., Hojjati, G. and Shahmorad, S. (2012) Super Implicit Multistep Collocation Methods for Nonlinear Volterra Integral Equations. Mathematical and Computer Modelling, 55, 590-607. http://dx.doi.org/10.1016/j.mcm.2011.08.034
[17]
Ketabchi, R., Mokhtari, R. and Babolian, E. (2015) Some Error Estimates for Solving Volterra Integral Equations by Using the Reproducing Kernel Method. Journal of Computational and Applied Mathematics, 273, 245-250. http://dx.doi.org/10.1016/j.cam.2014.06.016
[18]
Saberi-Nadjafi, J., Mehrabinezhad, M. and Akbari, H. (2012) Solving Volterra Integral Equations of the Second Kind by Wavelet-Galerkin Scheme. Computers & Mathematics with Applications, 63, 1536-1547. http://dx.doi.org/10.1016/j.camwa.2012.03.043
[19]
Yousefi, S.A. (2006) Numerical Solution of Abel’s Integral Equation by Using Legendre Wavelets. Applied Mathematics and Computation, 175, 574-580. http://dx.doi.org/10.1016/j.amc.2005.07.032
[20]
Wazwaz, A.M., Rach, R. and Duan J.S. (2013) Adomian Decomposition Method for Solving the Volterra Integral Form of the Lane-Emden Equations with Initial Values and Boundary Conditions. Applied Mathematics and Computation, 219, 5004-5019. http://dx.doi.org/10.1016/j.amc.2012.11.012
[21]
Costarelli, D. and Spigler, R. (2013) Solving Volterra Integral Equations of the 2nd Kind by Sigmoidal Functions Approximations. Journal of Integral Equations and Applications, 25, 193-222. http://dx.doi.org/10.1216/JIE-2013-25-2-193
[22]
Costarelli, D. and Spigler, R. (2014) A Collocation Method for Solving Nonlinear Volterra Integro-Differential Equations of the Neutral Type by Sigmoidal Functions. Journal of Integral Equations and Applications, 26, 15-52. http://dx.doi.org/10.1216/JIE-2014-26-1-15
[23]
Maleknejad, K., Hashemizadeh, E. and Ezzati, R. (2011) A New Approach to the Numerical Solution of Volterra Integral Equations by Using Bernstein’s Approximation. Communications in Nonlinear Science and Numerical Simulation, 16, 647-655. http://dx.doi.org/10.1016/j.cnsns.2010.05.006
[24]
Mathews, J. (1989) Symbolic Computational Algebra Applied to Picard Iteration. Mathematics and Computer Education, 23, 117-122.
[25]
Parker, G.E. and Sochacki, J.S. (1996) Implementing the Picard Iteration. Neural, Parallel and Scientific Computations, 4, 97-112.
[26]
Bailey, P.B., Shampine, L.F. and Waltman, P.E. (1968) Nonlinear Two Point Boundary Value Problems. Academic, New York/London.
