In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.
Abassy T. A. 2010. Improved Adomian decomposition method. Computers and Mathematics with Applications 59 42–54.
Adomian G. 1988. A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications 135 501–544.
Adomian G. 1990. A review of the decomposition method and some recent results for nonlinear equations. Mathematics Computer and Modelling 13 17–43.
Adomian G. and Rach R. 1996. Modified Adomian polynomials. Mathematics and Computer Modelling 24 39–46.
Babolian E. and Biazar J. 2002. Solution of nonlinear equations by modified Adomian decomposition method. Applied Mathematics and Computation 132 167–172.
Bahuguna D. Ujlayan A. and Pandey D. N. 2009. A comparative study of numerical methods for solving an integro-differential equation. Computers and Mathematics with Applications 57 1485–1493.
Bhatti M.I. and Bracken P. 2007. Solutions of differential equations in a Bernstein polynomial basis. Journal of Computational and Applied Mathematics 205 272–280.
Biazar J. Babolian E. and Islam R. 2004. Solution of the system of ordinary differential equations by Adomian decomposition method. Applied Mathematics and Computation 147 713–719.
Biazar J. Porshokuhi M. G. and Ghanbari B. 2010. Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method. Computers and Mathematics with Applications 59 622–628.
Bildik N. and Deniz S. 2015. Modified Adomian decomposition method for solving Riccati differential equations. Review of the Air Force Academy 30 21–26.
Bohm W. Farin G. and Kahmann J. 1984. A survey of curve and surface methods in cagd. Computer Aided Geometric Design 1 1–60.
Doan N. 2012. Solution of the system of ordinary differential equations by combined Laplace transform Adomian decomposition method. Mathematical and Computational Applications 17 203–211.
Evans D.J. and Raslan K. R. 2004. The Adomian decomposition method for solving delay differential equation. International Journal of Computer Mathematics 00 1–6.
Farouki R. T. 2012. The Bernstein polynomial basis: A centennial retrospective. Computer Aided Geometric Design 29 379–419.
Farouki R. T. and Rajan V. T. 1998. Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design 5 1–26.
Gabet L. 1994. The theoretical foundation of the Adomian method. Computers and Mathematics with Applications 27 41–52.
Ghazanfari B. and Sepahvandzadeh A. 2014. Adomian decomposition method for solving Bratu’s type equation. Journal of Mathematics and Computer Science 8 236–244.
Hashim I. 2006. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. Journal of Computational and Applied Mathematics 193 658–664.
Hosseini M. M. 2006. Adomian decomposition method for solution of nonlinear differential algebraic equations. Applied Mathematics and Computation 181 1737–1744.
Khuri S. A. 2001. A Laplace decomposition algorithm applied to a class of nonlinear differential equation. Journal of Applied Mathematics 1 141–155.
Kumar S. Kumar D. Abbasbandy S. and Rashidi M. M. 2014. Analytic solution of fractional Navier-Stokes equation by using modified Laplace decomposition method. Ain Shams Engineering Journal 5 569–574.
Li C. and Wang Y. 2009. Numerical algorithm based on Adomian decomposition for fractional differential equations. Computers and Mathematics with Applications 57 1672–1681.
Liu Y. 2009. Adomian decomposition method with orthogonal polynomials:Legendre polynomials. Mathematical and Computer Modelling 49 1268–1273.
Luo X.-G. Wu Q.-B. and Zhang B.-Q. 2006. Revisit on partial solutions in the Adomian decomposition method: solving heat and wave equations. Journal of Mathematical Analyis and Applications 321 353–363.
Maleknejad K. Hashemizadeh E. and Ezzati R. 2011. A new approach to the numerical solution of Volterra integral equations by using Bernsteins approximation. Communication in Nonlinear Science and Numerical Simulation 16 647–655.
Maleknejad K. and Najafi E. 2011. Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization. Communication in Nonlinear Science and Numerical Simulation 20 93–100.
Manafianheris J. 2012. Solving the integro-differential equations using the modified Laplace Adomian decomposition method. Journal of Mathematical Extension 6 41–55.
Marwat D. N. K. and Asghar S. 2008. Solution of the heat equation with variable properties by two-step Adomian decomposition method. Mathematical and Computer Modelling 48 83–90.
Quain W. Riedel M. D. and Rosenberg I. 2011. Uniform approximation and Bernstein polynomial with coefficients in the unit interval. European Journal of Combinatorics 32 448–463.
Singh N. and Kumar M. 2011. Adomian decomposition method for solving higher order boundary value problems. Mathematical Theory and Modeling 2 11–22.
Somali S. and Gokmen G. 2007. Adomian decomposition method for nonlinear Sturm-Liouville problems. Surveys in Mathematics and its Applications 2 11–20.
Venkatarangan S. N. and Rajalakshmi K. 1995. Modification of Adomian’s decomposition method to solve equations containing radicals. Computers and Mathematics with Applications 29 75–80.
Wazwaz A. M. 2010. The combined Laplace transform-Adomian demcomposition method for handling nonlinear Volterra integro-differential equations. Applied Mathematics and Computation 216 1304–1309.
Wazwaz A. M. 2011. Linear and Nonlinear Integral Equations: Methods and Applications. Springer.
Yousefi S. A. and Behroozifar M. 2010. Operational matrices of Bernstein polynomials and their applications. International Journal of System Science 41 709–716.
Zhang B. and Lu J. 2011. Exact solutions of homogenous partial differential equations by a new Adomian decomposition method. Procedia Environmental Science 11 440–446.
Rani D. and Mishra V. 2017. Approximate solution of boundary value problem with Bernstein polynomial Laplace decomposition method. International Journal of Pure and Applied Mathematics 114 823–833.