A computational solution of the multi-term nonlinear ODEs with variable coefficients using the integral-collocation-approach based on Legendre polynomials

  • Khadijah Mohammed Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia
  • M. M. Khader Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn SaudIslamic University (IMSIU), Riyadh: 11566, Saudi Arabia
Keywords: Integral collocation formulation, Legendre spectral method, Multi-term ODEs.


The main aim of this work is devoted to propose and analyze some schemes of the integral collocation formulation dependent on Legendre polynomials. We introduce these formulae to solve the multi-term ODEs with variable coefficients. The proposed technique is used to reduce the given problem to solve a system of algebraic equations. Numerical results are given to satisfy the accuracy and the applicability of the implemented approach.


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K. M. Abualnaja, A block procedure with linear multi-step methods using Legendre polynomials for solving ODEs, Applied Mathematics, 6, p.(717-723), 2015.

W. W. Bell, Special Functions for Scientists and Engineers, Great Britain, Butler and Tanner Ltd, Frome and London, 1968.

A. H. Bhrawy and A. S. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Applied Mathematics Letters, 26, p.(25-31), 2013.

R. L. Burden and J. D. Faires, Numerical Analysis, PWS, Boston, 1993.

M. M. Khader, On the numerical solutions for the fractional diffusion equation, Communications in Nonlinear Science and Numerical Simulation, 16, p.(2535-2542), 2011.

M. M. Khader, Introducing an efficient modification of the variational iteration method by using Chebyshev polynomials, Application and Applied Mathematics: An International Journal, 7(1),

p.(283-299), 2012.

M. M. Khader, On the numerical solution and convergence study for system of non-linear fractional diffusion equations, Canadian Journal of Physics, 92(12), p.(1658-1666), 2014.

M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudo-spectral method, International Journal of Pure and

Applied Mathematics, 74(3), p.(287-297), 2012.

M. M. Khader, A. M. S. Mahdy and M. M. Shehata, An integral collocation approach based on Legendre polynomials for solving Riccati, Logistic and delay differential equations, Applied Mathematics, 5, p.(2360-2369), 2014.

N. Mai-Duy, H. See and T. Tran-Cong, A spectral collocation technique based on integrated Chebyshev polynomials for biharmonic problems in irregular domains, Applied Mathematical Modelling,

, p.(284-299), 2009.

A. S. Momani and K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Applied Mathematics and Computation, 162, p.(1351-1365), 2005.

N. H. Sweilam, M. M. Khader and A. M. S. Mahdy, Numerical studies for fractional-order Logistic differential equation with two different delays, Journal of Applied Mathematics, 2012, p.(1-14), 2012.

How to Cite
Mohammed, K., & Khader, M. M. (2016). A computational solution of the multi-term nonlinear ODEs with variable coefficients using the integral-collocation-approach based on Legendre polynomials. Journal of Progressive Research in Mathematics, 9(3), 1406-1410. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/857