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Solution of third-order Emden–Fowler-type equations using wavelet methods

Arshad Khan (Department of Mathematics, Jamia Millia Islamia, New Delhi, India)
Mo Faheem (Department of Mathematics, Jamia Millia Islamia, New Delhi, India)
Akmal Raza (Department of Mathematics, Jamia Millia Islamia, New Delhi, India)

Engineering Computations

ISSN: 0264-4401

Article publication date: 26 January 2021

Issue publication date: 9 July 2021




The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.


Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.


This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.


This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.



The authors are obligated to the anonymous reviewers for their productive, beneficial remarks and suggestions in the advancement of the manuscript. The second author is supported by Junior Research Fellowship provided by CSIR, Govt. of India.


Khan, A., Faheem, M. and Raza, A. (2021), "Solution of third-order Emden–Fowler-type equations using wavelet methods", Engineering Computations, Vol. 38 No. 6, pp. 2850-2881.



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