Higher order Emden–Fowler type equations via uniform Haar Wavelet resolution technique

Swati, Swati; Singh, Kajal; Verma, Amrisha; Singh, Mandeep Singh Jit

doi:10.1016/j.cam.2020.112836

Cited by 34 publications

(17 citation statements)

References 26 publications

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“…The errors given in the last row of Table 7 are corresponding to J = 6, k = 4, M = 8 and N = 2 J = 64. We compare our results with the results of new cubic B-spline approximation (NCBSA) method by Iqbal et al (2018) and Haar wavelet resolution technique by Swati et al (2020) in Table 7. Table 8 shows the time taken to solve example 3 by our methods for different values of J .…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Mohammadi et al (2019), first time, used a collocation method based on Haar wavelet for the solution of singular and nonlinear fractional time-dependent Emden–Fowler equations. Recently, a numerical treatment for the solution of higher order Emden–Fowler-type equations via uniform Haar wavelet technique has been given by Swati et al (2020).…”

Section: Introductionmentioning

confidence: 99%

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

Khan

Faheem

Raza

2021

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Purpose 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. Design/methodology/approach 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. Findings 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. Originality/value 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.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Khan

Faheem

Raza

2021

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“…Several techniques (analytical/numerical) developed to deal with this type of equations, like monotone iterative technique coupled with upper/lower solutions (Singh and Verma, 2017;Singh et al, 2015;Verma, 2011Verma, , 2012Pandey and Verma, 2011), shooting method (Russell and Shampine, 1975), finite-difference method (Chawla and Katti, 1985;Pandey, 1997;Pandey and Singh, 2004), B-splines and cubic splines method (Chawla and Subramanian, 1988;Kanth and Bhattacharya, 2006), Haar wavelet (Verma and Tiwari, 2019;Swati et al, 2020;Verma et al, 2020b), etc.…”

Section: Introductionmentioning

confidence: 99%

A note on variation iteration method with an application on Lane–Emden equations

Verma

Kumar

Singh

et al. 2021

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PurposeIn this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, −u″(t)-(α/t) u′(t) = g(t, u), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equations.Design/methodology/approachThe authors improve the modified variation iteration method (VIM) proposed in [JAAC, 9(4) 1242–1260 (2019)], which greatly accelerates the convergence and reduces the computational task.FindingsThe findings revealed that either exact or highly accurate approximate solutions of Lane–Emden equations can be computed with the proposed method.Originality/valueNovel modification is made in the VIM that provides either exact or highly accurate approximate solutions of Lane-Emden equations, which does not exist in the literature.

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“…The Haar wavelets are the simplest family of wavelets. In this respect we mention the papers of Aziz et al (2013), Swati et al (2020), Raza and Khan (2019), Shah et al (2020), Azam et al (2019 and Singh et al (2020). Recently Rehman et al (2020) introduced a method based on Haar wavelet and Green function for numerical solution of generalized fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

Abdeen

Rehman

2021

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PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

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Higher order Emden–Fowler type equations via uniform Haar Wavelet resolution technique

Cited by 34 publications

References 26 publications

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

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

A note on variation iteration method with an application on Lane–Emden equations

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

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