Numerical Solution of Nonlinear Second Order Singular BVPs Based on Green’s Functions and Fixed-Point Iterative Schemes

Assadi, Reem; Khuri, Sami; Sayfy, A.

doi:10.1007/s40819-018-0569-8

Cited by 6 publications

(3 citation statements)

References 32 publications

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“…Therefore, several researchers have developed various analytical and numerical methods over the past decades to look for the numerical and approximate analytical solution to these problems. Some of the well-known methods in the literature are the Taylor wavelet method (Gümgüm 2020), Cubic splines (Kanth and Bhattacharya 2006;Chawla et al 1988), B-splines (Çaglar et al 2009), finite difference method (Chawla and Katti 1985), variational iteration method (VIM) (Kanth and Aruna 2010), Adomian decomposition method (ADM) (Inç and Evans 2003) and its modified versions (Kumar et al 2020;Wazwaz 2011), a combination of VIM and homotopy perturbation method (VIMHPM) (Singh and Verma 2016), Optimal homotopy analysis method (OHAM) (Singh 2018), a reproducing kernel method (Niu et al 2018), compact finite difference method (Roul et al 2019), fixed-point iterative schemes (Tomar 2021a;Assadi et al 2018) and nonstandard finite difference schemes (Verma and Kayenat 2018). It should be noted that the existing analytical methods require more iterations in order to obtain a relatively good precise solution, resulting in very high x powers and a high number of terms in successive approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

An effective method for solving singular boundary value problems with some relevant physical applications

2021

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In this paper, an effective iterative technique for analytical solutions of a class of nonlinear singular boundary value problems (SBVPs) occurring in different physical situations is presented. In constructing the recursive approach for the iterative solution components, a technique that relies on establishing a corresponding integral representation is used. This approach provides a highly accurate approximate solution with a few iterations. We also discussed the convergence of the methodology. Furthermore, we consider some numerical examples from various physical situations, including real-life problems, to demonstrate the efficacy of the technique. As seen in the presented numerical tests, our new proposal outperforms traditionally existing iterative methods in the literature. To demonstrate the comparable efficiency and robustness of the proposed iterative procedure, the numerical results are compared to existing results in the literature. It is apparently, an advanced approach to deal with various forms of highly nonlinear problems. This approach works well for SBVPs with nonlinear boundary conditions also as depicted by a numerical example.

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Section: Introductionmentioning

confidence: 99%

An effective method for solving singular boundary value problems with some relevant physical applications

2021

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“…Assadi et al [12] exploited a fixed point iterative scheme, Xin et al [13] a non-trivial solution of NSP-BVPs, El-Syed and Gaagar [14] provided the existence of a solution for non-linear singular differential equations, Wang et al [15] and Wang and Ru [16] a positive solution of periodic equations. The general form of the second order non-linear NSP-BVPs is written as [8]:…”

Section: Introductionmentioning

confidence: 99%

A Neuro-Swarming Intelligence-Based Computing for Second Order Singular Periodic Non-linear Boundary Value Problems

et al. 2020

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In the present investigation, a novel neuro-swarming intelligence-based numerical computing solver is developed for solving second order non-linear singular periodic (NSP) boundary value problems (BVPs), i.e., NSP-BVPs, using the modeling strength of artificial neural networks (ANN) optimized with global search efficacy of particle swarm optimization (PSO) supported with the methodology of rapid local search by interior-point scheme (IPS), i.e., ANN-PSO-IPS. In order to check the proficiency, robustness, and stability of the designed ANN-PSO-IPS, two numerical problems of the NSP-BVPs have been presented for different numbers of neurons. The outcomes of the proposed ANN-PSO-IPS are compared with the available exact solutions to establish the worth of the solver in terms of accuracy and convergence, which is further endorsed through results of statistical performance metrics based on multiple implementations.

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“…The presence of singularity in above problems create complications in obtaining solutions; different methods have been proposed to conquer these drawbacks. Some existing numerical methods are homotopy perturbation method (HPM) (Roul and Warbhe 2016), optimal homotopy analysis method (OHAM) (Danish et al 2012), B spline collocation method (Roul and Thula 2018), Bernstein operational matrix method (Pirabaharan and Chandrakumar 2016), neural networks approach (Yadav et al 2016), improved differential transform method (IDTM) (Xie et al 2016), combination of Pade approximation and B-spline collocation method (Tazdayte and Allouche 2019), domain decomposition optimal homotopy analysis method (Roul and Madduri 2018), Green's functions and fixed-point iterative schemes (Assadi et al 2018), Haar wavelets (Verma and Tiwari 2019), combination of HPM and Variational iteration method (VIM) (Singh and Verma 2016) and VIM (Singh et al 2019). Recently, these class of SBVPs have also been solved using NSFDs (Buckmire 2003(Buckmire , 2004Verma and Kayenat 2018).…”

Section: Introductionmentioning

confidence: 99%

Applications of modified Mickens-type NSFD schemes to Lane–Emden equations

Verma

Kayenat

2020

Comp. Appl. Math.

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If there is a jump discontinuity present in the forcing term of a boundary value problem (BVP), the nonstandard finite difference (NSFD) and finite difference (FD) methods do not approximate the solutions very well. Here we use fuzzy transforms (FTs) and derive fuzzy transformed NSFD schemes that are referred to as non-standard fuzzy transform methods (NFTMs). The convergence of the derived NFTMs is established. Numerical solutions of Lane-Emden type equations are obtained using NFTMs. We show that NFTMs provide better results than NSFD and FD methods when the forcing term has a jump discontinuity. Even for large jumps, the NFTMs provide more accurate results than the other methods.

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Numerical Solution of Nonlinear Second Order Singular BVPs Based on Green’s Functions and Fixed-Point Iterative Schemes

Cited by 6 publications

References 32 publications

An effective method for solving singular boundary value problems with some relevant physical applications

An effective method for solving singular boundary value problems with some relevant physical applications

A Neuro-Swarming Intelligence-Based Computing for Second Order Singular Periodic Non-linear Boundary Value Problems

Applications of modified Mickens-type NSFD schemes to Lane–Emden equations

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