Polynomial Least Squares Method for Fractional Lane–Emden Equations

Căruntu, Bogdan; Bota, Constantin; Lăpădat, Mărioara; Paşca, Mădălina Sofia

doi:10.3390/sym11040479

Cited by 14 publications

(10 citation statements)

References 15 publications

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“…In this section, we discuss the accuracy of the OAFM method by taking into consideration the first-order approximate solutions given by Eqs. ( 46), (47), where the index N max ∈ {10, 15, 25, 35} is an arbitrary fixed positive integer number.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…( 3), ( 4), ( 46), for β = 0, α 1 = 0, α 2 = 0, α 3 = 0, the Eqs. ( 8), ( 9), (47), for β = 0, α 1 = 0, α 2 = 0, α 3 = 0, the Eqs. ( 12), ( 14), (46), for α 1 = 0, α 2 = 0, α 3 = 0, the Eqs.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…In literature there are several analytical methods for solving the nonlinear differential problem given by Eqs. ( 5)-( 6), ( 10)-( 11), (15), (18), (21) such as: the Optimal Homotopy Asymptotic Method (OHAM) [40], [41], [42], the Optimal Homotopy Perturbation Method (OHPM) [43], [44], the Optimal Variational Iteration Method (OVIM) [45], the Optimal Iteration Parametrization Method (OIPM) [46], the Polynomial Least Squares Method [47], the Least Squares Differential Quadrature Method [48], the Multiple Scales Technique [49], the Function Method [50], the Homotopy Perturbation Method (HPM) and the Homotopy Analysis Method (HAM) [51], the Variational Iteration Method (VIM) [52].…”

Section: Closed-form Solutionsmentioning

confidence: 99%

See 2 more Smart Citations

Approximate Closed-form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

Ene¹,

Pop²,

Lăpădat³

2022

Preprint

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Based on some geometrical properties of the Rabinovich system the closed-form solutions of the equations has been established. Moreover the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied for more dynamical systems with these geometrical properties.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Closed-form Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Approximate Closed-form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

Ene¹,

Pop²,

Lăpădat³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The problem (1) was studied by using the Residual Power Series Method by Syam, M. (2018), Homotopy analysis method (HAM) by Huan et al (2017), Reproducing kernel Hilbert space method by Syam et al (2018), The fractional differential transformation (FDT) Rebenda and Smarda . (1978), Polynomial Least Squares Method by Caruntu et al (2019), Shifted Legendre Operational Matrix by Tripathi N. (2019), Chebyshev wavelets by Kazemi Nasab et al (2018), Orthonormal Bernoulli's polynomials by Sahu and Mallick. (2019), Orthonormal Bernstein polynomials by Abbas et al (2019).…”

Section: Introductionmentioning

confidence: 99%

A new operational matrix based on Boubaker polynomials for solving fractional Emden-Fowler problem

Bencheikh¹,

Chiter²

2022

Preprint

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In this paper the singular Emden-Fowler equation of fractional order is introduced and a computational method is proposed for its numerical solution. For the approximation of the solutions we have used Boubaker polynomials and defined the formulation for its fractional derivative operational matrix. This tool was not used yet, however, this area has not found many practical applications yet, and here introduced for the first time. The operational matrix of the Caputo fractional derivative tool converts these problems to a system of algebraic equations whose solutions are simple and easy to compute. Numerical examples are examined to prove the validity and the effectiveness of the proposed method to find approximate and precise solutions.

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“…This historical Lane-Emden model exists in astrophysics, quantum mechanics, and spherical cloud of gas, but considered stiff to solve due to the singular point at the origin. Many existing deterministic schemes have been applied to solve Lane-Emden model (Wazwaz 2001;He and Ji 2019;Singh et al 2019a,b;Asadpour et al 2019;Khalifa and Hassan 2019;Cȃruntu et al 2019) and its general form is written as follows (Farooq 2019;Hadian-Rasanan et al 2020;Sabir 2020):…”

Section: Introductionmentioning

confidence: 99%

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

et al. 2020

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In the present study, a novel fractional Meyer neuro-evolution-based intelligent computing solver (FMNEICS) is presented for numerical treatment of doubly singular multi-fractional Lane-Emden system (DSMF-LES) using combined heuristics of Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The design of novel FMNE-ICS for DSMF-LES is presented after derivation from standard Lane-Emden equation, and the singular points and shape factors along with fractional-order terms are analyzed. The MWNN modeling strength is used to represent the system model DSMF-LES in the meansquared error-based merit function and optimization of the networks is carried out with integrated optimization ability of GASQP. The verification, validation, and perfection of the FMNEICS for three different cases of DSMF-LES are established through comparative studies from reference solutions on convergence, robustness, accuracy, and stability measures. Moreover, the observations through the statistical analysis further authenticate the worth of proposed fractional MWNN-GASQP-based stochastic solver. Keywords Multi-fractional Lane-Emden model • Multi-singular systems • Artificial neural networks • Meyer wavelet neural networks • Sequential quadratic programming • Genetic algorithms Mathematics Subject Classification 34-XX • 34A08 • 34A34 • 82B31

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Polynomial Least Squares Method for Fractional Lane–Emden Equations

Abstract: This paper applies the Polynomial Least Squares Method (PLSM) to the case of fractional Lane-Emden differential equations. PLSM offers an analytical approximate polynomial solution in a straightforward way. A comparison with previously obtained results proves how accurate the method is.

Cited by 14 publications

References 15 publications

Approximate Closed-form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

Approximate Closed-form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

A new operational matrix based on Boubaker polynomials for solving fractional Emden-Fowler problem

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

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