An efficient numerical method for solving nonlinear Thomas-Fermi equation

Parand, Kourosh; Rabiei, Kobra; Delkhosh, Mehdi

doi:10.2478/ausm-2018-0012

Cited by 6 publications

(10 citation statements)

References 72 publications

(63 reference statements)

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“…Table 2 presents the absolute error of HSE with Adomian's method. A comparison of proposed technique is presented in Table 3 with the existing numerical techniques, Haar wavelet quasi-linearization method [29], a collocation method [52] and Chebyshev functions (B-GFCF) collocation method [53]. It is concluded that our proposed technique provides the better results on the basis of absolute errors.…”

Section: Casementioning

confidence: 93%

“…Furthermore, iterative methods like, variational iteration method (VIM), modified variational iteration method (VMIM), Adomian decomposition method (ADM), modified Adomian decomposition method (MADM) and Homotopy analysis method (HAM) are described [23]. The reciprocal transformations [24], Homotopy decomposition method [25], bivariate generalized fractional order of the Chebyshev functions (BGFCF) [26], cubic trigonometric B-Spline collocation method [27], collocation method [28], Harr wavelet quasilinearization approach [29], and Lipschitz metric [30], time marching scheme [31] are applied to study the diffusion of neumatic LCs. The generalized Hunter-Saxton equation is considered using integrability structures [32], Numerical solutions of HSE using Laguerre wavelet and by using efficient approach on time domains is presented in [33,34,35].…”

Section: Introductionmentioning

confidence: 99%

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Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

Ahmad

Ilyas

Kutlu

et al. 2021

Heliyon

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In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.

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Section: Casementioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

Ahmad

Ilyas

Kutlu

et al. 2021

Heliyon

View full text Add to dashboard Cite

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“…The Thomas-Fermi (T-F) equation is a nonlinear singular differential equation defined on a semi-infinite domain [23,24] as…”

Section: Thomas-fermi Equationmentioning

confidence: 99%

“…Using the routine PLeal2 from Appendix C, we obtain ̃1( ) and ̃2( ), using [0, 12,12] and [4, 10, 10] as the order for expansions, respectively; Considering the expansion points [0, 1, 5] for ̃1( ) and [5,10,25] for ̃2( ). The numerical values [23] employed as expansions points for LP, are…”

Section: Thomas-fermi Equationmentioning

confidence: 99%

“…Even-more, the missing derivatives required by LP are obtained using either a numerical method or the least squares method regarding the intrinsic parameters of the nonlinear problem. Hence, we will present as study cases some important nonlinear problems in physics and engineering: approximate Gelfand's equation [18,19,20], an equation of an isothermal reaction [21], a model for chronic myelogenous leukemia [22], Thomas-Fermi equation [23,24], and a high order nonlinear differential equations with discontinuities [25,26,27,28,29]; resulting in accurate approximations with a wide domain of convergence.…”

Section: Introductionmentioning

confidence: 99%

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The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations

Vázquez-Leal

Sandoval-Hernandez

Filobello-Nino

et al. 2020

Heliyon

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This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials. Such characteristic makes of LPs a handy and powerful tool to approximate different kind of differential equations including: singular problems, initial condition and boundary-valued problems, equations with discontinuities, coupled differential equations, high-order equations, among others. Additionally, we show how the process to obtain the polynomials is straightforward and simple to implement; generating a compact, and easy to compute, expression. Even more, we present the process to approximate Gelfand's equation, an equation of an isothermal reaction, a model for chronic myelogenous leukemia, Thomas-Fermi equation, and a high order nonlinear differential equations with discontinuities getting, as result, accurate, fast and compact approximate solutions. In addition, we present the computational convergence and error studies for LPs resulting convergent polynomials and error tendency to zero as the order of LPs increases for all study cases. Finally, a study of CPU time shows that LPs require a few nano-seconds to be evaluated, which makes them suitable for intensive computing applications.

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Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Vázquez-Leal

2020

Discrete Dynamics in Nature and Society

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This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

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An efficient numerical method for solving nonlinear Thomas-Fermi equation

Cited by 6 publications

References 72 publications

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

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