A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

Roul, Pradip

doi:10.1016/j.apnum.2020.03.018

Cited by 32 publications

(14 citation statements)

References 30 publications

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“…So, in the second estimate (k = 1) , we are looking for precision in the interval [35,36] or the accuracy of optimal to the first decimal place as shown Fig. 13 Figure 14 presents the evolution of the optimal shape parameter optimal with respect to number of steps for the new algorithm which shows that optimal varies versus the number of steps.…”

Section: Study On the Approximation Accuracy According To The Shape Parameter Values Of A Bi-dimensional Structure In Tensionmentioning

confidence: 99%

“…Also in the context to solve a class of Lane-Emden singular boundary value problems which describe several phenomena in theoretical physics and astrophysics. In the field of electrohydrodynamic flow of a fluid in a circular cylindrical conduit, Roul [35] presented a fourth-order non-uniform mesh optimal B-spline collocation method for a strongly nonlinear singular boundary value problem. We also used collocation and interpolation methods in our previous work for solving nonlinear elastic and elastoplastic problems [1] and viscoplastic problems [1,28,38].…”

Section: Introductionmentioning

confidence: 99%

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New collocation path-following approach for the optimal shape parameter using Kernel method

et al. 2021

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The goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given.

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Section: Study On the Approximation Accuracy According To The Shape Parameter Values Of A Bi-dimensional Structure In Tensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New collocation path-following approach for the optimal shape parameter using Kernel method

et al. 2021

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“…This assumption helped overcome the limitation of nonlinearity and singularity. Various numerical and analytical techniques such as the pseudospectral collocation method [26], spectral collocation method [27], discrete optimized homotopy analysis method (DOHAM) [28], DTM-Pade' approximation [29], least square method [30], Galerkin Method, Collocation Method [31] and optimal B-spline collocation method [32,33] have been used. The electrohydrodynamic (EHD) flow of fluid has been studied by various numerical and analytical approaches but due to the singularity and nonlinearity in the nature of its mathematical model, it is difficult for traditional techniques to find its approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Electrohydrodynamic Flow in a Circular Cylindrical Conduit by Using Neuro Evolutionary Technique

et al. 2021

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This paper analyzes the mathematical model of electrohydrodynamic (EHD) fluid flow in a circular cylindrical conduit with an ion drag configuration. The phenomenon was modelled as a nonlinear differential equation. Furthermore, an application of artificial neural networks (ANNs) with a generalized normal distribution optimization algorithm (GNDO) and sequential quadratic programming (SQP) were utilized to suggest approximate solutions for the velocity, displacements, and acceleration profiles of the fluid by varying the Hartmann electric number (Ha2) and the strength of nonlinearity (α). ANNs were used to model the fitness function for the governing equation in terms of mean square error (MSE), which was further optimized initially by GNDO to exploit the global search. Then SQP was implemented to complement its local convergence. Numerical solutions obtained by the design scheme were compared with RK-4, the least square method (LSM), and the orthonormal Bernstein collocation method (OBCM). Stability, convergence, and robustness of the proposed algorithm were endorsed by the statistics and analysis on results of absolute errors, mean absolute deviation (MAD), Theil’s inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE).

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“…In recent years, nonlinear singular boundary value problems (SBVPs) have been extensively studied in the literature 1‐5 . A wide variety of physical phenomena, such as the thermal behaviour of a spherical cloud of gas, oxygen diffusion in a cell, isothermal gas sphere, electrohydrodynamic flow of a fluid in a cylinder, and thermal explosion are modelled by SBVPs, see previous studies 6‐10 and the references therein. The main objective of this paper is to design an exponential cubic B‐spline collocation method for a general class of nonlinear SBVP subject to Neumann and Robin boundary conditions (BCs) of the following form:

{false(p false(x false) y^{'} false(x false) false)}^{'} = p false(x false) f false(x, y false(x false) false), 0 < x \leq 1,

y^{'} false(0 false) = 0, a y false(1 false) + b y^{'} false(1 false) = e,

with

p false(x false) = x^{α} g false(x false)

, α > 0 and g ( x ) is a nonnegative function and a > 0, b ≥ 0 and e are finite constants.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method based on exponential B‐spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

Roul

Kumari

Goura

2020

Math Methods in App Sciences

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In this paper, we develop a numerical scheme to approximate the solution of a general class of nonlinear singular boundary value problems (SBVPs) subject to Neumann and Robin boundary conditions. The original differential equation has a singularity at the point x=0, which is removed via L'Hospital's law with an assumption about the derivative of the solution at the point x=0. An exponential B‐spline collocation approach is then constructed to solve the resulting boundary value problem. Convergence analysis of the method is discussed. Numerical examples are provided to illustrate the applicability and efficiency of the method. Our results are compared with those obtained by other three numerical methods such as uniform mesh cubic B‐spline collocation (UCS) method, nonstandard finite difference method, and finite difference method based on Chawla's identity.

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A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid

Cited by 32 publications

References 30 publications

New collocation path-following approach for the optimal shape parameter using Kernel method

New collocation path-following approach for the optimal shape parameter using Kernel method

Numerical Analysis of Electrohydrodynamic Flow in a Circular Cylindrical Conduit by Using Neuro Evolutionary Technique

An efficient numerical method based on exponential B‐spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

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