Fourth order singularly perturbed boundary value problems via initial value techniques

Mishra, Hradyesh Kumar; Saini, Sonali

doi:10.12988/ams.2014.312676

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…In [19] Deniz and Bildik worked with the adomian decomposition approach to solve SPBVPs of the fourth-order. Development on the fourth-order SPBVPs employing initial value techniques are being done by Mishra and Saini [20]. Wang and Ni [21] talked about the contrast structure for the SSPBVP problem.…”

Section: Introductionmentioning

confidence: 99%

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Yousaf,

Srivastava,

Abbas

et al. 2023

Symmetry

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Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

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Section: Introductionmentioning

confidence: 99%

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Yousaf,

Srivastava,

Abbas

et al. 2023

Symmetry

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“…Raja and Tamilselvan [23] have designed a shooting method on a Shishkin mesh to solve reaction-diffusion type problems. Mishra and Saini [18] have used initial value technique for the numerical solution of fourth order singularly perturbed boundary value problems. Sarakhsi et al [25] have studied the existence of boundary layer problem.…”

Section: Introductionmentioning

confidence: 99%

Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems

Lodhi

Mishra

2018

Stud. Univ. Babes-Bolyai Math.

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In this communication, we have studied an efficient numerical approach based on uniform mesh for the numerical solutions of fourth order singular perturbation boundary value problems. Such type of problems arises in various fields of science and engineering, like electrical network and vibration problems with large Peclet numbers, Navier-Stokes flows with large Reynolds numbers in the theory of hydrodynamics stability, reaction-diffusion process, quantum mechanics and optimal control theory etc. In the present study, a quintic B-spline method has been discussed for the approximate solution of the fourth order singular perturbation boundary value problems. The convergence analysis is also carried out and the method is shown to have convergence of second order. The performance of present method is shown through some numerical tests. The numerical results are compared with other existing method available in the literature. (2010): 65L10. Keywords: Fourth order singular perturbation boundary value problem, quintic B-spline, quasilinearization, uniform mesh, convergence analysis. y (a) = η 1 , y (b) = η 2 , y (a) = η 3 , y (b) = η 4 . Mathematics Subject Classification(1.2) where η 1 , η 2 , η 3 and η 4 are finite real constants and ε is a small positive parameter, such that 0 < ε 1. Moreover, we assume that the functions p (t) ,q (t) ,r (t) and f (t) are sufficiently smooth. Further, the problem (1.1) is called non-turning point problem if p (t) ≥ α > 0 throughout the interval [a, b], where α is some positive constant and boundary layer will be in the neighbourhood of t = a [9]. In the same

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Solution of a class of fourth order singular singularly perturbed boundary value problems by quintic B-spline method

Lodhi

Mishra

2016

Journal of the Nigerian Mathematical Society

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Fourth order singularly perturbed boundary value problems via initial value techniques

Cited by 4 publications

References 17 publications

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems

Solution of a class of fourth order singular singularly perturbed boundary value problems by quintic B-spline method

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