A hybrid collocation method for solving highly nonlinear boundary value problems

Adewumi, A. O.; Akindeinde, Saheed Ojo; Aderogba, Adebayo Abiodun; Ogundare, B. S.

doi:10.1016/j.heliyon.2020.e03553

Cited by 6 publications

(2 citation statements)

References 29 publications

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“…The Darcy-Brinkman-Forchheimer equation has been solved analytically and approximately using a variety of methods, such as the homotopy analysis method [37], the finite difference method [38], the optimal asymptotic Galerkin homotopy method [36], and the Tau homotopy analysis method [34]. In particular, Adewumi et al [39] obtained the approximate solutions for the model by using the hybrid method in combination with the Chebyshev collocation method with Laplace and differential transform methods. Motsa et al [35] implemented the spectral homotopy analysis approach to obtain an accurate result for the model.…”

Section: The Mathematical Formulations Of Nonlinear Models 21 the Dar...mentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

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Section: The Mathematical Formulations Of Nonlinear Models 21 the Dar...mentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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“…Boundary value problems are essential to numerical physics and mathematical modeling (Adewumi et al, 2020;Henderson & Luca, 2016). They occur in many applications such as wave propagation (Egorova & Ye 2020), electrostatics (Duan et al, 2013), deflection of cantilever beams under concentrated load, the temperature distribution of the radiation fin of the trapezoidal profile, and potential theory among other engineering applications.…”

Section: Introductionmentioning

confidence: 99%

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

Bacar,

Rawhoudine

2024

JMR

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The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

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