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Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 19 June 2020

Chein-Shan Liu and Jiang-Ren Chang

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

Abstract

Purpose

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

Design/methodology/approach

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

Findings

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

Research limitations/implications

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

Practical implications

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

Originality/value

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

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Article
Publication date: 13 July 2010

A. Barari, B. Ganjavi, M. Ghanbari Jeloudar and G. Domairry

In the last two decades with the rapid development of nonlinear science, there has appeared ever‐increasing interest of scientists and engineers in the analytical…

Abstract

Purpose

In the last two decades with the rapid development of nonlinear science, there has appeared ever‐increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general boundary value problems, but also are used as mathematical models in viscoelastic and inelastic flows. The purpose of this paper is to present the application of the homotopy‐perturbation method (HPM) and variational iteration method (VIM) to solve some boundary value problems in structural engineering and fluid mechanics.

Design/methodology/approach

Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics.

Findings

Analytical solutions often fit under classical perturbation methods. However, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameters. In the present study, two powerful analytical methods HPM and VIM have been employed to solve the linear and nonlinear elastic beam deformation problems. The results reveal that these new methods are very effective and simple and do not require a large computer memory and can also be used for solving linear and nonlinear boundary value problems.

Originality/value

The results revealed that the VIM and HPM are remarkably effective for solving boundary value problems. These methods are very promoting methods which can be wildly utilized for solving mathematical and engineering problems.

Details

Journal of Engineering, Design and Technology, vol. 8 no. 2
Type: Research Article
ISSN: 1726-0531

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Article
Publication date: 7 November 2016

Abdul-Majid Wazwaz, Randolph Rach and Lazhar Bougoffa

The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions.

Abstract

Purpose

The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions.

Design/methodology/approach

The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions.

Findings

The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions.

Originality/value

This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 18 April 2017

L. Ahmad Soltani, E. Shivanian and Reza Ezzati

The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of…

Abstract

Purpose

The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of nonlinear boundary value problems (NBVPs).

Design/methodology/approach

A major problem with the previously modified HAM, namely, predictor homotopy analysis method, which is used to predict multiplicity of solutions of NBVPs, is a time-consuming computation of high-order HAM-approximate solutions due to a symbolic variable namely “prescribed parameter”. The proposed new technique which is based on traditional shooting method, and the HAM cuts the dependency on the prescribed parameter.

Findings

To demonstrate the computational efficiency, the mentioned method is implemented on three important nonlinear exactly solvable differential equations, namely, the nonlinear MHD Jeffery–Hamel flow problem, the nonlinear boundary value problem arising in heat transfer and the strongly nonlinear Bratu problem.

Originality/value

The more high-order approximate solutions are computable, multiple solutions are easily searched and discovered and the more accurate solutions can be obtained depending on how nonhomogeneous boundary conditions are transcribed to the homogeneous boundary conditions.

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Article
Publication date: 19 April 2013

Lazhar Bougoffa and Randolph C. Rach

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations…

Abstract

Purpose

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Design/methodology/approach

The authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.

Findings

The new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Originality/value

The paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

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Article
Publication date: 3 January 2017

Abdul-Majid Wazwaz

The purpose of this paper is to use the variational iteration method (VIM) for studying boundary value problems (BVPs) characterized with dual solutions.

Abstract

Purpose

The purpose of this paper is to use the variational iteration method (VIM) for studying boundary value problems (BVPs) characterized with dual solutions.

Design/methodology/approach

The VIM proved to be practical for solving linear and nonlinear problems arising in scientific and engineering applications. In this work, the aim is to use the VIM for a reliable treatment of nonlinear boundary value problems characterized with dual solutions.

Findings

The VIM is shown to solve nonlinear BVPs, either linear or nonlinear. It is shown that the VIM solves these models without requiring restrictive assumptions and in a straightforward manner. The conclusions are justified by investigating many scientific models.

Research limitations/implications

The VIM provides convergent series solutions for linear and nonlinear equations in the same manner.

Practical implications

The VIM is practical and shows more power compared to existing techniques.

Social implications

The VIM handles linear and nonlinear models in the same manner.

Originality/value

This work highlights a reliable technique for solving nonlinear BVPs that possess dual solutions. This paper has shown the power of the VIM for handling BVPs.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 1
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 31 March 2020

Amit K. Verma, Biswajit Pandit and Carlos Escudero

This paper aims to apply an iterative numerical method to find the numerical solution of the nonlinear non-self-adjoint singular boundary value problems that arises in the…

Abstract

Purpose

This paper aims to apply an iterative numerical method to find the numerical solution of the nonlinear non-self-adjoint singular boundary value problems that arises in the theory of epitaxial growth.

Design/methodology/approach

The proposed problem has multiple solutions and it is singular too; so not every technique can capture all the solutions. This study proposes to use variational iterative numerical method and compute both the solutions. The computed solutions are very close to the exact result.

Findings

It turns out that the existence or nonexistence of numerical solutions fully depends on the value of a parameter. The authors show that numerical solutions exist for small positive values of this parameter. For large positive values of the parameter, they find nonexistence of solutions. They also observe existence of solutions for negative values of the parameter and determine the range of parameter values which separates existence and nonexistence of solutions. This parameter has a clear physical meaning, as it describes the rate at which new material is deposited onto the system. This fact allows interpreting the physical significance of the results.

Originality/value

The authors could capture both the solutions and got accurate estimation of the parameter. This method will be a great tool to handle such types of nonlinear non-self-adjoint equations that have multiple solutions in engineering and mathematical sciences. The results in this paper will have an impact on the understanding of theoretical models of epitaxial growth in near future.

Details

Engineering Computations, vol. 37 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 8 August 2020

Amit K. Verma, Narendra Kumar and Diksha Tiwari

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and…

Abstract

Purpose

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations:

(tk1y(t))=tω1f1(t,y(t),z(t)),
(tk2z(t))=tω2f2(t,y(t),z(t)),
where t > 0, subject to the following initial values, boundary values and four-point boundary values:
y(0)=γ1, y(0)=0, z(0)=γ2, z(0)=0,
y(0)=0, y(1)=δ1, z(0)=0, z(1)=δ2,
y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),
where n1,n2,v1,v2(0,1) and k10,k20,ω1<1,ω2<1, γ1, γ2, δ1, δ2 are real constants.

Design/methodology/approach

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

Findings

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

Originality/value

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

Details

Engineering Computations, vol. 38 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 5 March 2018

Sudao Bilige and Yanqing Han

The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in…

Abstract

Purpose

The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics.

Design/methodology/approach

The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods.

Findings

First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions.

Originality/value

The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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