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Article
Publication date: 19 June 2020

## Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

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.

### Details

Engineering Computations, vol. 38 no. 1
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 3 May 2016

## Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary

### Abstract

#### Purpose

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

#### Design/methodology/approach

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

#### Findings

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

#### Originality/value

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 3/4
Type: Research Article
ISSN: 0961-5539

### Keywords

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Article
Publication date: 6 January 2012

## Numerical analysis of steady non‐Newtonian flows with heat transfer analysis, MHD and nonlinear slip effects

The purpose of this paper is to study the effects of nonlinear partial slip on the walls for steady flow and heat transfer of an incompressible, thermodynamically…

### Abstract

#### Purpose

The purpose of this paper is to study the effects of nonlinear partial slip on the walls for steady flow and heat transfer of an incompressible, thermodynamically compatible third grade fluid in a channel. The principal question the authors address in this paper is in regard to the applicability of the no‐slip condition at a solid‐liquid boundary. The authors present the effects of slip, magnetohydrodynamics (MHD) and heat transfer for the plane Couette, plane Poiseuille and plane Couette‐Poiseuille flows in a homogeneous and thermodynamically compatible third grade fluid. The problem of a non‐Newtonian plane Couette flow, fully developed plane Poiseuille flow and Couette‐Poiseuille flow are investigated.

#### Design/methodology/approach

The present investigation is an attempt to study the effects of nonlinear partial slip on the walls for steady flow and heat transfer of an incompressible, thermodynamically compatible third grade fluid in a channel. A very effective and higher order numerical scheme is used to solve the resulting system of nonlinear differential equations with nonlinear boundary conditions. Numerical solutions are obtained by solving nonlinear ordinary differential equations using Chebyshev spectral method.

#### Findings

Due to the nonlinear and highly complicated nature of the governing equations and boundary conditions, finding an analytical or numerical solution is not easy. The authors obtained numerical solutions of the coupled nonlinear ordinary differential equations with nonlinear boundary conditions using higher order Chebyshev spectral collocation method. Spectral methods are proven to offer a superior intrinsic accuracy for derivative calculations.

#### Originality/value

To the best of the authors' knowledge, no such analysis is available in the literature which can describe the heat transfer, MHD and slip effects simultaneously on the flows of the non‐Newtonian fluids.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 1
Type: Research Article
ISSN: 0961-5539

### Keywords

Content available
Article
Publication date: 3 August 2012

## A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011

### Details

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

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Article
Publication date: 18 April 2017

## Shooting homotopy analysis method: A fast method to find multiple solutions of nonlinear boundary value problems arising in fluid mechanics

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.

### Details

Engineering Computations, vol. 34 no. 2
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 7 April 2020

## Effect of the initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water: I: Wave dynamics

The purpose of this paper is to determine both analytically and numerically the solution to a new one-dimensional equation for the propagation of small-amplitude waves in…

### Abstract

#### Purpose

The purpose of this paper is to determine both analytically and numerically the solution to a new one-dimensional equation for the propagation of small-amplitude waves in shallow waters that accounts for linear and nonlinear drift, diffusive attenuation, viscosity and dispersion, its dependence on the initial conditions, and its linear stability.

#### Design/methodology/approach

An implicit, finite difference method valid for both parabolic and second-order hyperbolic equations has been used to solve the equation in a truncated domain for five different initial conditions, a nil initial first-order time derivative and relaxation times linearly proportional to the viscosity coefficient.

#### Findings

A fast transition that depends on the coefficient of the linear drift, the diffusive attenuation and the power of the nonlinear drift are found for initial conditions corresponding to the exact solution of the generalized regularized long-wave equation. For initial Gaussian, rectangular and triangular conditions, the wave’s amplitude and speed increase as both the amplitude and the width of these conditions increase and decrease, respectively; wide initial conditions evolve into a narrow leading traveling wave of the pulse type and a train of slower oscillatory secondary ones. For the same initial mass and amplitude, rectangular initial conditions result in larger amplitude and velocity waves of the pulse type than Gaussian and triangular ones. The wave’s kinetic, potential and stretching energies undergo large changes in an initial layer whose thickness is on the order of the diffusive attenuation coefficient.

#### Originality/value

A new, one-dimensional equation for the propagation of small-amplitude waves in shallow waters is proposed and studied analytically and numerically. The equation may also be used to study the displacement of porous media subject to seismic effects, the dispersion of sound in tunnels, the attenuation of sound because of viscosity and/or heat and mass diffusion, the dynamics of second-order, viscoelastic fluids, etc., by appropriate choices of the parameters that appear in it.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 11
Type: Research Article
ISSN: 0961-5539

### Keywords

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Article
Publication date: 13 July 2010

## Assessment of two analytical methods in solving the linear and nonlinear elastic beam deformation problems

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

### Keywords

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Article
Publication date: 1 January 1992

## STATIC CONTACT PROBLEMS—A REVIEW

Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years…

### Abstract

Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years. The finite element method has been widely used to solve contact problems with various grades of complexity. Great progress has been made on both theoretical studies and engineering applications. This paper reviews some of the main developments in contact theories and finite element solution techniques for static contact problems. Classical and variational formulations of the problem are first given and then finite element solution techniques are reviewed. Available constraint methods, friction laws and contact searching algorithms are also briefly described. At the end of the paper, a bibliography is included, listing about seven hundred papers which are related to static contact problems and have been published in various journals and conference proceedings from 1976.

### Details

Engineering Computations, vol. 9 no. 1
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 5 March 2018

## Nonlinear impedance boundary condition for 2D BEM

The purpose of this paper is the derivation and efficient implementation of surface impedance boundary conditions (SIBCs) for nonlinear magnetic conductors.

### Abstract

#### Purpose

The purpose of this paper is the derivation and efficient implementation of surface impedance boundary conditions (SIBCs) for nonlinear magnetic conductors.

#### Design/methodology/approach

An approach based on perturbation theory is proposed, which expands to nonlinear problems the methods already developed by the authors for linear problems. Differently from the linear case, for which the analytical solution of the diffusion equation in the semi-infinite space for the magnetic field is available, in the nonlinear case the corresponding nonlinear diffusion equation must be solved numerically. To this aim, a suitable smooth map is defined to reduce the semi-infinite computational domain to a finite one; then the diffusion equation is solved by a Galerkin method relying on basis functions constructed via the push-forward of a Lagrangian polynomial basis whose degrees of freedom are collocated at Gauss–Lobatto nodes. The use of such basis in connection with a suitable under-integration naturally leads to mass-lumping without impacting the order of the method. The solution of the diffusion equation is coupled with a boundary element method formulation for the case of parallel magnetic conductors in terms of E and B fields.

#### Findings

The results are validated by comparison with full nonlinear finite element method simulations showing very good accordance at a much lower computational cost.

#### Research limitations/implications

Limitations of the method are those arising from perturbation theory: the introduced small parameter must be much less than one. This implies that the penetration depth of the magnetic field into the magnetic and conductive media must be much smaller than the characteristic size of the conductor.

#### Originality/value

The efficient implementation of a nonlinear SIBC based on a perturbation approach is proposed for an electric and magnetic field formulation of the two-dimensional problem of current driven parallel solid conductors.

### Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 37 no. 2
Type: Research Article
ISSN: 0332-1649

### Keywords

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Article
Publication date: 1 June 2006

## New methods for solving of nonlinear weakly singular integral equations

Aims to present a boundary integral equation method for solving Laplace's equation Δu=0 with nonlinear boundary conditions.

### Abstract

#### Purpose

Aims to present a boundary integral equation method for solving Laplace's equation Δu=0 with nonlinear boundary conditions.

#### Design/methodology/approach

The nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the collocation method on smooth or nonsmooth boundary; the singularities of solution degrade the rates of convergence.

#### Findings

Variants of the methods for finding numerical solutions are suggested. So these methods have been compared with respect to number of iterations.

#### Practical implications

Numerical experiments show the efficiency of the proposed methods.

#### Originality/value

Provides new methods to solve nonlinear weakly singular integral equations and discusses difficulties that arise in particular cases.

### Details

Kybernetes, vol. 35 no. 5
Type: Research Article
ISSN: 0368-492X

### Keywords

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