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Article
Publication date: 1 July 1996

Y.C. LI

This paper presents approximate analytical solutions for the diffusion problems of a cylindrical hole in an infinite medium and a slot in an infinite medium with properly…

Abstract

This paper presents approximate analytical solutions for the diffusion problems of a cylindrical hole in an infinite medium and a slot in an infinite medium with properly prescribed boundary conditions and initial conditions. These solutions have much simpler forms than those of exact analytical solutions, and asymptotically approach the exact solutions with increasing time or the material point moving away from the internal boundary. The approximate analytical solution for the diffusion problem of a slot in an infinite medium is applied to establish a shape function for the infinite elements. Good agreement is found in comparison of our results with those presented by Li and Huang and Cinco‐Ley et al. Finally, an example simulating a primary recovery procedure in hydraulic fracturing technique for an oil field is presented.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 6 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 1994

Michael M. Grigor’ev

The paper gives the description of boundary element method(BEM) with subdomains for the solution ofconvection—diffusion equations with variable coefficients and Burgers’equations…

Abstract

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 May 2011

Manel Labidi and Khaled Omrani

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified…

Abstract

Purpose

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified Degasperis‐Procesi (mDP) equations.

Design/methodology/approach

Perturbation method is a traditional method depending on a small parameter which is difficult to be found for real‐life nonlinear problems. To overcome the difficulties and limitations of the above method, two new ones have recently been introduced by He, i.e. VIM and HPM. In this paper, mCH and mDP equations are solved through these methods.

Findings

To assess the accuracy of the solutions, the comparison of the obtained results with the exact solutions reveals that both methods are tremendously effective.

Originality/value

The paper shows that VIM and HPM can be implemented to solve mCH and mDP equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 6 January 2012

Precious Sibanda, Sandile Motsa and Zodwa Makukula

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to…

Abstract

Purpose

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to correct the inaccurate results presented in Siddiqui et al., where the homotopy perturbation method was used, and to demonstrate the computational efficiency and accuracy of the spectral‐homotopy analysis methods (SHAM and MSHAM) in solving problems that arise in fluid mechanics.

Design/methodology/approach

Exact and approximate analytical series solutions of the non‐linear equations that govern the flow of a steady laminar flow of a third grade fluid through a horizontal channel are constructed using the homotopy analysis method and two new modifications of this method. These solutions are compared to the full numerical results. A new method for calculating the optimum value of the embedded auxiliary parameter ∼ is proposed.

Findings

The “standard” HAM and the two modifications of the HAM (the SHAM and the MSHAM) lead to faster convergence when compared to the homotopy perturbation method. The paper shows that when the same initial approximation is used, the HAM and the SHAM give identical results. Nonetheless, the advantage of the SHAM is that it eliminates the restriction of searching for solutions to the nonlinear equations in terms of prescribed solution forms that conform to the rule of solution expression and the rule of coefficient ergodicity. In addition, an alternative and more efficient implementation of the SHAM (referred to as the MSHAM) converges much faster, and for all parameter values.

Research limitations/implications

The spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. It however remains to be generalized and verified for more complicated nonlinear problems.

Originality/value

The spectral‐HAM has already been proposed and implemented by the authors in a recent paper. This paper serves the purpose of verifying and demonstrating the utility of the new spectral modification of the HAM in solving problems that arise in fluid mechanics. The MSHAM is a further modification of the SHAM to speed up converge and to allow for convergence for a much wider range of system parameter values. The utility of these methods has not been tested and verified for systems of nonlinear equations. For this reason as much emphasis has been placed on proving the reliability and validity of the solution techniques as on the physics of the problem.

Details

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

Keywords

Article
Publication date: 8 June 2012

Mehdi Dehghan and Jalil Manafian Heris

This paper aims to show that the variational iteration method (VIM) and the homotopy perturbation method (HPM) are powerful and suitable methods to solve the Fornberg‐Whitham…

Abstract

Purpose

This paper aims to show that the variational iteration method (VIM) and the homotopy perturbation method (HPM) are powerful and suitable methods to solve the Fornberg‐Whitham equation.

Design/methodology/approach

Using HPM the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. Also, by using VIM the analytical results of this equation have been obtained in terms of convergent series with easily computable components.

Findings

Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy.

Originality/value

Also the results show that the introduced methods are efficient tools for solving the nonlinear partial differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 May 2015

Hossein Aminikhah

The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the…

Abstract

Purpose

The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

Design/methodology/approach

The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions.

Findings

The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences.

Originality/value

The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 May 2015

Maria Tchonkova

– The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity.

Abstract

Purpose

The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity.

Design/methodology/approach

The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. The proposed theoretical formulation is given the general three-dimensional case and is tested numerically on the solution of one-dimensional wave equations.

Findings

To test the performance of the method, it has been implemented in an original computer code, using object-oriented logic and written from scratch. Two one-dimensional problems from the mathematical physics, with well-known exact analytical solutions, have been solved. The numerical examples include a forced vibrating spring, fixed at its both ends and a rod, vibrating under its own weight, when one of its ends is fixed and the other is traction-free. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values.

Originality/value

The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications to two and three-dimensional problems in dynamic elasticity.

Details

Engineering Computations, vol. 32 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 26 August 2014

Antonio Campo, Abraham J. Salazar, Diego J. Celentano and Marcos Raydan

The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an…

Abstract

Purpose

The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables.

Design/methodology/approach

The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense.

Findings

In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent.

Originality/value

Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 June 2020

Zhivko Georgiev, Ivan Trushev, Todor Todorov and Ivan Uzunov

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these…

Abstract

Purpose

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

Design/methodology/approach

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Findings

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

Originality/value

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 40 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 15 August 2019

Emad H. Aly and Ioan Pop

The purpose of this study is to present both effective analytic and numerical solutions to MHD flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid…

Abstract

Purpose

The purpose of this study is to present both effective analytic and numerical solutions to MHD flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. Water (base fluid) nanoparticles of alumina and copper were considered as a hybrid nanofluid.

Design/methodology/approach

Proper-similarity variables were applied to transform the system of partial differential equations into a system of ordinary (similarity) differential equations. Exact analytical solutions were then presented for the dimensionless stream and temperature functions. Further, the authors introduce a very nice analytic and numerical solutions for both small and large values of the magnetic parameter.

Findings

It was found that no/unique/two equal/dual physical solutions exist for the investigated boundary value problem. The physically realizable practice of these solutions depends on the range of the governing parameters. For a stretching/shrinking sheet, it was deduced that a hybrid nanofluid works as a cooler on increasing some of the investigated parameters. Moreover, in the case of a shrinking sheet, the first solutions of hybrid nanofluid are stable and physically realizable rather than the nanofluid, while those of the second solutions are not for both hybrid nanofluid and nanofluid.

Originality/value

The present results for the hybrid nanofluids are new and original, as they successfully extend (generalize) the problems previously considered by different authors for the case of nanofluids.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

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