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 nonlinear…
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
Keywords
S. Abbasbandy, Elyas Shivanian, K. Vajravelu and Sunil Kumar
The purpose of this paper is to present a new approximate analytical procedure to obtain dual solutions of nonlinear differential equations arising in mixed convection flow in a…
Abstract
Purpose
The purpose of this paper is to present a new approximate analytical procedure to obtain dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain. This method, which is based on Padé-approximation and homotopy–Padé technique, is applied to a model of magnetohydrodynamic Falkner–Skan flow as well. These examples indicate that the method can be successfully applied to solve nonlinear differential equations arising in science and engineering.
Design/methodology/approach
Homotopy–Padé method.
Findings
The main focus of the paper is on the prediction of the multiplicity of the solutions, however we have calculated multiple (dual) solutions of the model problem namely, mixed convection heat transfer in a porous medium.
Research limitations/implications
The authors conjecture here that the combination of traditional–Pade and Hankel–Pade generates a useful procedure to predict multiple solutions and to calculate prescribed parameter with acceptable accuracy as well. Validation of this conjecture for other further examples is a challenging research opportunity.
Social implications
Dual solutions of nonlinear differential equations arising in mixed convection flow in a semi-infinite domain.
Originality/value
In this study, the authors are using two modified methods.
Details
Keywords
R Ellahi, E Shivanian, S Abbasbandy and T. Hayat
The purpose of this paper is to study the generalized Couette flow of Eyring-Powell fluid. The paper aims to discuss diverse issues befell for the heat transfer…
Abstract
Purpose
The purpose of this paper is to study the generalized Couette flow of Eyring-Powell fluid. The paper aims to discuss diverse issues befell for the heat transfer, magnetohydrodynamics and slip.
Design/methodology/approach
A hybrid technique based on pseudo-spectral collocation is applied for the solution of nonlinear resulting system.
Findings
Viscous fluid results which are yet not available can be taken as a limiting case of presented problem. The results for the case of Hartmann flow can be obtained as a special case when plate velocity is zero, i.e. pressure gradient induced flow. The results for the zero fluid slip and no thermal slip also become special cases of this work, and the results can be recovered by setting, and to zero. These solutions are valid not only for small but also for large values of all emerging parameters.
Originality/value
This model is investigated for the first time, as the authors know.
Details
Keywords
The purpose of this paper is to develop pseudospectral meshless radial point Hermit interpolation (PSMRPHI) for applying to the Motz problem.
Abstract
Purpose
The purpose of this paper is to develop pseudospectral meshless radial point Hermit interpolation (PSMRPHI) for applying to the Motz problem.
Design/methodology/approach
The author aims to propose a kind of PSMRPHI method.
Findings
Based on the Motz problem, the author aims also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods.
Originality/value
Although the PSMRPHI method has been infrequently used in applications, the author proves it is more accurate and trustworthy than the PSMRPI method.
Details
Keywords
Nasibeh Karamollahi, Ghasem Barid Loghmani and Mohammad Heydari
In this paper, a numerical scheme is provided to predict and approximate the multiple solutions for the problem of heat transfer through a straight rectangular fin with…
Abstract
Purpose
In this paper, a numerical scheme is provided to predict and approximate the multiple solutions for the problem of heat transfer through a straight rectangular fin with temperature-dependent heat transfer coefficient.
Design/methodology/approach
The proposed method is based on the two-point Taylor formula as a special case of the Hermite interpolation technique.
Findings
An explicit approximate form of the temperature distribution is computed. The convergence analysis is also discussed. Some results are reported to demonstrate the capability of the method in predicting the multiplicity of the solutions for this problem.
Originality/value
The duality of the solution of the problem can be easily predicted by using the presented method. Furthermore, the computational results confirm the acceptable accuracy of the presented numerical scheme even for estimating the unstable lower solution of the problem.
Details
Keywords
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
Keywords
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
Keywords
This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover…
Abstract
Purpose
This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique.
Design/methodology/approach
First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs.
Findings
The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods.
Originality/value
Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.
Details
Keywords
The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations.
Abstract
Purpose
The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations.
Design/methodology/approach
The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times.
Findings
Numerical simulations corroborate the efficiency and accuracy of the proposed method.
Originality/value
Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.
Details
Keywords
In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational…
Abstract
Purpose
In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method.
Design/methodology/approach
A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method.
Findings
The optimal variational iteration method is found to be useful for heat and fluid flow problems.
Originality/value
The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.