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In this article, the semi-analytical method known as the Differential Transform Method (DTM) for solving different types of differential equations is reviewed. First, basic definitions and formulas of DTM and Differential Transform-Padé approximation (DTM-Padé), which are used to increase the convergence and accuracy of DTM approximations, are discussed. Then both techniques of DTM and DTM-Padé, which have been successfully applied to partial differential equations, as well as the application of these methods in fluid mechanic and heat transfer are presented. In addition, the extension of DTM for integral differential equations and the fuzzy differential transformation method (FDTM) for fuzzy problems are discussed.

Two new functions on the semi-infinite interval, namely Rational Gegenbauer (R) and Exponential Gegenbauer (E) functions are proposed to solve the heat transfer problem. The considered problem is flow of MHD micropolar over a moving plate with suction and injection boundary conditions. For applying Tau method efficiently, two matrices of derivative and product for both of rational and exponential Gegenbauer whose enable us to solve a system of nonlinear algebraic equations on the semi-infinite interval were introduced, and an error bound of these functions approximation was estimated which led to have an exponential convergence rate in this method. Moreover, the influence of the important physical parameters on heat and mass transfer phenomena are studied with details. Comparing the results of Rational Gegenbauer Tau and Exponential Gegenbauer Tau methods with available analytical and numerical solutions shows that the present methods are efficient and have fast convergence rate and high accuracy. This method can solve a set of coupled nonlinear and high-order differential equations on a semi-infinite domain by converting to a set of linear equations.

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditions on both ends of the domain. The proposed method is based on spectral method. We construct some new operational matrices which have the ability to handle integral type non-local boundary constrains. These operational matrices can be effectively applied to convert the FPDEs together with its integral types boundary conditions to easily solvable matrix equation. The accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems. The proposed method has the ability to solve non-local FPDEs with high accuracy and low computational cost. Different aspects of presented approach are compared with two other recently developed methods, Haar wavelets collocation method and a family of collocation methods which are based on Radial base functions. It is observed that the proposed method is highly accurate, robust, efficient and stable as compared to these methods.

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

In the current article, we investigated a three-dimensional nanofluid bio-convection model near a stagnation attachment. The set of governing equations, with a set of similarity variables, expressing the conservation of momentum, total mass, microorganisms, thermal energy and nanoparticles are reduced to nonlinear ODEs set. The obtained nonlinear system is tackled via collocation-based Gegenbauer wavelets technique. The purpose of this extension is to reduce the computational work as compared to the traditional Gegenbauer wavelets method. Additionally, the suggested trial solutions must satisfy the boundary conditions. Convergence analysis shows an excellent agreement. The graphical plots for various physical parameters are illustrated. Additionally, the significant physical quantities of viable interests and the impact of different physical parameters on the dissemination of the motile microorganisms are also included in our study. The error and convergence analyses are endorsing that the proposed method is well efficient and could be extended to other nonlinear problems.

The present work explores the analytical study of nanofluid flow above a stretching medium with the heat source and viscous dissipation. Additional radiative effects are also incorporated. The main physical problem is offered and changed into an arrangement of combined nonlinear differential equations with appropriate transformations. Optimal homotopy analysis method is used to attain the analytical solutions of the set of nonlinear differential equations. Important predictions of the flow phenomena are explored and deliberated by means of graphs and numerical tables. Moreover, the accurateness of the existing findings is verified by equating them with the previously available work.

In this paper, the first and second-law analysis for the thermodynamic air-standard Atkinson cycle with an account for heat transfer is performed using finite-time thermodynamics. In order to have more accurate evaluations, the effects of thermodynamic and design key parameters on the performance characteristics of Atkinson cycle are shown. Further, artificial neural network and imperialist competition algorithm are employed to predict and optimize the net work output value versus the minimum cycle temperature and also the compression ratio. The results obtained show that the heat loss is an effective factor of the performance of the cycle and it should be considered in the analysis and comparison of practical internal combustion engines.

In this paper, stagnation point flow over a permeable shrinking sheet under the influence of Magnetohydrodynamics (MHD) is analyzed using Successive linearization method. The governing equations are simplified with the help of similarity variables. The resulting nonlinear coupled differential equations have been solved with a combination of Successive linearization method and Chebyshev spectral collocation method. The impact of various pertinent parameters are demonstrated numerically and graphically. Comparison has also been made with the existing literature and found that the present results are in good agreement. It is also found that the present methodology converges more rapidly.

An efficient Spectral Collocation method based on the shifted Legendre polynomials was applied to get solution of heat transfer of a micropolar fluid through a porous medium with radiation. A similarity transformation is applied to convert the governing equations to a system of non-linear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an approximate solution for this system of non-linear differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution procedure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed Spectral Collocation method and those of previously published ones.

The fluid mechanics of incompressible turbulent boundary layers air flow over a flat plate is investigated using an open-ended suction wind tunnel. The wall shear stress is measured by a distinct method using Thin-Oil-film technique in order to determine skin friction of the plate. On theoretical side, the governing partial differential equations are transformed to an ordinary differential equation with inconsistent coefficients using similarity variables and they are solved by variational iteration method. The distribution of the velocity, friction coefficient and thickness of the boundary layer are obtained analytically and experimentally, and compared with the previously reported results, where good agreements are observed. New correlations for skin friction coefficient and boundary layer thickness of turbulent flow over flat plate are proposed.

The key goal of this article is to present an efficient hybrid computational technique, namely homotopy analysis transform method (HATM), to investigate Jeffery–Hamel flow. The HATM is an innovative and efficient amalgamation of homotopy analysis technique, standard Laplace transform scheme and homotopy polynomials. The effect of Reynolds number on velocity profile is studied graphically. The obtained results are compared with existing results and it is noticed that the outcomes are in an excellent agreement. The outcomes of the suggested method reveal that the technique is easy to handle and computationally very fantastic.

The magneto hydrodynamic boundary layer flow with heat and mass transfer of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity under the radiation effect is examined. It is assumed that the sheet is non-flat. The governing partial differential equations are reduced to nonlinear coupled ordinary differential equations by applying the suitable similarity transformations. These nonlinear coupled ordinary differential equations, subject to the appropriate boundary conditions, are then solved by using spectral quasi-linearisation method (SQLM). The effects of the physical parameters on the flow, heat transfer and nanoparticle concentration characteristics of the problem are presented through graphs and are discussed in detailed. Numerical values of skin friction co-efficient and Nusselt number with different parameters were computed and analysed. Keywords: MHD, SQLM, Variable thermal conductivity, Variable thickness, Williamson fluid

Purpose The purpose of this paper is to present investigation of the flow, heat and mass transfer of a nanofluid over a suddenly moved flat plate using Buongiorno’s model. This study is different from some of the previous studies as the effects of Brownian motion and thermophoresis on nanoparticle fraction are passively controlled on the boundary rather than actively. Design/methodology/approach The partial differential equations governing the flow are reduced to a system of non-linear ordinary differential equations. Viable similarity transforms are used for this purpose. A well-known numerical scheme called Runge-Kutta-Fehlberg method coupled with shooting procedure has been used to find the solution of resulting system of equations. Discussions on the effects of different emerging parameters are provided using graphical aid. A table is also given that provides the results of different parameters on local Nusselt and Sherwood numbers. Findings A revised model for Stokes’ first problem in nanofluids is presented in this paper. This model considers a zero flux condition at the boundary. Governing equations after implementing the similarity transforms get converted into a system of non-linear ordinary differential equations. Numerical solution using RK-Fehlberg method is also carried out. Emerging parameters are analyzed graphically. Figures indicate a quite significant change in concentration profile due to zero flux condition at the wall. Originality/value This work can be extended for other problems involving nanofluids for the better understanding of different properties of nanofluids.

In this paper, an efficient spectral collocation method based on the shifted Legendre polynomials is applied to study the unsteady boundary-layer flow and heat transfer due to a stretching sheet. A similarity transformation is used to reduce the governing unsteady boundary-layer equations to a system of non-linear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an analytical approximate solution of this system of non-linear ordinary differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution procedure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed spectral collocation method and those of previously published ones.

To localize objects in Web images using an invariant descriptor is crucial. The HOG (histogram of oriented gradients) descriptor is used to increase the accuracy of localization. It is a shape descriptor that considers frequencies of gradient orientation in localized portions of an image. This well known descriptor does not cover rotation variations of an object in images. This paper introduces a rotation invariant feature descriptor based on HOG. The proposed descriptor is used in a top-down searching technique that covers the scale variation of the objects in images. The efficiency of this method is validated by comparing the performance with existing research in a similar domain on the Caltech-256 Web dataset. The proposed method not only provides robustness against geometrical transformations of objects but also is computationally more efficient. HighlightsGeometrical invariant Object localization/recognition in Web images.The Rotation Invariant HOG (RIHOG).Fast scale invariant object localization/recognition in images.The Rapid Ranking Method (RRM) is based on Top-Down searching under RIHOG.Geometrical invariant image ranking method.

In this paper, the Homotopy Analysis Method (HAM) with two auxiliary parameters and Differential Transform Method (DTM) are employed to solve the geometric nonlinear vibration of Euler-Bernoulli beams subjected to axial loads. A second auxiliary parameter is applied to the HAM to improve convergence in nonlinear systems with large deformations. The results from HAM and DTM are compared with another popular numerical method, the shooting method, to validate these two analytical methods. HAM and DTM show excellent agreement with numerical results (the maximum errors in our calculations are about 0.002%), and they additionally provide a simple way to conduct a parametric analysis with different physical parameters in Euler-Bernoulli beams. To show the benefits of this method, the effect of different physical parameters on the amplitude is discussed for a cantilever beam with a cyclically varying axial load.

The effects of axisymmetric flow of a Powell-Eyring fluid over an impermeable radially stretching surface are presented. Characteristics of the heat transfer process are analyzed with a more realistic condition named the convective boundary condition. Governing equations for the flow problem are derived by the boundary layer approximations. The modeled highly coupled partial differential system is converted into a system of ordinary differential equations with acceptable similarity transformations. The convergent series solutions for the resulting system are constructed and analyzed. Optimal values are obtained and presented in a numerical form using an optimal homotopy analysis method (OHAM). The rheological characteristics of different parameters of the velocity and temperature profiles are presented graphically. Tabular variations of the skin friction coefficient and the Nusselt number are also calculated. It is observed that the temperature distribution shows opposite behavior for Prandtl and Biot numbers. Furthermore, the rate of heating/cooling is higher for both the Prandtl and Biot numbers. 2016, Shanghai University and Springer-Verlag Berlin Heidelberg. Scopus

In this article, entropy generation on MHD nanofluid towards a stagnation point flow over a permeable stretching surface has been investigated numerically. The governing equations of nanofluid are simplified using similarity variables with the help of momentum, energy and concentration equations. The resulting highly nonlinear coupled differential equations are solved with the help of successive linearization method and Chebyshev spectral collocation method. The impact of all the pertinent parameters such as Hartmann number, suction/injection parameter, heat source/sink parameter, Lewis number, Prandtl number, Brownian motion parameter, thermophoresis parameter are demonstrated graphically. Furthermore, the effect of Brinkman number and Reynolds number are also presented for entropy generation. It is analyzed that the velocity of the fluid increases due to greater influence of magnetic field and porosity parameter. Moreover, it is also observed that the entropy generation number increase due to the increment in Brinkman number and Reynolds number. Numerical comparison is also given with the existing published literature and found that the present results are in good agreement.

In this article, a new numerical method is employed to solve the stagnation-point flow problem over a permeable stretching/shrinking sheet through porous media. The effects of magnetohydrodynamics (MHD) and heat transfer are also taken into account. The governing flow problem is based on momentum equation and energy equation which are further simplified with the help of similarity transformations. The reduced resulting highly non-linear coupled ordinary differential equations are solved using the successive linearization method (SLM) and Chebyshev spectral collocation method. The impact of the physical parameters of interest is sketched for velocity and temperature profiles. The numerical comparison is also presented with the existing literature which shows that the present results are in good agreement and also confirms the validity of SLM for the present problem.

In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear) under $\hat{m}$-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the boundary value problem to a system of easily solvable algebraic equations. We show the applicability of the scheme by solving some test problems. The scheme is computer oriented.

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.

In this article, stagnation point flow of a Maxwell fluid over a shrinking porous sheet is considered. The governing partial differential equations are reduced to ordinary differential equations by using similarity transformations. The solution of the resulting nonlinear boundary value problem is calculated with the help of Successive linearization method (SLM) using computational software Matlab. The present analysis confirmed the existence of dual solution for shrinking sheet, while for stretching case the solution is unique. The effects of suction parameter S on the velocity profiles are shown through graphs and analyzed in detail.

In the previous work (Akram and Rehman Numer. Algor. 62 527---540 2013), Akram and Rehman presented the reproducing kernel method (RKM) for solving various eighth order boundary value problems. However, an effective error estimation for this method has not yet been discussed. This work is devoted to deal with this problem. Some other aspects of the RKM will be considered such as convergence analysis and numerical implementations.

This study is dedicated to analyze the combined effects of partial slip and prescribed surface heat flux when the fluid is moving due to stretching cylinder. A very moderate and powerful technique Chebyshev Spectral Newton Iterative Scheme is used to determine the solution of the present mathematical model. Involved physical parameters, namely the slip parameter, Casson fluid parameter, curvature parameter and Prandtl number are utilized to control the fluid moments and temperature distribution. The results show that the fluid velocity and the skin friction coefficient on the stretching cylinder are strongly influenced by the slip parameter. It is further analyzed that hydrodynamic boundary layer decreases and thermal boundary layer increases with the slip parameter. Influence of Casson fluid parameter on temperature profile provides the opposite behavior as compared to the slip parameter. The comparison of numerical values of skin friction coefficient and the local Nusselt number is made with the results available in the literature. The accuracy and convergence of Chebyshev Spectral Newton Iterative Scheme is compared with finite difference scheme (Keller box method) through tables. The CPU time is calculated for both schemes. It is observed that CSNIS is efficient, less time consuming, stable and rapid convergent.

This paper presents the successful application of the Jensen-Shannon divergence (JSD) for the overall condition assessment of bridge deck structures using nondestructive evaluation (NDE) data. Periodic (time-lapsed) results from electrical resistivity, half-cell potential, ground penetrating radar, ultrasonic surface wave, and impact-echo measurements on the Haymarket Bridge deck (located in Virginia, U.S.) are represented in the form of probability distributions, and the deviation from original distributions is calculated using the square root of the Jensen-Shannon divergence ( J S D ). The effects of bin size on the probability distribution, and of instruments varying in resolution on J S D are discussed. Finally, by assuming ideal probability distributions representing an intact condition of the bridge deck, an NDE index for condition assessment of bridge decks is suggested and deterioration curves are developed. Results demonstrate the effectiveness of J S D to evaluate relative changes in the bridge deck condition.

This article deals with the laminar flow of a viscous fluid and heat transfer analysis due to a porous stretching/shrinking cylinder with partial slip condition. The flow equations corresponding to momentum and energy equations are transformed into a set of highly nonlinear ordinary differential equations by means of similarity transformations, which are then, solved numerically using Runge–Kutta–Fehlberg method. The physical significance of the various involved parameters on the flow velocity and temperature distribution is discussed through graphs and tables in detail. It is found that the dual solutions exist for the shrinking cylinder, while a unique solution exists for stretching cylinder. Comparison of the present results with the existing previous results is given and found to be in good agreement.

Mathematical modeling of many autonomous physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear evolution equations plays a significant role in the study of nonlinear physical phenomena. In this article, the enhanced (G′/G)-expansion method has been applied for finding the exact traveling wave solutions of longitudinal wave motion equation in a nonlinear magneto-electro-elastic circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. It has been shown that the applied method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields.

In this study, a comparison among three semi-analytical numerical integration algorithms for solving stiff ODE systems is presented. The algorithms are based on Differential Transform Method (DTM) which are Multiple-Step DTM (MsDTM), Enhanced MsDTM (E-MsDTM) and MsDTM with PadA© approximants (MsDTM-P). These methods can be classified as explicit one step semi-analytical numerical integration methods. The error and stability analysis of each method is presented. New important relationships among the methods are introduced. To demonstrate our results, a comparison of the accuracy, stability and computational efficiency of the methods is presented through solving some linear and nonlinear problems arising in applied science and engineering.

The similar solution on the equations of the revised Cheng-Minkowycz problem for natural convective boundary layer flow of nanofluid through a porous medium gives (using an analytical method), a system of non-linear partial differential equations which are solved by optimal homotopy analysis method. Effects of various drastic parameters on the fluid and heat transfer characteristics have been analyzed. A very good agreement is observed between the obtained results and the numerical ones. The entropy generation has been derived and a comprehensive parametric analysis on that has been done. Each component of the entropy generation has been analyzed separately and the contribution of each one on the total value of entropy generation has been determined. It is found that the entropy generation as an important aspect of the industrial applications has been affected by various parameters which should be controlled to minimize the entropy generation.

In this paper, the problem of the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field is investigated numerically. Using a Berman’s similarity transformation, the two-dimensional momentum conservation partial differential equations can be written as a system of nonlinear ordinary differential equations incorporating Lorentizian magneto-hydrodynamic body force terms. A new computational method based on the operational matrix of derivative of orthonormal Bernstein polynomials for solving the resulting differential systems is introduced. Moreover, by using the residual correction process, two types of error estimates are provided and reported to show the strength of the proposed method. Graphical and tabular results are presented to investigate the influence of the Hartmann number (Ha) and the transpiration Reynolds number (Re on velocity profiles in the channel. The results are compared with those obtained by previous works to confirm the accuracy and efficiency of the proposed scheme.

In this paper, we introduce controlled frames in Hilbert [Formula: see text]-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert [Formula: see text]-module. Also multiplier operators for controlled frames in Hilbert [Formula: see text]-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert [Formula: see text]-modules and verify their relations to controlled frames and multiplier operators.

Analysis has been conducted to present the generalized magnetic field effects on the flow of a Burgers' nanofluid over an inclined wall. Mathematical modelling for hydro-magnetics reveals that the term "[Formula: see text]" is for the Newtonian model whereas the generalized magnetic field term (as mentioned in Eq 4) is for the Burgers' model which is incorporated in the current analysis to get the real insight of the problem for hydro-magnetics. Brownian motion and thermophoresis phenomenon are presented to analyze the nanofluidics for the non-Newtonian fluid. Mathematical analysis is completed in the presence of non-uniform heat generation/absorption. The constructed set of partial differential system is converted into coupled nonlinear ordinary differential system by employing the suitable transformations. Homotopy approach is employed to construct the analytical solutions which are shown graphically for sundr5y parameters including Deborah numbers, magnetic field, thermophoresis, Brownian motion and non-uniform heat generation/absorption. A comparative study is also presented showing the comparison of present results with an already published data.

This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. In this paper, it can be observed that the auxiliary parameter ℏ , which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples.

In the present study, the combination of differential transform method (DTM) and Pade approximant, called DTM-Pade, is employed to investigate entropy generation in magnetohydrodynamic (MHD) stagnation-point flow with heat transfer in a porous medium. An entropy generation equation is derived as a function of velocity and temperature gradients. A very good agreement is observed between some of the obtained results of the current study and those of the shooting method results. The results are given to survey the effect of various parameters like magnetic interaction number, Brinkman number, Reynolds number, and Hartmann number on the entropy generation rate and Bejan number. In addition, the rates of two irreversibility mechanisms are investigated for each of the involved parameters.

In this paper, numerical techniques are developed for solving two-dimensional Bratu equations using different neural network models optimized with the sequential quadratic programming technique. The original two-dimensional problem is transformed into an equivalent singular, nonlinear boundary value problem of ordinary differential equations. Three neural network models are developed for the transformed problem based on unsupervised error using log-sigmoid, radial basis and tan-sigmoid functions. Optimal weights for each model are trained with the help of the sequential quadratic programming algorithm. Three test cases of the equation are solved using the proposed schemes. Statistical analysis based on a large number of independent runs is carried out to validate the models in terms of accuracy, convergence and computational complexity.

The present research derives analytical expressions for the heat transfer in a porous annulus with pulsating pressure gradient imposed across the annulus. The governing partial and coupled equations are transformed to ordinary ones by utilizing the similarity variables and then solved analytically using Homotopy Analysis Method (HAM). The analytic solution is expressed in the form of series, and its convergence is analyzed. The resulting temperature distribution, Nusselt number and entropy generation profiles are shown graphically for the different values of the parameters of the problem. A numerical solution has also been obtained to verify the solution by HAM. Good agreement is observed between the two methods.

In this letter, it is shown that the applied relations in the paper by Hakan Ozcan [H. Ozcan, The effects of heat transfer on the exergy efficiency of an air-standard Otto cycle, Heat and Mass Transfer (2011) 47:571–577] are erroneous and thus the reported results are invalid. These incorrect relations [Eqs. (8), (9), (10), (14) and (16) of HO2011] are replaced by correct ones. Moreover, the obtained results (graphs and tables) are modified based on the correct relations. Finally, to achieve more realistic results, the internal irreversibility described by using the compression and expansion efficiencies is added to the analysis.

The similarity solution for the steady stagnation flow towards an off-centered rotating disc is gives a systemof non-linear partial differential equations. These nonlinear differential equations are numerically solved by applyingwell known Keller-Box Method. After finding the velocity distributions, the important designing subject, entropy generation of this system has been analyzed. Graphical results are presented to investigate effects of the rotation ratio α, off-centering, Reynolds number and axial height on the radial and azimuthal velocities and entropy generation. In order to show the effectiveness of the Keller- Box method, the obtained results are compared with available solutions obtained using DTM. The obtained results demonstrate the reliability of the algorithm and the Keller- Box method is an attractive method in solving the systems of nonlinear partial differential equations, and also the entropy generation is an important parameter depends on design andwork conditions that should be in the attention of designers of these rotating systems.