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Gross domestic product (GDP) is a monetary measure of the market value of overall final goods and services produced in a given year, and serves as a gauge of the economy’s overall health and size. The GDP prediction is significant, as it can capture and understand the future developments of a country’s economy. In this paper, three different mathematical models have been used to predict Malaysia’s gross domestic product using regressions. The models discussed in this paper are linear, exponential and parabolic regressions. In developing the models, data from year 1970 to 2014 has been employed and data from year 2015 to 2019 has been used to examine the models' accuracy. The models are then observed to identify the most appropriate to express the relationship between the years and Malaysia’s gross domestic product. In this study, it is found that the parabolic regression model is more accurate compared to the linear and exponential regression models. The parabolic regression model is also the most appropriate since it is adjusted to the real conditions of Malaysia's gross domestic product which is the main subject of this paper. Finally, it is obtained that the prediction values of GDP in Malaysia will increase for the next ten years (2020 - 2029). Keywords: Gross domestic product, Linear regression, Exponential regression, Parabolic regression

In this paper, we studied the 2D steady laminar boundary layer flow and heat transfer of Casson based nanofluid over an exponentially vertical stretching and shrinking sheet using one phase model. The thermal radiation and heat source/sink parameters are incorporated in the heat transfer equation and the slip parameters for the velocity and temperature are considered in the boundary conditions. The similarity variables have been used to convert the governing equations as a system of partial differential equations to the ordinary differential equations. The transformed equations are then solved by applying shooting technique, shootlib in Maple software. The numerical solutions for the governing equations indicate that the triple solutions arise when high suction is imposed for both stretching/shrinking sheet at certain ranges of the pertinent parameters. To examine the stability of the solutions, stability analyses is done by using BVP4c in Matlab software. The first solution is found to be a stable solution and physical realizable while the remaining solutions are not stable. The inclusion of three different nanoparticles shows that Copper – Casson nanofluid has the highest heat transfer rate compared to Aluminum-Casson and Graphite Oxide-Casson nanofluid.

The aim of a present article is to investigate the laminar unsteady two‐dimensional boundary layer flow of a nanofluid with Stefan blowing and slip effect. First, governing boundary layer equations are converted in the ordinary form of the differential equations (ODEs) from partial differential equations using appropriate coordinate transformations. The obtained ODEs are then solved by applying a shooting method with the Runge‐Kutta fourth order method by implementation of the Maple software. The influences of different controlling dimensionless parameters over the dimensionless velocity, temperature, concentration, friction factor, local heat as well as mass transfer have been discussed and represented by plots. It is found that there exist dual solutions for the different applied nanofluid parameters along with the blowing parameter. The results reveal that by increasing the values of the Brownian motion (Nb), thermophoresis (Nt) and blowing parameters (fw), the skin friction increases (decreases) in the first (second) solution

PurposeThe purpose of this paper is to investigate different branches of the solution of micropolar fluid in a channel with permeable walls. Moreover, the intention of the study is to examine the effect of different physical parameters on fluid flow.Design/methodology/approachThe mathematical modeling is performed on the basis of law of conservation of mass, momentum and angular momentum. The governing partial differential equations were transformed into ordinary differential equations by applying suitable similarity transformation. Afterwards, the set of nonlinear ordinary differential equations was solved numerically by a shooting method.FindingsThe study reveals that various branches of the solution of the proposed problem exist only in the case of strong suction.Originality/valueThe investigation of new branches of the solution of non-Newtonian micropolar fluid is relatively difficult as far as the single solution is concern. This study explores the new branches of the solution of a micropolar fluid in a channel with suction/injection. Simultaneous effect of suction Reynolds number and vortex viscosity parameter on velocity and micro-rotation profile is examined for different branches of solution in order to make the analysis more interesting.

The purpose of the present paper is to study the unsteady two dimensional laminar incompressible flow of non-Newtonian fluid between two orthogonally moving porous disks using the Casson fluid model that is used to characterize the non-Newtonian fluid behavior. Governing partial differential equations are reduced into nonlinear ordinary differential equations by using suitable similarity transformation and then solved numerically by Runge-Kutta-Fehlberg technique. The numerical results are found excellent agreement with the literature published before. Trusting to this validity, effects of different physical parameters are discussed. The study reveals that velocity of the fluid particles increases with an increase in the disk expanding ratio.

Dynamics of nanofluid comprising a base fluid (water) with copper (Cu) nanoparticles have been considered in channel with porous walls under magnetic field influence. The channel walls are considered to be permeable in order to analyze the wall mass transfer phenomenon. Relevant mathematical modelling has been performed and the derived PDEs are converted into coupled nonlinear ODEs by using suitable transformations. Computations have been made numerically by employing the shooting technique. It is noted that multiple solutions occur for the variation of suction Reynolds number, solid volume fraction and magnetic parameters which are interpreted in detail.

A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver (Ag) and copper (Cu) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of Ag and Cu are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver Ag nanoparticles show the greater rate of heat transfer enhancement as compared to copper Cu nanoparticles when suspended in Casson fluid., Mechanical engineering; Casson nanofluid; Shooting method; Dual solution; Stability analysis; Stretching/shrinking surface.

This paper deals with heat transfer analysis on MHD nanofluids in a porous channel with stretching walls. An incompressible Copper (Cu)–water nanofluid is filled in the porous channel. The governing partial differential equations are first transformed into ordinary differential equations by using suitable similarity transformation. The resulting nonlinear ordinary differential equations are then solved numerically by using shooting method. It is found that the different values of nanoparticle volume fraction, Reynolds number, magnetic field and Prandtl number affects the profiles of velocity, temperature, skin friction and heat transfer.

Purpose The purpose of this paper is to investigate some multiple solutions for carbon nanotubes (CNTs) in a porous channel with changing walls. Moreover, the intention of the study is to examine the physicochemical and rheological properties of titania and CNTs. Design/methodology/approach The mathematical modeling is performed for the laws of conservation of mass, momentum and energy profiles. Governing partial differential equations are transformed into ordinary differential equations by applying suitable similarity transformation and then solved numerically with the help of shooting technique. Findings The effects of different physical parameters on the rheology of nanofluids are discussed graphically and numerically, in order to make the analysis more interesting. The present study revealed that multiple solutions of nanofluids in a channel with changing walls are obtained only for the case of suction. Originality/value Some new branches of the solution for nanofluids in a channel with changing walls are obtained in this research, which has never been reported before. Combined effects of physicochemical and rheology of titania and carbon nanotubes are investigated briefly. The effects of physical parameters on velocity and temperature profile are examined in detail which is more realistic than real-world problem.

A numerical study is carried out in order to investigate the multiple solutions of micropolar fluid in a channel with changing walls. Mathematical modeling of laws of conservation of mass, momentum, angular momentum and energy is performed and governing partial differential equations are converted into self-similar ordinary differential equations by applying suitable similarity transformation and then solved numerically by shooting method. A new branch of solutions is found and presented graphically and numerically for the various values of parameters, which has never been reported.

An investigation has been carried out to present the hydro-magnetic three-dimensional flow of a nanofluid in a rotating channel. Simultaneous effects of energy and concentration of the nanoparticles have been considered and discussed. Velocity, thermal and concentration slip conditions are utilized in order to make the analysis more interesting. Brownian motion and thermophoresis effects are analyzed in order to discuss the nanofluids for the rotating flow situations. The mathematical modeling is performed for the laws of conservation of mass, momentum, energy and mass fraction profiles. The computations are made numerical and the obtained results are presented graphically and numerically in details.

This paper proposes a new generalized half-step hybrid block method with one off-step point to find the direct solution of second order ordinary differential equations. In developing this method, a power series adopted as an approximate solution is interpolated at \( x_{n} \) and \( x_{n + p} \) points while its second derivatives collocated at all points in the interval i.e. \( x_{n} \), \( x_{n + p} \) and \( x_{{n + \frac{1}{2}}} \) to obtain the main continuous scheme, where \( 0 < p < \frac{1}{2} \). The analysis of the method such as order, zero stability, consistency and convergence is also discussed. The derived method is then compared with the existing methods in terms of accuracy. The numerical results suggest that the new method can be served as a viable alternative to solve the initial value problems of second order.

A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver ( A g ) and copper ( C u ) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of A g and C u are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver A g nanoparticles show the greater rate of heat transfer enhancement as compared to copper C u nanoparticles when suspended in Casson fluid.

The purpose of the present paper is to investigate the micropolar nanofluid flow on permeable stretching and shrinking surfaces with the velocity, thermal and concentration slip effects. Furthermore, the thermal radiation effect has also been considered. Boundary layer momentum, angular velocity, heat and mass transfer equations are converted to non-linear ordinary differential equations (ODEs). Then, the obtained ODEs are solved by applying the shooting method and in the results, the dual solutions are obtained in the certain ranges of pertinent parameters in both cases of shrinking and stretching surfaces. Due to the presence of the dual solutions, stability analysis is done and it was found that the first solution is stable and physically feasible. The results are also compared with previously published literature and found to be in excellent agreement. Moreover, the obtained results reveal the angular velocity increases in the first solution when the value of micropolar parameter increases. The velocity of nanofluid flow decreases in the first solution as the velocity slip parameter increases, whereas the temperature profiles increase in both solutions when thermal radiation, Brownian motion and the thermophoresis parameters are increased. Concentration profile increases by increasing N t and decreases by increasing N b .

This paper investigate theoretically the problem of steady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid at a stagnation point over an exponentially shrinking vertical sheet with suction. In this study, we assume that the shrinking velocity and wall temperature have specific exponential function form. An appropriate similarity transformation is employed to transform the governing equations in partial differential equations form to similarity equations in ordinary differential equations form. The resulting equations are then solved numerically using shooting technique with Maple implementation. The influence of mixed convection/buoyancy parameter λ , suction parameter s and shrinking variable c / a on the flow and heat transfer characteristics is examined and discussed. Numerical results indicate that the presence of buoyancy force would contribute to the presence of triple solutions to the flow and heat transfer for particular value of pertinent parameters.

The unsteady flow over a continuously shrinking surface with wall mass suction in a nanofluid using the model proposed by Buongiorno [7] is investigated. By using an appropriate similarity transformation, similarity equations are obtained and the shooting method is used to solve these equations for different values of the wall mass suction, unsteadiness and nanofluid parameters. It is found that dual solutions exist for a certain range of wall mass suction, unsteadiness and nanofluid parameters.

In this paper, we investigate theoretically the problem of steady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of buoyancy force over an exponentially shrinking vertical sheet with suction. The shrinking velocity and wall temperature are assumed to have specific exponential function forms. The governing equations are first transformed to similarity equations using an appropriate similarity transformation. The resulting equations were then solved numerically using shooting technique involving fourth-order Runge–Kutta method and Newton–Raphson method. The influence of mixed convection/buoyancy parameter λ, suction parameter s and Prandtl number Pr on the flow and heat transfer characteristics is examined and discussed. Numerical results indicate that the presence of buoyancy force would contribute to the existence of triple solutions to the flow and heat transfer for particular value of pertinent parameters. It is different for the non-buoyant flow case i.e. when the buoyancy force is absent, the problem admits only dual solutions. Further, this study also reveals that the features of flow and heat transfer characteristics are significantly affected by buoyancy parameter λ, suction parameter s and Prandtl number Pr.

This study focuses on Casson fluid flow in a channel with slowly expanding and contracting walls. Main structures of partial differential equations were taken in the form of continuity, momentum and energy equations. Governing partial differential equations were converted into set of nonlinear ordinary differential equations by applying suitable similarity transformation and then solved numerically by employing shooting technique and triple solutions are obtained depending on Reynolds, Casson, expanding or contracting ratio and magnetic parameters considered.

The present study is focused on the presentation of a numerical solution for copper-water nanofluid through a stretching channel with spherical and cylindrical shape nanoparticles. The analysis of nanofluid in a channel with stretching walls under slip effects is made by introducing the conservation equation of nanoparticle volume fraction into Hamilton-Crosser’s nanofluid model. Governing partial differential equations are transformed into nonlinear ordinary differential equations by applying similarity transformation and then solved with the help of shooting method. The effects of different physical parameters on the rheology of nanofluids’ particles are presented in tabulation and pictorial representation. The study reveals that the thermal boundary layer thickness increases by increasing the solid volume fraction.

The steady mixed convection boundary-layer flow on a vertical circular cylinder embedded in a porous medium filled by a nanofluid is studied for both cases of a heated and a cooled cylinder. The governing system of partial differential equations is reduced to ordinary differential equations by assuming that the surface temperature of the cylinder and the velocity of the external (inviscid) flow vary linearly with the axial distance x measured from the leading edge. Solutions of the resulting ordinary differential equations for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction $${\phi}$$ , the mixed convection or buoyancy parameter λ and the curvature parameter γ. Results are presented for the specific case of copper nanoparticles. A critical value λ c of λ with λ c λ c for both aiding, λ > 0 and opposing, λ

The effects of suction/blowing and viscous dissipation on the steady non-linear viscous flow and heat transfer over a horizontal shrinking permeable surface of variable temperature or heat flux with a power-law velocity is numerically studied. The results complete those recently reported by Cortell [10] to the case of a shrinking surface. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of multiple solutions. Two cases are considered, namely prescribed surface temperature (PST) and prescribed heat flux (PHF). Dual and triple solutions are found to exist for certain parameters.

A problem derived previously (Rohni et al., Transp Porous Media 92:1–14, 2012) for unsteady mixed convection flow in a porous medium involving a ‘temperature slip’ boundary condition and fluid transfer through the boundary is considered. It is shown that the solution to this problem can be directly related to the solution of the corresponding problem for a prescribed surface temperature, involving a mixed convection parameter λ, an unsteadiness parameter A and transpiration parameter s. This latter problem is discussed in detail, particular attention being given to the steady analogue, A = 0, allowing for fluid transfer through the surface, and to the unsteady problem, A > 0, for an impermeable surface, s = 0. Asymptotic results are obtained for large fluid transfer rates, \({s \gg 1}\) and \({s

The unsteady flow over a continuously shrinking surface with wall mass suction in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3 and TiO2 is numerically studied. Similarity equations are obtained through the application of similarity transformation techniques. The shooting method is used to solve the similarity equations for different values of the wall mass suction, the unsteadiness and the nanoparticle volume fraction parameters. The results of skin friction coefficient f ″ ( 0 ) and heat transfer rate −θ′(0) are presented in tables and graphs. It is found that dual solution exists for a certain range of wall mass suction s, volume fraction φ and unsteadiness parameters A. The results of velocity and temperature profiles are also presented. It is seen that two values of boundary layer thickness η∞ are obtained, which gives two different velocity and temperature profiles that satisfy the boundary conditions. It is also found that the nanoparticle volume fraction parameter φ and types of nanofluid play an important role to significantly determine the flow behaviour.

The unsteady flow over a continuously shrinking sheet with wall mass suction in a nanofluid is numerically studied. The governing boundary layer equations are transformed into a set of nonlinear ordinary differential equations by using similarity transformation. The resulting similarity equations are then solved by the shooting method for three types of nanofluid: copper-water, alumina-water and titania-water to investigate the effect of nanoparticle volume fraction parameter ɸ to the flow in nanofluid. The skin friction coefficient and velocity profiles are presented and results show that dual solutions exist for a certain range of unsteadiness parameter A. It is also found that the nanoparticle volume fraction parameter ɸ and types of nanofluid play an important role to significantly determine the flow behaviour.

The unsteady mixed convection boundary-layer flow near the two-dimensional stagnation point on a vertical permeable surface embedded in a fluid-saturated porous medium with suction and a temperature slip effect is studied numerically. Similarity equations are obtained through the application of a similarity transformation technique. The shooting method is used to solve these similarity equations for different values of the mixed convection, wall mass suction, the unsteadiness and the slip parameters. Results show that multiple solutions exist for certain ranges of these parameters. Some limiting forms are then discussed, namely strong suction, the free convection limit, the situation when there is a large temperature slip and when the time dependence dominates.

PurposeThe purpose of this paper is to theoretically investigate the steady two‐dimensional boundary‐layer flow past a moving semi‐infinite flat plate in a water‐based nanofluid containing three different types of nanoparticles: copper (cuprum) Cu, alumina (aluminium oxide) Al2O3, and titania (titanium dioxide) TiO2. The effects of moving parameter λ as well as solid volume fraction parameter φ on the flow and heat transfer characteristics are studied. Taking into account the rising demands of modern technology, including chemical production, power stations and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance.Design/methodology/approachA similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using Keller‐box method.FindingsThere is a region of unique solutions for λ>0, however, multiple (dual) solutions exist for λc<λ≤0 and no solutions for λ<λcλResearch limitations/implicationsThe solutions can be obtained up to a certain value of the moving parameter (critical value or turning point). The boundary layer separates from the plate beyond the turning point hence it is not possible to get the solution based on the boundary‐layer approximations after this point. To obtain further solutions, the full Navier‐Stokes equations have to be solved.Originality/valueThe present results are original and new for the boundary‐layer flow and heat transfer of a moving flat plate in a nanofluid. Therefore, this study would be important for the scientists and engineers in order to become familiar with the flow behaviour and properties of such nanofluids, and the way to predict the properties of this flow for the process equipments.

A numerical investigation is made to determine the occurrence of the multiple solutions of MHD Casson fluid in a porous channel. Governing partial differential equation of the proposed problem converted into nonlinear ordinary differential equations by using similarity transformation. Numerical technique known as shooting method is used to investigate the existence of the multiple solutions for the variations of different parameters. Effects of physical parameters on velocity profile, temperature, concentration, and skin friction are presented in pictorial and tabulation representation.

The aim of this Letter is to show that a further exact solution of the problem on unsteady mixed convection boundary layer in a porous medium with temperature slip can be obtained in addition of that reported very recently by Fang et al. (2012).

A study has been carried out to examine the occurrence of multiple solutions for Copper-Water nanofluids flows in a porous channel with slowly expanding and contracting walls. The governing equations are first transformed to similarity equations by using similarity transformation. The resulting equations are then solved numerically by using the shooting method. The effects of wall expansion ratio and solid volume fraction on velocity and temperature profile have been studied. Numerical results are presented graphically for the variations of different physical parameters. The study reveals that triple solutions exist only for the case of suction.

The most challenging task dealing with permutation is when the element is large. In this paper, a new algorithm for listing down all permutations for n elements is developed based on distinct starter sets. Once the starter sets are obtained, each starter set is then cycled to obtain the first half of distinct permutations. The complete list of permutations is achieved by reversing the order of the first half of permutation. The new algorithm has advantages over the other methods due to its simplicity and easy to use.

The Keller box method has been employed for free convection over a vertical plate subjected to a step change in surface temperature in micropolar fluid. Numerical results presented include the reduced angular velocity profiles, development of wall shear stress or skin friction and development of the rate of change of the gyration component at the wall for various values of Prandtl numbers and temperature ratios. The study shows that the present results obtained in micropolar fluids, when temperature ratio 0w2=1, agree very well with the previous study without temperature change.

The mathematical model for a boundary layer flow due to a moving flat plate in micropolar fluid is discussed. The plate is moving continuously in the positive x-direction with a constant velocity. The governing boundary-layer equations are solved numerically using an implicit finite-difference scheme. Numerical results presented include the reduced velocity profiles, gyration component profiles and the development of wall shear stress. The results obtained, when the material parameter K = 0 (Newtonian fluid) showed excellent agreement with those for viscous fluids. Further, the wall shear stress increases with increasing K. For fixed K, the wall shear stress decreases and the gyration component increases with increasing values of n, in the range 0 ≤ n ≤ 1 where n is a ratio of the gyration vector component and the fluid shear stress at the wall.