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In the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

The aim of the present study is to investigate the structural, wear and thermal behaviour of Cu–Al2O3–graphite hybrid metal matrix composites. Copper matrix composites with Al2O3–graphite reinforcement (0.5-0.5, 1.0-1.0, 1.5-1.5 and 2.0-2.0 wt%) were prepared by stir casting process. Phase, microstructure, density, hardness, wear, compressive strength and specific heat of prepared samples have been investigated. X-ray diffraction revealed that there is no intermediate phase formation between matrix and reinforcement phase as a result of interfacial bonding between them. Microstructure study shows the uniform distribution of Al2O3–graphite particles in the Cu-matrix. Density and hardness were found to decrease with increase in reinforcements percentage whereas the compressive strength was found to increase as the amount of reinforcements was increased. Composite containing 2.0 wt% reinforcements showed the maximum resistance to wear. Specific heat was found to increase with addition of reinforcements; however, this increase was very marginal. Structural, wear and thermal properties of these Cu matrix-based hybrid metal matrix composites were found to be dependent on the reinforcements concentration. It is expected that the present composite will be useful for heat exchanger and heat sink applications.

A rigorous analysis of unsteady electrically conducting nanofluid with MHD effect is presented. First, the governing partial differential equations for momentum and energy conservation are converted to couple nonlinear ordinary differential equations by means of exact similarity transformation. The Tiwari-Das nanofluid model is employed to obtain the analytical approximations for flow velocity and temperature distributions of alumina-sodium alginate nanofluid using HAM. The solution is found to be dependent on some parameters including the nanoparticle volume fraction, unsteadiness parameter, magnetic parameter, mixed convection parameter and the generalized Prandtl number. A systematic study is carried out to illustrate the effects of these parameters on the velocity and temperature distributions. Also, the value of skin friction coefficient and local Nusselt number are evaluated with variation of Prandtl number and compared with different nanoparticles.

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