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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.