In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.

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In this paper, the study of the problem of simultaneous approximation by the Szász-Mirakjan-Stancu-Durrmeyer type operators is carried out. An upper bound for the approximation to the r th-derivative of a function by these operators is established.

In this paper, we introduced a summation-integral type modification of Szász-Mirakjan operators. Calculation of moments, density in some space, a direct result and a Voronvskaja-type result, are obtained.

Gas turbines play a vital role in the today's industrialized society, and as the demands for power increase, the power output and thermal efficiency of gas turbines must also increase. Modern high-speed aero-engines operate at elevated temperatures about 2000 K to achieve better cycle efficiencies. However, the presently available alloys cannot resist temperatures much higher than 1350 K. Internal cooling techniques for gas turbine blades have been studied for several decades. The internal cooling techniques of the gas turbine blade includes: jet impingement, rib turbulated cooling, and pin-fin cooling which have been developed to maintain the metal temperature of turbine vane and blades within acceptable limits in this harsh environment. The designers need detailed hot gas path heat transfer and temperature distributions along with the detailed flow and heat transfer data to understand the flow physics and to improve the current internal cooling designs. Gas turbine blades are cooled internally by passing the coolant through several artificially roughened serpentine passages to remove heat conducted from the outside surface. The cooling passages located in the middle of the airfoils are often lined with rib turbulators. Near the leading edge of the blade, jet impingement (coupled with film cooling) is commonly used. Pin-fins and dimples can be used in the trailing edge portion of the blades. These techniques have also been combined to further increase the heat transfer from the airfoil walls. For internal cooling, focus is now placed on the effect of rotation on rotor blade coolant passage heat transfer. To better understand the complex three-dimensional flow physics in the complicated blade internal coolant passage geometry, the computational flow and heat transfer results are presented and reviewed at improving the internal cooling of gas-turbine blades.