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In this work, the Chebyshev collocation scheme is extended for the Volterra integro-differential equations of pantograph type. First, we construct the operational matrices of pantograph and derivative based on Chebyshev polynomials. Also, the obtained operational matrices are utilized to approximate the derivatives of unknown functions. Furthermore, a detailed analysis of convergence is discussed in the weighted square norm. We conduct some numerical experiments to verify the high performance of the suggested numerical approach. The results show that the computational scheme is accurate.

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.

The Laplace transform and Laguerre polynomials are applied to the solution of the Bagley–Torvik equation. We first convert the fractional differential equation into the algebraic equation by Laplace transform. Then, we find the numerical inversion of Laplace transform by Laguerre polynomials. The fractional derivative is described in the Caputo sense. Numerical examples are provided to demonstrate the accuracy and efficiency of the method.

This study investigates the heat transfer characteristics of a horizontal tube-in-tube heat exchanger with a helical wire inserted in the inner tube. The influence of the pitch (or helix angle) of the wire on the heat transfer performance and pressure drop during condensation (having all other geometric parameters the same) was investigated experimentally. Tests were conducted for condensing refrigerants R22, R134a, and R407C at an average saturation temperature of 40°C, with mass fluxes ranging from 300–800 kg/m2s and with vapor qualities ranging from 0.85–0.95 at condenser inlet to 0.05–0.15 at condenser outlet. Measurements were made for three helical wire-inserted tubes with different pitches of 5, 7.77, and 11 mm. The local and average heat transfer coefficients were compared not only with the measured data of a smooth tube, but also with the results of micro-fin tubes. The tube with a helical wire pitch of 5 mm inserts was found to have the highest enhancement factor, which can be elucidated by the ...