Computational Mathematics and Modeling, Vol. 28, No. 2, April, 2017 ANALYTICAL AND NUMERICAL SOLUTIONS OF OSCILLATING FLOW AT A TRAPEZOIDAL PRESSURE WAVEFORM M. Abdulhameed 1 , R. Roslan 2 , D. Vieru 3 , and S. Shafie 4 This study aimed to develop a mathematical model of an unsteady Burgers’ fluid in a circular cylinder with a trapezoidal pressure waveform described by an infinite Fourier series. An analytical solution was ob- tained for the governing equation using the Bessel transform method together with similarity arguments. The validity of the solution was verified using a numerical inversion method based on Stehfest’s method. Limiting cases were considered to examine the fluid flow performance of different fluids. Our results show that the Newtonian and Oldroyd-B fluids performed similar velocity time variation for the trape- zoidal waveform of oscillating motion, whereas the velocity time variation was different for Maxwell and Burger’s fluids. Moreover, it is evident that the material constant of a Burgers’ fluid is another important factor that affects flow performance in an oscillating flow. Keywords: Trapezoidal waveform, oscillating motion, Bessel transform, Stehfest’s method. Nomenclature a mn constant defined by Eq. (23); A mn constant defined by Eq. (25); A i , i =1, 2, Revlin–Ericksen tensors; b mn constant defined by Eq. (24); B mn constant defined by Eq. (26); c p specific heat at constant pressure; d dt material time derivative; I identity tensor; J n Bessel function of order n; p pressure, N/m 2 ; r radius, m; r m ,q m eigenvalues; S extra-stress tensor; 1 School of Science and Technology, The Federal Polytechnic, Bauchi, Nigeria. 2 Centre for Research in Computational Mathematics, University Tun Hussein Onn, Malaysia. 3 Department of Theoretical Mechanics, Technical University of Iasi, Romania. 4 Department of Mathematical Sciences, Faculty of Science, University Teknology, Skudai, Malaysia. 1046-283X/17/2802–0195 c  2017 Springer Science+Business Media New York 195 DOI 10.1007/s10598-017-9357-2