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In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order α∈(3,4] with variable coefficients. The method of solution is based on utilizing the fractiona...

A fully analytical solution of the steady, laminar and axisymmetric flow of a Newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet has been derived using the extended homotopy perturbation method. The solution differs from that obtained by the classical homotopy perturbation method in that it is capable of generating a totally analytical solution up to any desired degree of accuracy and is not limited to the first-order correction terms. For an eight-decimal accuracy, it is sufficient to take 12 terms in the power series in the perturbation parameter, provided that use is made of Shanks’ transformation. Unlike other similar problems involving mass transfer across the sheet and/or the presence of a transverse magnetic field, the solution for the present problem is relatively insensitive to the velocity slip parameter.

The extended homotopy perturbation method (EHPM), which is an extension to the well-known homotopy perturbation method (HPM) due to He, is applied to derive the analytical solution of the boundary layer flow of falling vapour due to the condensation on a cold, porous vertical plate. The EHPM calculates the solution automatically by adjusting the scaling factor of the independent similarity variable normal to the plate. The results obtained by the EHPM are in excellent agreement with the exact numerical solution. Also an asymptotic solution, valid for a large suction parameter is developed which matches very well with the exact solution even for moderate values of the suction velocity. Finally, it is shown that the EHPM solution is also applicable to the moderate values of blowing across the plate.

The differential equation of Sturm-Liouville problems is generalized into fractional form by replacing the first-order derivative by a fractional derivative of order α, 0

The laminar unsteady periodic flow motion found in the wake of a rotationally oscillating circular cylinder is investigated numerically at a Reynolds number of 855 over a wide range of oscillation amplitude and frequency ratio. The governing Navier–Stokes equations are integrated to determine the flow field structure for large values of the time using an accurate spectral-finite difference method, but with the boundary vorticity calculated using global vorticity conditions rather than local finite-difference approximations. The lock-on phenomenon has been predicted and its effect on the flow hydrodynamics has been determined. The numerical scheme is verified by applying it to the special case of a steadily rotating cylinder (no forced oscillations) and good qualitative agreements are found.