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Formulation and Solution of Space -Time Fractional KdV - Burgers Equation

Abulwafa E. M., Elgarayhi A. M., Mahmoud A. A., Tawfik A. M.

Computational Methods in Science and Technology

2013

Vol. 19, No. 4

235--243

The space-time fractional KdV-Burgers equation has been derived using the semi-inverse method and Agrawal’s variational method. The modified Riemann-Liouville definition is used for the fractional differential operators. The derived fractional equation is solved using the fractional sub-equation method.

Free vibrations of a stepped beam with two uniform and/or tapered parts

El-Din S. S., Mahmoud A. A., Nassar M. A.

Mechanics and Mechanical Engineering

2000

Vol. 4, nr 2

165--181

Closed form solutions for free vibrations of a stepped beam of two parts are given. Each part is of rectangular cross-sectional area. The parts may be of uniform cross-section and/or tapered with both equal and different tapered ratio in the horizontal and vertical planes. General constraints at the ends are possible at the three ends of the beam. The equations of motion of the beam are given in terms of trigonometric functions, hyperbolic functions, and the well known Bessel functions. Various special cases are deduced from the present solution and showed complete agreement with previous closed form solutions for these special cases.