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1

Transformations and Soliton Solutions for a Variable-coefficient Nonlinear Schrödinger Equation in the Dispersion Decreasing Fiber with Symbolic Computation

Zeng Z.-F., Liu J.-G., Jiang Y., Nie B.

Fundamenta Informaticae

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2016

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Vol. 145, nr 2

207--219

EN

Describing the dispersion decreasing fiber, a variable-coefficient nonlinear Schrödinger equation is hereby under investigation. Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrödinger equations with the relevant constraints on the variable coefficients presented, which turn out to be more general than those previously published in the literature. Meanwhile, several families of exact dark-soliton-like and bright-soliton-like solutions are constructed. Also, we obtain some similarity solutions, which can be illustrated in terms of the elliptic and the second Painlevé transcendent equations.

2

Flow and heat transfer at a nonlinearly shrinking porous sheet: the case of asymptotically large powerlaw shrinking rates

Prasad K. V., Vajravelu K., Pop I.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 3

779--791

EN

The boundary layer flow and heat transfer of a viscous fluid over a nonlinear permeable shrinking sheet in a thermally stratified environment is considered. The sheet is assumed to shrink in its own plane with an arbitrary power-law velocity proportional to the distance from the stagnation point. The governing differential equations are first transformed into ordinary differential equations by introducing a new similarity transformation. This is different from the transform commonly used in the literature in that it permits numerical solutions even for asymptotically large values of the power-law index, m. The coupled non-linear boundary value problem is solved numerically by an implicit finite difference scheme known as the Keller- Box method. Numerical computations are performed for a wide variety of power-law parameters (1 < m < 100,000) so as to capture the effects of the thermally stratified environment on the velocity and temperature fields. The numerical solutions are presented through a number of graphs and tables. Numerical results for the skin-friction coefficient and the Nusselt number are tabulated for various values of the pertinent parameters.

3

Vishwakarma J. P., Pathak P.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 2

603-619

EN

Similarity solutions are obtained for an adiabatic flow behind a magnetogasdynamic cylindrical shock wave propagating in a rotational axisymmetric flow of a perfect gas in the presence of a variable azimuthal magnetic field. The initial medium is considered to have a variable density and a variable azimuthal velocity component in addition to a variable axial velocity. Initial velocities, density and the magnetic field are assumed to obey power laws. Distributions of the fluid velocities, density, pressure, magnetic field and vorticity components are obtained in the flow-field behind the shock front. Effects of an increase in the value of the index for variation of an ambient azimuthal magnetic field, in the value of the index for ambient density and in the strength of the initial magnetic field are obtained. It is found that the presence of the magnetic field has a decaying effect on the shock wave.

4

Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect

Al-Odat M. Q., Damseh R. A., Al-Azab T. A.

International Journal of Applied Mechanics and Engineering

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2006

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Vol. 11, no 2

289-299

EN

The thermal boundary layer on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field effect is investigated numerically. The local similarity solution is applied to the governing equationes. The obtained results are verified with the results of earlier authors. Temperature distribution and local Nusselt number have been presented at different values of the governing parameters. In particular, the magnetic field decreases the temperature difference at the wall of the stretching surface while the Nusselt number decreases with it.