Bounds, monotonicity, uniqueness, and analytical calculation of a class of similarity solutions for the fluid flow over a nonlinearly stretching sheet
Abbreviated Journal Title
Math. Meth. Appl. Sci.
nonlinearly stretching sheet; existence result; uniqueness result; shooting parameter; analytical method; CONTINUOUS SOLID SURFACES; VISCOUS-FLOW; DIFFERENTIAL-EQUATION; SHRINKING SHEET; LAYER EQUATIONS; HEAT-TRANSFER; BRANCH; PLATE; DIFFUSION; EXISTENCE; Mathematics, Applied
Invoking some estimates obtained in [F.T. Akyildiz et al., Mathematical Methods in the Applied Sciences 33 (2010) 601-606] (which presented an alternate method of proof for the present problem), we correct the parameter regime considered in [R.A. Van Gorder, K. Vajravelu, and F. T. Akyildiz, Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet, Applied Mathematics Letters 24 (2011) 238-242] and add some details, which were omitted in the original proof. After this is done, we formulate a more elegant method of proof, converting the nonlinear ODE into a first nonlinear order system. This gives us a more natural way to view the problem and lends insight into the behavior of the solutions. Finally, we give a new way to approximate the shooting parameter =f(0) analytically, through minimization of the L-2([0,)) norm of residual errors. This approximation demonstrates the behavior of the parameter we expect from the proved theorems, as well as from numerical simulations. In this way, we obtain a concise analytical approximation to the similarity solution. In summary, from this analysis, we find that monotonicity of solutions and their derivatives is essential in determining uniqueness, and these monotone solutions can be approximated analytically in a fairly simple way. Copyright (c) 2014 John Wiley & Sons, Ltd.
Mathematical Methods in the Applied Sciences
"Bounds, monotonicity, uniqueness, and analytical calculation of a class of similarity solutions for the fluid flow over a nonlinearly stretching sheet" (2015). Faculty Bibliography 2010s. 6845.