Page 56 - Mathematical Techniques of Fractional Order Systems

46 Mathematical Techniques of Fractional Order Systems

that should be determined. The existence of unique or multiple solutions for
the original boundary value problem (2.1) depends on the fact of whether the
condition u(a) 5 b admits unique or multiple values for the formally intro-
duced parameter E in the boundary conditions (2.17). The condition u(a) 5 b
satisfies the series solutions (2.27) and (2.34) for the problem then
U M ða; h; EÞ 5 b; u m11 ða; h; EÞ 5 b: ð2:36Þ
The above equations have two unknown parameters h and E which con-
trol the convergence of the obtained series solutions. If the value of E is con-
stant value, then the series solutions (2.27) and (2.34) converge at E 5 r only
in that range of h, where the parameter E does not change with the variation
of h. According to Eq. (2.36), E is a function of h (E(h)), by drawing the
Eq. (2.36) gives the famous way of the HAM, the so-called h-curve for each
method. The number of such horizontal plateaus in h-curve where E(h) con-
stant will predict the multiplicity of the solution. The horizontal plateaus
indicate the convergence because if the formally introduced E(h) is a constant
value then a horizontal line segment in h-curve which corresponds to the
valid region of h for the series solutions approaches the exact solution for
the problem (2.1).

2.3 APPLICATIONS

2.3.1 Mixed Convection Flows in a Vertical Channel
The aim of this section is to apply the PHAM and controlled Picard’s
method (CPM) to detect the multiple solutions of a kind of model in mixed
convection flows in the asfractional order domain. The model in integer
order domain is combined forced and free flow in the fully developed region
of a vertical channel with isothermal walls having the same temperature
(Barletta, 1999; Barletta et al., 2005). In this model, the viscous dissipation
effect is taken into account and the fluid properties are assumed to be con-
stant. The set of governing have led to a fourth order ordinary differential
equation for the dimensionless velocity field and reads
4
d u E du 2
5 ; ð2:37Þ
dt 16 dt
with the boundary conditions

ð 1
0
u ð0Þ 5 u vð0Þ 5 uð1Þ 5 0; uðtÞdt 5 1; ð2:38Þ
0
0
where E is dimensionless parameter coincides with the product of the
Prandtl number, the Gebhart number, and the Reynolds number. In special
case E-0, it can be easily verified that, the models (2.37) and (2.38) admit
a unique solution, namely