Page 49 - Mathematical Techniques of Fractional Order Systems

Nonlinear Fractional Order Boundary-Value Problems Chapter | 2 39

TABLE 2.1 Symbols
Symbol Name
D α t Caputo fractional derivative operator
ℬ,ℬ 0 Boundary operators
α
J Riemann Liouville fractional integral operator
α Order
ϕ(t, p), φ(t, p) Homotopy series
h Control parameter
L Auxiliary linear operator
p Homotopy parameter
u 0 (t) Initial guess
u m (t) Successive approximation
H(t) Auxiliary function
U M (t) mth-order approximate solution
d k Initial conditions values
λ(τ) A general Lagrange multiplier
Γ Gamma function
E Unknown parameter
E Dimensionless parameter
u(t) Unknown function
ψ Convective-conductive parameter
λ Frank Kamenetskii parameter
β Dimensionless parameter

2.2 THE METHODS PROCEDURES
To illustrate the methods procedures consider the following nonlinear frac-
tional differential equation:
α
D uðtÞ 1 f ðt; uðtÞÞ 5 0; n 2 1 , α # n; ð2:1Þ
t
with boundary conditions
@u

ℬ u; 5 0; ð2:2Þ
@n
α
where ℬ is a boundary operator and D denotes Caputo fractional derivative
t
which is defined as (Podlubny, 1999)