Throughout this section, we let be such that for some value of the parameter s, the Laplace transform converges.
Theorem 2.1 Let and . Then the fractional differential equation
(2.1)
with the initial conditions
and
has its solution given by
(2.2)
Proof Applying the Laplace transform (see Preliminary 3) and taking into account, we have
(2.3)
Equation (2.3) yields
(2.4)
since
(2.5)
Thus, from Equation (2.4), we derive the following solution by the inverse Laplace transform to Equation (2.2):
□
Example 2.1 The fractional differential equation of a generalized viscoelastic free damping oscillation (cf. [1])
(2.6)
with the initial conditions and has its solution given by
(2.7)
In particular, if and , then the equation
(2.8)
with the initial conditions and has its solution given by
(2.9)
Theorem 2.2 Let and . Then the fractional differential equation
(2.10)
with the initial conditions
and
has its solution given by
(2.11)
Proof Applying the Laplace transform (see Preliminary 3) and taking into account, we have
That is,
(2.12)
Equation (2.12) yields
(2.13)
since
(2.14)
Thus, from Equation (2.13), we derive the following solution by the inverse Laplace transform to Equation (2.11):
This solution can be expressed by the Wright function as
□
Example 2.2 If we let , and in Theorem 2.2, then the equation
has a solution
Theorem 2.3 Let and . Then the equation
(2.15)
with the initial condition
has its solution given by
(2.16)
Proof Applying the Laplace transform to Equation (2.15), that is,
we have
□
Remark 2.1 If in Equation (2.10), then the equation
(2.17)
with the initial conditions and has its solution given by
(2.18)
Theorem 2.4 A nearly simple harmonic vibration equation (cf. [1])
(2.19)
with the initial conditions
and
has its solution given by
(2.20)
Proof We complete this proof by putting in Equation (2.17). □
In fact, by applying the Laplace transform to a linear fractional differential equation with the initial conditions, we can easily derive its solutions as the previous forms in this paper.