3.1. Preliminary
Let be the following second-order nonlinear differential equation:
subject to the initial conditions
with
a linear operator,
a nonlinear operator,
a boundary operator,
g a known function,
u an unknown smooth function depending on the independent variable
t and
.
If an analytic function
F is expanded in Taylor series it is obtained:
where
denotes the partial derivative of the function
F with respect to
u, for
,
,
real values. Instead of solving the nonlinear differential Equation (
11), one can solve another equation, using Equation (40) and Optimal Parametric Iteration Method (OPIM), developed by Marinca et al. [
20]:
where
,
, and
are auxiliary continuous functions;
,
,
(obtained from Taylor series expansion of the nonlinear operator
);
is the (
n + 1)-th-order approximate solution of Equations (
11) and (
12), denoted by
or
and
is the initial approximation, a solution of the linear differential problem:
The unknown convergence-control parameters
,
, and
that appears in Equation (
14) will be optimally computed.
The
-order approximate solution of Equations (
11) and (
12) is well determined if the convergence-control parameters are known.
In OPIM, the linear operator is arbitrarily chosen, not the physical parameters. There are situations when the choice of physical parameters give rise to chaotic behavior of the dynamic system. This happened in the case of choosing higher values of damping factor or for exceeding the optimal resonance conditions. Likewise in the case of the arbitrary choice of the initial conditions.
Let
be an initial approximation of Equation (
15). The nonlinear operators
,
,
and
that appear in Equation (
14) have the form
where
is a positive integer, and
and
are known functions that depend on
.
For
an
-order approximate solution of Equations (
11) and (
12), the validation of this procedure is highlighted by computing the residual function given by
such that
, for all
.
Some mathematical notions as: OPIM sequence of the Equation (
11), OPIM functions of Equation (
11),
-approximate OPIM solutions of Equation (
11), weak
-approximate OPIM solutions of Equation (
11) on the real interval
are defined in [
29]. The existence of weak
-approximate OPIM solutions is built by the theorem presented in [
29].
Remark 1. The integration of the Equation (14) produces secular terms of the form: , , , , , , and so on. For the nonlinear oscillator the secular terms that appear through integration generate the resonance phenomenon. Consequently, the secular terms have to be avoided. 3.2. Semi-Analytical Solutions via OPIM Technique
The applicability of the OPIM scheme for the Equation (
6) using only one iteration is presented in details below.
By means of the linear operator given by Equation (
18), the initial approximation
, solution of Equation (
15) is
with
,
,
unknown parameters at this moment.
Using Equation (
18), a simple computation yields the following expressions:
For the Equation (
14), there are more possibilities to choose the following auxiliary functions:
or
,
,
, and so on.
By means of the Equations (
19), (
20) and (
21) respectively, the following expression which appears in Equation (
14) becomes:
This expression contains some terms depending on the elementary functions
and
. By integration of the Equation (
17), for
, the first-order approximate solution
will contain secular terms of the form
and
which have to go to zero. Therefore, avoiding these secular terms in the Equation (
22) involves:
For
, for example
Thus, the expression
contains a linear combination of the elementary functions of the functions set
In this way, by integration of the Equation (
17), for
, the first-order approximate solution
has the following form:
where the unknown control-convergence parameters
,
,
,
depend on the
,
,
,
and will be optimally computed.
In the case when
the expression
becomes:
Thus, the expression
contains a linear combination of the elementary functions of the functions set
As in the previous case, the first-order approximate solution
is a linear combination of the elementary functions of the functions set
and so on.
Generally, for
,
a fixed number the first approximation
is a linear combination of the functions set
where the unknown control-convergence parameters
,
,
depend on the
,
,
and will be optimally computed.
Thus, by using only one iteration, the OPIM solution is well determined as
by Equation (
27).
Furthermore, using two iterations, the OPIM solution is computed as
by Equation (
27), and so on.
For the first-order approximate solution
given by Equation (
27) imposing the initial data from Equation (
6) yields:
In the second case of the Equation (
8) by the same manner, the OPIM procedure is applied using only one iteration.
The linear and nonlinear operators, respectively, are:
Taking into consideration the linear operator given by Equation (
29), the initial approximation
, solution of Equation (
15) is
with
,
,
unknown parameters at this moment.
From Equation (
29), by a trivial computation, results the expressions:
Returning to Equation (
14), there are a lot of possibilities to choose the following auxiliary functions:
or
,
,
, and so on.
By means of the Equations (
30), (
31) and (
32) respectively, the following expression from Equation (
14) becomes:
This expression contains some terms depending on the elementary functions
and
. By integration of the Equation (
17), for
, the first-order approximate solution
will contain secular terms of the form
and
which have to go to zero. Therefore, avoiding these terms in the Equation (
33) involves:
Thus, the expression
contain a linear combination of the elementary functions of the functions set
In this way, by integration of the Equation (
17), for
, the first-order approximate solution
has the following form:
with unknown control-convergence parameters
,
depending on the
,
,
,
and will be optimally computed.
In the case when
the expression
become:
Thus, the expression
contain a linear combination of the elementary functions of the functions set
As in the previous case, the first-order approximate solution
is a linear combination of the elementary functions of the functions set
and so on.
Generally, for
,
a fixed number, the first approximation
is a linear combination of the functions set
where the unknown control-convergence parameters
,
,
,
depend on the
,
,
,
,
and will be optimally computed.
Thus, by using only one iteration, the OPIM solution is well determined as
by Equation (
38).
Furthermore, using two iterations, the OPIM solution is computed as
by Equation (
38), and so on.
Let
be a first-order approximate solution given by Equation (
38). Imposing the initial data from Equation (
8) yields:
The applicability of the OPIM procedure for the nonlinear differential problem given by Equation (
10) using only one iteration is presented in details below.
For the third case
,
the following well-known series expansions are used:
The linear and nonlinear operators could be chosen as:
Taking into consideration the linear operator given by Equation (
41), the initial approximation
, solution of Equation (
15) is
with
,
,
unknown parameters at this moment.
Using Equation (
41), a simple computation yields the following expressions:
In this case the following auxiliary functions are proposed:
or
,
,
, and so on.
A semi-analytical solution of the Equation (
10) can be obtain by OPIM procedure choosing an arbitrary value of the index
in the Equation (40).
Thus, for
, using Equations (40) and (
43), the expression
has the form:
where
Imposing the conditions
and
(by avoiding the secular terms in
and
) involves:
Therefore, the expression
becomes a linear combination of the functions set
Consequently, the first-order approximate solution
, solution of the Equation (
14) becomes a linear combination of the functions set
Analogously, for
the first-order approximate solution
, solution of the Equation (
14) becomes a linear combination of the functions set
and so on.
For
a fixed arbitrary number the first-order approximate solution
, solution of the Equation (
14) has the form
where the unknown control-convergence parameters
,
,
,
depend on the
,
,
and will be optimally computed.
If
is a first-order approximate solution given by Equation (
47), then imposing the initial data from Equation (
10) yields: