1. Introduction
Fractional calculus, a generalization of conventional integer calculus, has been found to be more appropriate for describing important phenomena in the fields of dynamics [
1], physics [
2,
3], medicine [
4], chemistry [
5], and other scientific areas [
6]. There have been plenty of works, especially those considering real-world physical systems, where the Caputo-type fractional derivative and the Riemann–Liouville fractional integral have been used to describe complex dynamical systems. As most fractional order equations cannot be resolved analytically, numerical methods have been developed to give their numerical solutions [
7,
8,
9,
10]. In recent years, multi-term fractional order equations have received increasing attention due to their wider applicability. Many papers have been devoted to numerical methods for approximating multi-term fractional differential equations, such as the Bernstein polynomials method [
11] for deriving the operation matrix of the Caputo derivative based on the collocation method, and the B-spline method [
12] for constructing an operation matrix based on linear B-spline functions. In contrast, only a few papers have focused on the development of numerical techniques for multi-term fractional order equations with fractional integral and derivative. In [
13], Kojabad and Rezapour theoretically discussed the existence of solutions of multi-term fractional order equations, then numerically verified that the differences between the results of the Legendre method and the Chebyshev method are negligible; however, they did not provide a method with greater accuracy and speed for multi-term fractional order equations. Zheng et al. [
14] studied linear multi-term fractional initial problems using the discontinuous Galerkin finite element method; however, the study of numerical methods for non-linear multi-term fractional order equations, which can describe material transport behaviors in complex physical systems, is scarce. This motivated us to propose an efficient and accurate numerical method.
In this paper, we consider the following Problem I about multi-term fractional order equations:
with
boundary conditions in
,
and
where
,
,
, and
. The functions
f and
are non-linear functions of
and its derivatives
. The operator
is the Caputo-type fractional derivative operator of order
and
is the Riemann–Liouville fractional integral operator of order
. The definitions of the Caputo fractional derivative and the Riemann–Liouville fractional integral are introduced in the next section.
Wavelet methods, which are relatively novel approaches, have been applied to solving problems in fractional calculus. Due to the structural characteristics that the basis of wavelets can achieve through the dilation and translation of mother wavelet functions, wavelet methods can enhance numerical efficiency. Some orthogonal wavelet methods have been developed to estimate approximate solutions for fractional order equations [
15,
16]. Compared with orthogonal wavelet methods, the semi-orthogonal B-spline wavelet method (SOBWM) has the following advantages: compact support, explicit analytical form, and finite basis functions in any wavelet subspaces [
17]. Due to its accuracy and efficiency, SOBWM has been applied in several kinds of differential and integral equations. Maleknejad et al. [
18] adopted this kind of method to solve non-linear Fredholm–Hammerstein integral equations of the second kind, Aram et al. [
19] used the method to deal with integro-difference equations, while Liu et al. [
20] approximated multi-term linear fractional differential equations by this method.
The quasilinearization method [
21,
22] and the homotopy method [
23,
24] are common approaches for linearizing non-linear functions. For multi-term fractional order equations, however, our simulation results show that the solutions obtained by the homotopy method easily diverge. By contrast, the quasilinearization method is more suitable for non-linear multi-term fractional order equations. In this paper, we utilize the quasilinearization method to linearize the non-linear fractional order equations in Problem I. Then, we solve the linearized fractional order equations using the semi-orthogonal B-spline wavelet collocation method (SOBWCM).
The remainder of this article is structured as follows: In
Section 2, we introduce some basic notation, definitions, and lemmas in fractional calculus. The definition of the semi-orthogonal B-spline wavelets (SOBW) and related theorems and properties are given in
Section 3.
Section 4 presents the implementation process of the quasilinearized semi-orthogonal B-spline wavelet method (QSOBWM). The convergence of QSOBWM is analyzed in
Section 5. In
Section 6, the validity of the presented scheme is examined through illustrative examples. In
Section 7, we draw a concise conclusion.
4. Function Approximation
The quasilinearization approach [
29] is applied to approximate
in Problem I. Concisely, we note
, such that Problem I is transformed into Problem II as
with linearized non-linear boundary conditions
and
where
,
and
,
. The initial value,
, can be selected from mathematical or physical conditions, and
is further obtained by iteration.
Equations (
24)–(
26) are linear equations in
and, so, Problem II consists of multi-term linear fractional order equations that can be addressed efficiently by SOBWCM. For simplicity, Problem II is reformulated into the equivalent Problem III as
with the boundary conditions
where
,
,
, and
are related to the functions and values of
r-th iteration by the quasilinearization approach.
A function
in
can be expanded by semi-orthogonal B-spline scaling functions and wavelets [
27] as
To meet the needs of practical applications, the higher frequency components are truncated at
M, such that the infinite series in Equation (
30) is approximated as
where
C and
are the
vectors
Substituting
from Equation (
31) into Equations (
25)–(
29) of Problem III, we obtain:
with the boundary conditions
In order to increase the computational efficiency, Equation (
32) can be rewritten, in matrix form, as:
where
and
The terms
and
can be implemented on a computer, based on Lemmas 3 and 4. Therefore, Problem III is converted into the solution of a system of linear equations, which consists of Equations (
33)–(
35).
5. Convergence Analysis
The QSOBWM is a hybrid numerical method that combines the quasilinearization method and SOBWCM. Thus, we analyzed the convergence of the two basic methods, followed by the convergence of the method proposed in this paper, obtained in the process of function approximation.
Theorem 1. Suppose , , , and the difference function is considered in the quasilinearization method for Problem II. Then, there exists a positive constant such thatwhere is the maximum value of any of in , . Proof. By iteration of Equations (
24)–(
26), we have
with the corresponding boundary conditions:
and
According to the mean value theorem in [
30],
where
is between
and
,
. Substituting Equation (
42) into Equation (
39) yields
In view of the
n-term linear fractional Green’s function properties in [
26], there exists a Green function
such that
Therefore,
where
is the
norm. Due to the properties of Green functions and the boundedness of
, there exists a positive constant
depending on
,
, and
p such that
The convergence of the quasilinearization method is shown in Theorem 1. Then, to estimate the error of SOBWCM, we use the following theorem from [
20]:
Theorem 2. Suppose is approximated by SOBWCM of order m in Problem III. Then, the truncation error for is Theorem 2 implies that when , reflecting the convergence of SOBWCM. As each iteration step (i.e., from to ) and the process of approximating are convergent, QSOBWM proposed in this paper is convergent.
6. Numerical Examples
This section presents some numerical examples to demonstrate the validity of the proposed scheme. Computation was carried out on a personal computer with an Intel(R) Core(TM) i5-7500 CPU @ 3.40 GHz with 8.00 GB RAM and the codes were written in Matlab 2014a. We denote u as the exact solution and as the numerical solution. To assess the performance of the method, we calculated the absolute error, error, and error.
The absolute error in
is
the
error is defined as
and the
error is defined as
where
is the number of collocation points in
.
Example 1. Consider the fractional integro-differential equation with weakly singular kernel [8,9]:with the initial conditionwhere , . The exact solution of the problem is . We adopted the QSOBWM with
for several truncation values
M, and the absolute errors for each case are exhibited in
Table 1. The numerical results of the proposed scheme in
Table 1 illustrate that the absolute errors decreased when the value of truncation
M increased. More intuitively, we describe the absolute errors of the present scheme for various values of
M in
Figure 1. The results were in accordance with the convergence analysis of the present scheme.
In order to compare with the second-kind Chebyshev polynomials method (SKCPM) in [
8] and the fractional order Euler functions method (FEFsM) in [
9], we computed the
errors of the present scheme with various values of
M and list the results in
Table 2, where
N denotes the maximal degree of the polynomials in the space spanned by all polynomials for SKCPM and FEFsM, and
M indicates the truncation in the present scheme. The degree of all polynomials was no more than three when
in the present scheme.
Table 2 shows that the
errors of the present scheme with
were smaller.
Example 2. Consider the following fractional Langevin equation [14]:with the initial conditionhere and . The exact solution is given by . We solved this problem by using the QSOBWM with
and
for fixed step size
.
Table 3 exhibits the absolute errors, which demonstrates that the absolute errors of various values of
,
, and
were less than
, and that the computation took only 4.10 s. Furthermore, the results confirmed the effectiveness of this method.
Figure 2 and
Figure 3 depict the absolute errors of the present scheme for different
and
, respectively.
Figure 2 presents the absolute errors of the present scheme with
and
for various values of
, while
Figure 3 displays the absolute errors of the present scheme with
and
for several values of
. These graphs demonstrate that the approximate solutions agreed closely with the analytical result for various values of
and
, with absolute errors less than
.
In order to verify the performance of QSOBWM, as shown in
Table 4, we calculated the
error and
error of the present scheme and the method in [
14] for
, and
with different grid sizes. The results demonstrated that the approximate results for the present scheme were closer to the analytical solutions than the method in [
14].
Some interesting phenomena are shown in
Figure 1,
Figure 2 and
Figure 3. There may be two reasons: First, when
, the error is much closer to zero as the value at
satisfies both the original equation and the initial value; however, the values with
only satisfy the original equation, which is more likely to have bigger errors. Second, it might be related to the reduced algebraic equations, which needs more exploration.
Example 3. Consider the following fractional Duffing–Holmes model for a non-linear oscillator [31]:with the initial value: From [31], is the given exact solution. We approximated the solution using QSOBWM with
and different
M (
). This problem was also solved by the spectral collocation method (SCM) in [
31] with
and
where
N denotes the maximal degree of polynomials in the space formed by the basis of the method. In
Table 5, we compare the
errors and
errors of the two methods mentioned above. The degree
N of the polynomials in the present scheme was three. The
and
errors of the present scheme were much smaller than those of the SCM from
Table 5. Moreover, the
errors and
errors of the present scheme decreased when
M increased. Thus, the results were in accordance with the convergence analysis discussed in the previous section.
Figure 4 depicts the analytical and numerical results of the present scheme with
for
and
. The figure demonstrates that the numerical results tended to the analytical solution for different values of
.
Example 4. Consider the following multi-term fractional non-linear boundary value problem [11,12]:with boundary conditions: The exact solution of this problem is .
The problem was resolved by the B-spline operational matrix method (BSOMM) [
12] and the Bernstein operational matrix method (BOMM) [
11], respectively. We also solved the example by applying the present scheme with
and list the values of the
and
errors of the three methods in
Table 6. The
errors of the present scheme achieved
within 7.98 s. As shown in
Table 6, the present scheme was the most accurate.