The theory of fractional calculus was first raised in the year 1695 by Marquis
de L Hopital and from now on many studies were done and many important
books were published in this field wherein we can point out to the study of
Oldham and Spanier (1974), Miller and
Ross (1993), Samko et al. (1993) and Podlubny
(1999). Most of the scientific problems and phenomena are modeled by Fractional
Ordinary Differential Equations (FODEs) and Fractional Partial Differential
Equations (FPDEs). For instance; in mathematical physics (Podlubny,
1999), in fluid and continuum mechanics (Carpinteri and
Mainardi, 1997), coloured noises (Sun et al.,
1984), biology, chemistry, acoustics and psychology (Ahmad
and El-Khazali, 2007). Some of FPDEs have been studied and solved including
the classic Fractional PDEs (Vanani and Aminataei, 2011a),
the fractional KdV equation (Momani, 2005) and linear
and nonlinear space and time-fractional diffusion-wave equation (Momani
et al., 2007; Jafari and Seifi, 2009).
In most cases, these problems do not admit analytical solution, so these equations
should be solved using special techniques. In the last decade, several computational
methods have been applied to solve FDEs, prominent among which are the Homotopy
Perturbation Method (HPM) (Momani and Odibat, 2007a;
Jafari and Seifi, 2009), the Adomian Decomposition Method
(ADM) (El-Sayed and Gaber, 2006; Momani
and Odibat, 2006), the Variational Iteration Method (VIM) (Momani
and Odibat, 2007b), the Generalized Differential Transformation Method (GDTM)
(Momani and Odibat, 2008) and the Fractional Difference
Method (FDM) (Momani and Odibat, 2007a; Ghorbani,
Out of these methods, we would like to present a simple and efficient method for solving FODEs. The MQ approximation scheme is an useful method for the numerical solution of ordinary and partial differential equations (ODEs and PDEs). It is a grid-free spatial approximation scheme that converges exponentially for the spatial terms of ODEs and PDEs.
The MQ approximation scheme was first introduced by Hardy
(1971) who successfully applied this method to approximate surfaces and
bodies from field data. Hardy (1990) has written a detailed
review article summarizing its explosive growth since it was first introduced.
In 1972, Franke (1982) published a detailed comparison
of 29 different scattered data schemes for analytic problems. Of all the techniques
tested, he concluded that MQ performed the best in accuracy, visual appeal and
ease of implementation, even against various finite element schemes. To the
best of our knowledge, this is the first demonstration of the application of
the MQ approximation scheme to FODEs.
BASIC DEFINITIONS OF THE FRACTIONAL CALCULUS
Here, we state some preliminaries and definitions of fractional calculus (Podlubny,
Definition 1: A real function u (x), x>0 is said to be in the space Cμ, μεR, if there exists a real number p>μ such that u (x) = xpv (x), where v(x)εC (0, ∞) and it is said to be in the space Cμm iff u(m) (x)εCμ, mεN.
Definition 2: The Riemann--Liouville fractional integral operator of order α≥0, of a function u (x)εCμ, μ≥-1, is defined as:
where, Γ is the Gamma function. Some of the most important properties of operator Jα for u (x)εCμ, μ≥-1, α, β≥0 and γ>-1, are as follows:
||Jα Jβu (x) = J(α+β)
||Jα Jβu (x) = Jβ Jα
Definition 3: The fractional derivative of u (x) in the Caputos
sense is defined as:
where, m-1<α≤m, mεN, x>0, u (x)εCm-1.
MQ APPROXIMATION SCHEME
The basic MQ approximation scheme assumes that any function can be expanded as a finite series of upper hyperboloids that are written as follows:
where, N is the total number of data centers under consideration and:
(x-xj)2 is the square of Euclidean distances in R and R2>0 is an input shape parameter. Note that, the basis function h, which is a type of spline approximation, is continuously differentiable. The expansion coefficients aj are found by solving a set of full linear equations, which are written as follows:
Zerroukut et al. (1998) found that a constant
shape parameter (R2) achieves better accuracy. Mai-Duy
and Tran-Cong (2001) have developed new methods based on Radial Basis Function
Networks (RBFNs) to approximate both functions and their first and higher derivatives.
The Direct RBFN (DRBFN) and Indirect RBFN (IRBFN) methods have been studied
and it has been found that the IRBFN methods yield consistently better results
for both functions and their derivatives. Recently, Aminataei
and Mazarei (2005) stated that, in the numerical solution of elliptic PDEs
using DRBFN and IRBFN methods, the IRBFN method is more accurate than other
methods with very small error. They have shown that, especially, on one-dimensional
equations, IRBFN method is more accurate than DRBFN method. Furthermore, Aminataei
and Mazarei (2008) used the DRBFN and IRBFN methods on the polar coordinate
and have achieved better accuracy.
Micchelli (1986) proved that MQ schemes belong to a
class of conditionally positive definite RBFNs. He showed that equation is always
solvable for distinct points. Madych and Nelson (1990)
proved that MQ interpolation always produces a minimal semi-norm error and that
the MQ interpolant and derivative estimates converge exponentially as the density
of data centers increases.
In contrast, the MQ interpolant is continuously differentiable over the entire domain of data centers and the spatial derivative approximations were found to be excellent, especially in very steep gradient regions where traditional methods fail. This ability to approximate spatial derivatives is due in large part to a slight modification of the original MQ scheme that permits the shape parameter to vary with the basis function.
Instead of using the constant shape parameter defined in Eq.
3, we have used variable shape parameters (Kansa, 1990a,
b; Vanani and Aminataei, 2009,
2008, 2011b) as follows:
R2max and R2min are two input parameters chosen so that the following ratio is in the given range:
Madych (1992) has proven that very large values of
a shape parameter are desirable in certain circumstances. Equation
5 is one way to have at least one very large value of a shape parameter
without incurring the onset of severe ill-conditioned problems.
Now, in order to apply the MQ approximation scheme for solving FODEs, let us
consider a FODE in the form:
Dαu (x)+Lu (x) = f (x),
u(k) (0) = λk,
k = 0,1,
where, L is the differential operator, Dα is the fractional differential operator of order α that operates on the interior and f: [0, b]→R is a known function.
be the N-[α] collocation points in [0, b]. Substituting the collocation
points into Eq. 8, we obtain:
Therefore, we have the following system:
Therefore, imposing the [α] initial conditions, the system of N equations with N unknowns is available. Then, we must solve this system to make distinct the unknown coefficients. Hence, we have used the Gauss elimination method with total pivoting to solve such a system.
Remark: It is noticeable that collocating points can be scattered. This is one of the most important advantages of the MQ approximation scheme [0, 0]. The numerical results show this issue easily and the applicability of the MQ approximation scheme in this sense, is observable.
Here, three experiments including linear and nonlinear problems on regular
and irregular domains are solved using MQ approximation scheme. Since, most
of FDEs have not exact solution, therefore we must compare them with known numerical
methods such as HPM, VIM, FDM and ADM. The results and their comparison with
several powerful methods illustrate the validity and capability of MQ approximation
scheme. If the exact solution of the problem exists, the accuracy of an approximate
solution is measured by means of the discrete relative L2 norm defined
where, u and
are the exact and computed solutions, respectively and N is the number of unknown
nodal values of u. The computations associated with the experiments were also
performed in Maple 14 on a PC, CPU 2.8 GHz.
Experiment 1: Consider the following FODE:
Dα u (x)+u″ (x)+u (x) = 8, x>0, 0<α≤2
with the initial conditions: u (0) = u (0) = 0
This problem is chosen from Momani and Odibat (2007a)
and Arikoglu and Ozkol (2007). It was solved by FDM,
ADM and VIM methods. Here, we have solved it by MQ approximation scheme with
Rmax = 250, Rmin = 0.5, N = 30 for different α and
have compared it with the closed form series solution of the exact solution
(Momani and Odibat, 2007b; Arikoglu
and Ozkol, 2007) and the three aforesaid mentioned methods. The results
show a good agreement with the same results obtained by using the MQ approximation
scheme. The comparison is shown in Table 1.
In the above experiment, we have compared the MQ approximation scheme with
the FDM, ADM and VIM for solving a FODE. We observe that higher-order accuracy
can be achieved by MQ-RBF approximation scheme than using the same terms in
the other methods. The discrete relative L2 norm (Ne)
given by Eq. 12 for α = 0.5, 1 and 1.5 is 0.000009,
0.0 and 0.000467, respectively. These results, demonstrate another capability
of the proposed method. Furthermore, when α is an integer, the MQ approximation
scheme is so much accurate than the other methods.
Experiment 2: Consider the following FODE (Abdulaziz
et al., 2008):
Dα u (x) = -u (x), 0<α<1, x>0,
u (0) = 1
The close form series solution of the exact solution is obtained by HPM  as:
We have tested it for seven scattered data points (N = 7) for α = 0.75, 0.85, 0.95 with Rmax = 500, Rmin = 0.3 and have compared it with the HPM. The comparison is given in Table 2.
||Comparison of the solutions of FDM, ADM, VIM and MQ-RBF approximationscheme
with the exact solution for different α and x of the experiment 1
The computed results in Table 2, demonstrate the approximate
solution obtained using the MQ approximation scheme, is in good agreement with
the approximate solution obtained using the HPM for all values of scattered
Experiment 3: Consider the following nonlinear FODE:
Dα u (x)+eu (x) = 0, 0≤x≤1,
The exact solution of this initial value problem for α = 1, is u (x) = -In (x+1).
In this experiment, we use from the following expansion to linearize eu (x) as:
We have solved this experiment for N = 11 and α = 0.75, 0.85, 0.95, 0.99, 0.999.
Figure 1, shows the numerical results and illustrates a good
agreement with the same results obtained by using the VIM and ADM in .
||The graphs of the experiment 4.3 for N = 11 and different
|| Comparison of the solutions of HPM and MQ-RBF approximation
scheme for different α and scattered data points of X of the experiment
Also, the results show that when α→1, then approximate solution tends
to exact solution, rapidly.
In this study, a MQ approximation scheme is proposed to solve FODEs. The results reveal that the technique introduced here is effective and convenient in solving FODEs. This method is also easy to implement and yields the desired accuracy with only a few terms. Other advantages of the present method are its low run time, a minimal number of data points in the required domain and its applicability to scattered collocated points. All of these advantages of the MQ approximation scheme suggest that the method is a valid and powerful tool.