Comparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations Farhad Farokhi, Mohammad Haeri, and Mohammad Saleh Tavazoei
Abstract This paper is a result of comparison of some available numerical methods for solving nonlinear fractional order ordinary differential equations. These methods are compared according to their computational complexity, convergence rate, and approximation error. The present study shows that when these methods are applied to nonlinear differential equations of fractional order, they have different convergence rate and approximation error.
1 Introduction Differentialequations of fractional order have been the focus of many studies due to their frequent appearance in various applications in physics, fluid mechanics, biology, and engineering. Consequently, considerable attention has been given to the solutions of fractional order ordinary differential equations, integral equations and fractional order partial differential equations of physical interest. Number of literatures concerning the application of fractional order differential equations in nonlinear dynamics has been grown rapidly in the recent years [2, 3, 5, 12–14, 20]. Most fractional differential equations do not have exact analytic solutions and therefore, approximating or numerical techniques are generally applied. There are many different numerical methods such as Predictor Corrector Method (PCM) [8], Quadrature Methods (QM) [22], Kumar-Agrawal Method (KAM) [15], and Lubich Method [17] which have been developed to solve the fractional differential equations. Many new ideas which try to solve these kinds of problems faster and in more convenient way are Nested Memory Principle (NMP) and Fixed Length Integral Principle (FLIP) [7, 10]. These methods are relatively new and provide an approximated solution both for linear and nonlinear equations. There are several papers in
F. Farokhi, M. Haeri ( ), and M.S. Tavazoei Advanced Control System Lab., Electrical Engineering Department, Sharif University of Technology, Azadi Ave., P.O. Box 11155-9363, Tehran, Iran e-mail:
[email protected];
[email protected]; m
[email protected]
D. Baleanu et al. (eds.), New Trends in Nanotechnology and Fractional Calculus Applications, DOI 10.1007/978-90-481-3293-5 13, c Springer Science+Business Media B.V. 2010
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which some of these methods have been comparatively studied [1, 18, 19, 21, 23]. In most of these works, the existing methods were compared by their implementation difficulties or convergence mean error. Also, these papers discuss issues mostly related to the linear equations and they have paid less attention to the nonlinear cases more specifically chaotic ones. In this paper, we implement the above mentioned methods to solve the nonlinear differential equation of fractional order and then compare them through some numerical examples. The paper is organized as follows. In Sect. 2, there is a review on mathematical concepts of fractional order differential equations and then some different methods to solve the nonlinear versions of these equations are described. The mentioned approaches are applied to some common nonlinear systems and results are discussed. Finally some concluding remarks are given in Sect. 3.
2 Numerical Methods In this section we want to discuss on the numerical solution of differential equations of fractional order with arbitrary initial conditions; D˛ y.x/ D f .x; y.x//;
(1)
where ˛ > 0 (but not necessarily ˛ 2 N ), D˛ is Caputo differential operator of order ˛ [4]. We combine our fractional differential equation (1) with initial conditions; y .k/ .0/ D y0.k/
k D 0; 1; : : : ; m
1:
(2)
Existence and uniqueness of solution of system of fractional differential equations (1) for a given initial condition (2) have been proven in [6, 9]. In the following sections, methods are introduced and compared in accordance to their computational complexity, convergence rate, and the approximating error. For each method, one example is implemented and the mentioned properties are investigated. These methods can easily be extended to multi-term equations or a system of equations [11].
2.1 Quadrature Method (QM) This approach is based on the analytical property that the initial value problem (1) and (2) is equivalent to the Volterra integral equation;
y.x/ D
d˛e X1 kD0
.k/ x
y0
k
kŠ
C
1 .˛/
Z
x
.x 0
t/˛
1
f .t; y.t//dt;
(3)
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in the sense that a continuous function is a solution of the initial value problem if and only if it is a solution of (3). This method is based on using Quadratures [22] like Simpson in integrating (3) and finding the answer. According to mathematical concepts and numerical methods [22], the following relation exists between the (step size) and the number of flops. 8˛ W Flops D O.h
2
/;
(4)
and the following relation exists between the approximation error and the step size; error D O.h˛ /;
(5)
According to (4) and (5), this method in too slow and it is not so accurate and we cannot improve this method because of the theorems in [22].
2.2 Predictor Corrector Method (PCM) In this section the algorithm is a generalization of the classical Adams–Bashforth– Moulton integrator that is well known for the numerical solution of the first-order problems [8]. According to [8], the following relation can be seen between the h (step size) and the number of flops. 8˛ W Flops D O.h
2
/;
(6)
The approximation error is related to step size as follows; error D O.h˛ /;
(7)
According to (6) and (7), this algorithm is too slow and its approximation error is high but we can improve its performance (approximation error) [8].
2.3 Fixed Length Integral Principle (FLIP) For performing numerical computation, the simplest approach is to integrate only over a fixed period of recent history [7, 10]. If we can do this, then the computational cost at each step is reduced to O.1/ , and the total amount of computational cost is reduced to O.h 1 / [7, 10]. But this method causes a loss of order and changes the nature of the fractional derivative from a non-local operator into a local operator.
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2.4 Nested Memory Principle (NMP) The idea of nested memory concept was introduced by Ford and Simpson. It can be well applied to numerical approximation of (3) in case of ˛ 2 .1; 1/, thus the computational cost at each step is reduce to O.h 1 log.h 1 // [7, 10].
2.5 Higher Order Methods (Lubich) This method is based on a new method for approximating the fractional order derivative which has more accurate answers in comparison to the last ones [16, 17]. It can be seen that the numerical method (p-HOFLMSM) is in excellent agreement with the exact solution, and the error between p-HOFLMSM and exact solution is O.hp / [16, 17]. This method is the most accurate one with computational cost of O.h 2 / [16]. It can be applied on most of fractional order equations but the implementation of the algorithm is complex.
2.6 Kumar Agrawal Method (KAM) This method is based on integrating the Voltra equation with translating it in to a nonlinear set of equations and then solving it with nonlinear solver. According to [15], the approximation error is O.h˛ / and it has a computational cost of O.h 2 /.
3 Numerical Simulations The numerical results of these methods are discussed in this section based on two nonlinear fractional differential equations. Example 1. The following nonlinear fractional-order ordinary differential equation with ˛ D 1:5 is considered: D˛ y.t/ C y 2 .t/ D f .t/;
y .i / .0/ D 0; i D 1; 2;
(8)
where, f .t/ D
.6
.6/ 5 t ˛/
˛
3 .5/ 4 t .5 ˛/
˛
C
2 .4/ 3 t .4 ˛/
˛
C.t 5 3t 4 C2t 3 /2 : (9)
The exact solution for y is y.t/ D t 5
3t 4 C 2t 3 :
(10)
Numerical results for exact and approximated solutions are illustrated in Figs. 1 and 2 for ˛ D 1:5.
Comparing Numerical Methods for Solving Nonlinear FODEs
a
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b 2 y, z
y, z
2 0 −2
0
0.5
1 Time
1.5
0 −2
2
c
0
0.5
1 Time
1.5
2
0
0.5
1 Time
1.5
2
0
0.5
1 Time
1.5
2
d
0 −2
e
2 y, z
y, z
2
0
0.5
1 Time
1.5
−2
2
f
2
y, z
y, z
2 0 −2
0
0
0.5
1 Time
1.5
0 −2
2
Fig. 1 The exact and the numerical solutions for system in (8) for ˛ D 1:5 with (a) QM (h D 0:002) (b) PCM (h D 0:002) (c) FLIP (h D 0:002) (d) NMP (h D 0:002) (e) Lubich (h D 0:02 and p D 3) and (f) KAM (h D 0:002)
a
b 2 z−y
z−y
0.01 0 −0.01
0
0.5
1 Time
1.5
1 Time
1.5
2
0.5
1 Time
1.5
2
0.5
1 Time
1.5
2
0.1 z−y
z−y
0.5
d 0.05 0 −0.05
0
0.5
1 Time
1.5
0
−0.1 0
2
e
f x 10−4
6 z−y
4 z−y
0 −2 0
2
c
x 10−3
2 0
0
0.5
1 Time
1.5
2
x 10−3
4 2 0
Fig. 2 Error between the exact and the numerical solutions for system in (8) for ˛ D 1:5 with (a) QM (h D 0:002) (b) PCM (h D 0:002) (c) FLIP (h D 0:002) (d) NMP (h D 0:002) (e) Lubich (h D 0:02 and p D 3) and (f) KAM (h D 0:002)
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Example 2. Consider the following fractional differential equation; cD˛ y.t/ D
40320 8 t .9 ˛/ C .1:5t ˛=2
˛
.5 ˛=2/ 4 t .5 C ˛=2/
3
t 4 /3
˛=2
C 2:25 .˛ C 1/
y.t/3=2
(11)
with the following initial conditions; y.0/ D 0; y 0 .0/ D 0:
(12)
The exact solution of this initial value problem is; y.t/ D t 8
t 4C˛=2 C 2:25t ˛ :
(13)
The given equation is solved for ˛ D 0:9 and results are shown in Figs. 3 and 4 .
b
2
y, z
y, z
a
1 0 0
0.5 Time
1
0.5 Time
1
0.5 Time
1
2 y, z
y, z
0.5 Time
d 2 1 0 0
0.5 Time
1 0 0
1
e
f 2 y, z
2 y, z
1 0 0
1
c
2
1 0 0
0.5 Time
1
1 0 0
Fig. 3 The exact and the numerical solutions for system in (11) for ˛ D 0:9 with (a) QM (h D 0:002) (b) PCM (h D 0:002) (c) FLIP (h D 0:002) (d) NMP (h D 0:002) (e) Lubich (h D 0:02 and p D 3) and (f) KAM (h D 0:002)
Comparing Numerical Methods for Solving Nonlinear FODEs
a
177
b 0.01 z−y
z−y
0.02 0 −0.02 0
z−y
5
−0.01 0
1
0
1
0.5 Time
1
0.5 Time
1
0.1
0 −5
0.5 Time
d
x 10−3
z−y
c
0.5 Time
0
0.5 Time
0 −0.1 0
1
e
f x 10−3
5 z−y
z−y
3 2 1
0
0.5 Time
x 10−3
0 −5 0
1
Fig. 4 Error between the exact and the numerical solutions for system in (11) for ˛ D 0:9 with (a) QM (h D 0:002) (b) PCM (h D 0:002) (c) FLIP (h D 0:002) (d) NMP (h D 0:002) (e) Lubich (h D 0:02 and p D 3) and (f) KAM (h D 0:002) Table 1 Comparison of some methods in solving fractional order systems Method Computational Cost Approximation Error Quadrature method Predictor corrector method Lubich (pth order) method Kumar Agarwal method
O.h O.h O.h O.h
2
Fixed length integral principle
O.h
1
Nested memory principle
O.h
1
O.h˛ / O.h˛ / O.hp / O.hp ( /
/ / 2 / 2 /
2
ED
/ log.h
1
//
M T ˛ 1h .˛/ ˛C1 ˛ 1 M .t T /h .˛/ nC1
˛ 2 .0; 1/ ˛ 2 .1; 1/
O.h˛ /
4 Conclusions The present analysis exhibits the applicability of the Quadrature Method (QM), Predictor Corrector Method (PCM), Lubich Method, and Kumar-Agrawal Method (KAM) to solve ordinary differential equations of fractional order. In addition to them, we have introduced some new methods which reduce the computational cost of these solving methods, such as, Fixed Length Integral Principle (FLIP) and Nested Memory Principle (NMP). Table 1 compares the studied methods based on
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their computational load and the approximating error level. The Lubich method is the most reliable method because of its simplicity and low approximating error but its speed is too low. At the second step one can choose the PCM to solve the fractional order differential equations and as the last chance, one can choose the QM. As one sees the computational cost of all these methods are high and these methods are very slow, but one can use FLIP and NMP in order to reduce the computational cost. NMP is more accurate than FLIP because it is better mapped with fractional order nature and preserves their non-locality property.
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