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An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives

Odetunde OS1* and Taiwo OA2

1Department of Mathematical Sciences, Olabisi Onabanjo University of Ago-Iwoye P.M.B 2002, Ago-Iwoye, Ogun State, Nigeria

2Department of Mathematics, University of Ilorin, P.M.B 1515, Ilorin, Nigeria

*Corresponding Author:
Odetunde OS
Department of Mathematical Sciences
Olabisi Onabanjo University of Ago-Iwoye P.M.B 2002
Ago-Iwoye, Ogun State, Nigeria
Tel: 9647804274839
E-mail: [email protected]

Received: March 14, 2015; Accepted: April 28, 2015; Published: August 22, 2015

Citation: Odetunde OS, Taiwo OA (2015) An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives. J Appl Computat Math 4:242. doi:10.4172/2168-9679.1000242

Copyright: © 2015 Odetunde OS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, we propose an algorithm to obtain approximate solutions of Fractional Differential-Algebraic Equations with Caputo-type derivatives. The method, Iterative Decomposition Method presents solutions as rapidly convergent infinite series of easily computable terms. Numerical examples are considered to highlight the significant features of the IDM, as well as illustrate the efficiency and accuracy of the method, when compared with known methods.

Keywords

Fractional differential-algebraic equations; Caputo derivatives; Iterative decomposition method

2000 Subject Classifications: 4A08,34K28, 34 B05, 34 B15, 56 L10, 745 S30

Introduction

Many physical phenomena have been successfully modeled by the use of fractional order differential equations. Such physical phenomena abound in rheology, fluid flow, polymer physics, viscoelasticity, mathematical biology and several areas of science, and technology [1-5]. A review of some other applications are given in [1,3,6]. There have been many research works in the literature on Fractional Differential Equations, principally to derive efficient methods for finding solutions to them. However, most nonlinear fractional differential equations cannot be solved analytically. This has further generated further and more intense interest in finding numerical methods which accurately and efficiently solve nonlinear fractional differential equations. Some of the methods include Adomian Decomposition Method (ADM) [7-13], the Variational Iteration Method (VIM) [13-16], Homotopy Analysis Method (HAM) [17-19], Homotopy perturbation Method (HPM) [13,20]. Many physical problems are governed by a system of differential-Algebraic Equations (DAE’s), and the solution of these equations has been a subject of many investigations in recent years [7-9,14-16,18,19,21,22]. So far, no analytic methods has been found to yield an exact solution for nonlinear Differential-Algebraic Equations. Numerical methods for approximating Differential- Algebraic Equations have been presented by several authors in [7-9,14,15,19,22]. Recently, important mathematical models have been expressed in terms of fractional order differential-algebraic equations. Consequently, many known approximation techniques have been applied to solve them. In [17,19], the Homotopy Analysis Method (HAM) was applied for Fractional Differential-Algebraic Equations (FDAE’s). The Adomian Decomposition Method (ADM) was applied in [7,9-11,20,23] while in [16] the Variational Iteration Method was applied and the Differential Transform was applied in [14,21].

In this paper, the Iterative Decomposition Method (IDM) is applied to solve Fractional Differential-Algebraic Equations of the form

equation (1)

equation (2)

subject to the initial conditions

equation (3)

The Iterative Decomposition Method (IDM) has been applied extensively to solve integer order differential equation of various classes [4,5]. The results obtained compared favorably with other known results.

Basic Definitions

There are several definitions of a fractional derivative order α > 0 e. g. Riemann Liouville, Grunwald-Letnikow, Caputo and Generalized Functions Approach. The most commonly used definitions are the Riemann-Liouville and Caputo. However, the Caputo derivative seems to be more favoured because of it’s ease in adaptability to initial conditions for physical problems [3,24]. We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 4.1: A real function equation , is said to be in the space equation if there exists a real number equation, such that equation, where equation. Clearly equation if equation

Definition 4.2: A function equation, is said to be in the space equation if equation

Definition 4.3: The Riemann-Liouville fractional integral operator defined of order α ≥ 0 of a function equationis defined as

equation (4)

equation (5)

Properties of the operator Jα can be found in [3], we mention only the following:

For equation and equation

equation (6)

equation (7)

equation (8)

the Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena using fractional differential equations. Therefore, we will introduce a modified fractional differential operator proposed by Caputo [6].

Definition 4.4: The fractional derivative of f (x) in the Caputo sense is defined as

equation (9)

for equation and equation Also, we need here two of its basic properties.

Lemma 1: If equation and equation, then

equation (10)

equation (11)

Iterative Decomposition Method

We consider the approximation of the IDM [4,5] to FDAE’s. Applying the inverse operator (4) to both sides of (1), we have

equation (12)

By the IDM, the solution x(t) could be decomposed into the infinite series convergent terms

equation (13)

for each equationrom [4,5], the Iterative Decomposition Method suggests that tthe nonlinear operator could be decomposed as

equation (14)

where N is a nonlinear operator, which in this case can be replaced by the Riemann-Liouville integral operator (4).

From (13) and (12) is equivalent to

equation(15)

We then define

equation

equation

The solution for each xiis then obtained by substituting (16) into 12 in the form

equation

Numerical Examples

We now consider some numerical examples to illustrate the suitability of the proposed method.

Example 1: Consider the Fractional DAE

equation (17)

equation (18)

with initial conditions

equation(19)

For the special case α =1 the exact solution is

equation

From (17) and (18), and

equation

equation

equation

equation

Applying the inverse operator to both sides,

equation

equation

equation

equation

Taking

equation

equation

we have

equation

equation

equation

equation

equation

Taking

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

Then, x(t) is approximated as

equation

equation

equation

equation

equation

equation

For the particular case α =1

equation

equation

equation

equation

Table 1 shows the Comrarison of Numerical results with Exact solution.

     x(t) for α=1         α=0.5       α=0.75  
t Exact solution IDM Ref. [21] IDM Ref. [21] IDM Ref. [21]
0.0 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
0.1 0.9148208 0.9148208 0.9148208 0.7542922 0.7642925 0.8492992 0.8492996
0.2 0.8584646 0.8584646 0.8584646 0.7545094 0.7545096 0.8016695 0.8016697
0.3 0.8294743 0.8294743 0.8294743 0.7903160 0.7903162 0.7978997 0.7978999
0.4 0.8260874 0.8260874 0.8260874 0.8524947 0.8524950 0.8250871 0.8250871
0.5 0.8462434 0.8462434 0.8462434 0.9323244 0.9323247 0.8760144 0.8760144
0.6 0.8875971 0.8875971 0.8875971 1.0242047 1.0242052 0.9454581 0.9454582
0.7 0.9475377 0.9475377 0.9475377 1.1237904 1.1237906 1.0290752 1.0290757
0.8 1.0232138 1.0232138 1.0232138 1.2273288 1.2273291 1.1229593 1.1229595
0.9 1.1115639 1.1115639 1.1115639 1.3313914 1.3313915 1.2234352 1.2234368
1.0 1.2093504 1.2093504 1.2093504 1.4327550 1.4327552 1.3269755 1.3269767

Table 1: Comparison of Numerical results with exact solution.

Example 2: Consider the following Fractional Differential- Algebraic Equations

equation (20)

equation (21)

equation (22)

with the initial conditions

equation (23)

From (21) and (22), we have

equation

Applying the inverse operator,

equation

equation

Taking equation we have

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

then, z(t) can be approximated as

equation

equation

For the particular caseα2 =1 ,we have

equation

equation

For (1), we have

equation

equation

By IDM, we have

equation

For the particular caseα1 =1 , we have

equation

which in closed form gives equation.

Tables 2 and 3 shows the Numerical Results of x(t) and y(t) in Example 2.

       x(t) for α=1      x(t) for α=0.5      x(t) for α=0.75
t Exact Solution IDM  Ref. [21] IDM  Ref. [21] IDM Ref. [21]
0.0 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
0.1 1.1051709 1.1051709 1.1051709 1.4678849 1.4678849 1,2187069 1,2187069
0.2 1.2214028 1.2214028 1.2214028 1.7411322 1.7411322 1.4000280 1.4000280
0.3 1.3498588 1.3498588 1.3498588 1.9927891 1.9927891 1.5841270 1.5841270
0.4 1.4918247 1.4918247 1.4918247 2.2392557 2.2392557 1.7769089 1.7769089
0.5 1.6487213 1.6487213 1.6487213 2.4871415 2.4871415 1.9813870 1.9813870
0.6 1.8221188 1.8221188 1.8221188 2.7401183 2.7401183 2.1997453 2.1997453
0.7 2.0137527 2.0137527 2.0137527 3.0006469 3.0006469 2.4338838 2.4338838
0.8 2.2255409 2.2255409 2.2255409 3.2706055 3.2706054 2.6856249 2.6856249
0.9 2.4596031 2.4596031 2.4596031 3.5515669 3.5515666 2.9568131 2.9568131
1.0 2.7182818 2.7182818 2.7182818 3.8450351 3.8450346 3.2493750 3.2493750

Table 2: Numerical Results x(t) in Example 2.

       y(t) for α=1   y(t) for α=0.5    y(t) for α=0.75
t Exact Solution IDM  Ref. [21] IDM  Ref. [21] IDM  Ref. [21]
0.0 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
0.1 1.2214028 1.2214027 1.2214028 2.1546862 2.1546862 0.4852465 0.4852465
0.2 1.4918247 1.4918246 1.4918247 3.0315412 3.0315412 1.9600784 1.9600784
0.3 1.8221188 1.8221188 1.8221188 3.9712084 3.9712084 2.5094586 2.5094586
0.4 2.2255409 2.2255410 2.2255409 5.0142660 5.0142660 3.1574047 3.1574052
0.5 2.7182815 2.7182813 2.7182815 6.1858731 6.1858731 3.9258940 3.9258942
0.6 3.3201169 3.3201168 3.3201169 7.5082482 7.5082482 4.8388784 4.8388794
0.7 4.0552000 4.0552005 4.0552000 9.0038821 9.0038821 5.9237912 5.9237903
0.8 4.9530324 4.9530323 4.9530324 10.6968505 10.6968505 7.2125789 7.2125809
0.9 6.0496475 6.0496477 6.0496475 12.6037326 12.6037326 8.7427134 8.7427398
1.0 7.3890561 7.3890560 7.3890561 14.7830711 14.7830711 10.5581211 10.558395

Table 3: Values of y(t) in Example 2.

Example 3: Consider the equations

equation (24)

equation (25)

with initial conditions

equation (26)

For the special case α =1 , the exact solution is

equation

From (1),

equation(27)

Putting (4) and (5) in (2), we have

equation (28)

equation (29)

equation (30)

Applying the inverse operator, we have

equation

equation

we take

equation

We have by IDM,

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

Then, we may approximate x(t) as

equation

equation

equation

equation

equation

For the particular case α =1

equation

equation

equation

equation

For the particular case α =1

equation

equation

equation

equation

From equation (27), y(t) may then be approximated.

Table 4 shows the Table of Values of x(t) for Example 3.

  x(t) for α=1   x(t) for α=0.5    x(t) for α=0.75
t Exact Solution IDM  Ref. [21] IDM  Ref. [21] IDM  Ref. [21]
0.0 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
0.1 0.9048374 0.9048374 0.9048374 0.7608910 0.7608910 0.8373931 0.8373931
0.2 0.8187308 0.8187308 0.8187308 0.6909262 0.6909262 0.7494379 0.7494391
0.3 0.7408182 0.7408182 0.7408182 0.6396502 0.6396502 0.6816126 0.6816129
0.4 0.6703201 0.6703201 0.6703201 0.5970819 0.5970878 0.6250301 0.6250322
0.5 0.6065307 0.6065307 0.6065307 0.5599903 0.5599926 0.5760117 0.5760122
0.6 0.5488116 0.5488118 0.5488116 0.5268829 0.5268894 0.5326228 0.5326238
0.7 0.4965853 0.4965853 0.4965853 0.4969620 0.4969640 0.4937106 0.4937128
0.8 0.4493290 0.4493290 0.4493290 0.4697012 0.4697024 0.4585181 0.4585197
0.9 0.4065697 0.4065697 0.4065697 0.4447408 0.4447444 0.4265076 0.4265076
1.0 0.3678794 0.3678794 0.3678795 0.4218500 0.4218206 0.3972736 0.3972736

Table 4: Table of Values of x(t) for Example 3.

Conclusion

In this paper we have applied an Iterative Decomposition Method to the solution of Fractional Differential-Algebraic Equations.The solutions obtained are in appreciable agreement with the exact solutions as well as those obtained using other known approximation techniques. the relative ease of handling of the present method is its strength . In general the method is quite promising as the terms of the approximating series are easily computable.

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