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

^{2}Department 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

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

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

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

(1)

(2)

subject to the initial conditions

(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.

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 , is said to be in the space if there exists a real number , such that , where . Clearly if

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

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

(4)

(5)

Properties of the operator *J*^{α} can be found in [3], we mention only the following:

For and

(6)

(7)

(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

(9)

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

**Lemma 1: **If and , then

(10)

(11)

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

(12)

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

(13)

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

(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

(15)

We then define

The solution for each *x*_{i}is then obtained by substituting (16) into 12 in the form

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

**Example 1:** Consider the Fractional DAE

(17)

(18)

with initial conditions

(19)

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

From (17) and (18), and

Applying the inverse operator to both sides,

Taking

we have

Taking

Then, *x(t)* is approximated as

For the particular case *α* =1

**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

(20)

(21)

(22)

with the initial conditions

(23)

From (21) and (22), we have

Applying the inverse operator,

Taking we have

then, *z(t)* can be approximated as

For the particular case*α*_{2} =1 ,we have

For (1), we have

By IDM, we have

For the particular caseα_{1} =1 , we have

which in closed form gives .

**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

(24)

(25)

with initial conditions

(26)

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

From (1),

(27)

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

(28)

(29)

(30)

Applying the inverse operator, we have

we take

We have by IDM,

Then, we may approximate *x(t)* as

For the particular case *α* =1

For the particular case *α* =1

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.

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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