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Approximate solution of integro-differential equation of fractional (arbitrary) order
Accepted Manuscript Original article Approximate Solution of Integro-Differential Equation of Fractional (Arbitrary) Order Asma A. Elbeleze, Adem Kilicman, Bachok M. Taib PII: DOI: Reference:
S1018-3647(15)00041-5 http://dx.doi.org/10.1016/j.jksus.2015.04.006 JKSUS 312
To appear in:
Journal of King Saud University - Science
Please cite this article as: A.A. Elbeleze, A. Kilicman, B.M. Taib, Approximate Solution of Integro-Differential Equation of Fractional (Arbitrary) Order, Journal of King Saud University - Science (2015), doi: http://dx.doi.org/ 10.1016/j.jksus.2015.04.006
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APPROXIMATE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATION OF FRACTIONAL (ARBITRARY) ORDER Asma A. Elbeleze1, Adem Kilicman2* and Bachok M. Taib3 1,3
Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), 71800 Nilai, Malaysia 2
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia emails:[email protected], [email protected], [email protected] *Corresponding author
APPROXIMATE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATION OF FRACTIONAL (ARBITRARY) ORDER
Abstract In the present paper, we study the integro-differential equations which are combination of differential and Fredholm-Volterra equations that having the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented. Keywords: Fractional calculus, Homotopy perturbation method, The variational iteration method, Fractional integro-differential equations
1
Introduction
During the past decades, the topic of fractional calculus has attracted many scientists and researchers due to its applications in many areas, see [5, 6, 15, 21]. Thus several researchers have investigated existence results for solutions to fractional differential equations due to the fact that many mathematical formulation of physical phenomena lead to integro-differential equations, for instance, mostly these types of equations arise in continuum and statistical mechanics and chemical kinetics , fluid dynamics, and biological models, for more details see [3, 17, 20]. Integro-differential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. The homotopy perturbation method and variational iteration method which are proposed by He [7, 8] are of the methods which have received much concern. These methods have been successfully applied by many authors such as [1, 2, 22]. In this work, we study the Integro-differential equations which are combination of differential and Fredholm-Volterra equations that have the fractional order. In particular, we applied the HPM and VIM for fractional Fredholm Integro-differential equations with constant coefficients of the form ∞ ∑
∫a Pk D∗α u(t)
= g(t) + λ
k=0
H(x, t)u(t)dt,
a ≤ x,
,t ≤ b
(1.1)
0
under the initial-boundary conditions D∗α u(a)
= u(0)
D∗α u(0)
= u (a)
′
(1.2)
where a is constant and 1 < α ≤ 2 and D∗α is the fractional derivative in the Caputo sense.
2
Preliminaries
In this section, we give some basic definitions and properties of fractional calculus theory which used in this paper. Definition 2.1. A real function f (x), x > 0 is said to be in space Cµ, µ ∈ R if there exists a real number p > µ, such that f (x) = xp f1 (x) where f1 (x) ∈ C(0, ∞), and it is said to be in the space Cµn if f n ∈ Rµ , n ∈ N .
2 Definition 2.2. The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f ∈ Cµ, µ ≥ −1 is defined as: 1 J f (x) = (Γ(α)
∫x α−1
(x − t)
α
f (t)dt,
α>0,
t>0
(2.1)
0
in particular J 0 f (x) = f (x) For β ≥ 0 and γ ≥ −1, some properties of the operator J α 1. J α J β f (x) = J α+β f (x) 2. J α J β f (x) = J β J α f (x) 3. J α xγ =
Γ(γ+1) α+γ Γ(α+γ+1) x
m Definition 2.3. The Caputo fractional derivative of f ∈ C−1 , m ∈ N is defined as:
1 D f (x) = Γ(m − α)
∫x m−α−1 m
(x − t)
α
f (t)dt, m − 1 < α ≤ m
(2.2)
0
Lemma 2.4. if m − 1 < α ≤ m ,m ∈ N, f ∈ Cµm , µ > −1 then the following two properties hold 1. Dα [J α f (x)] = f (x) 2. J α [Dα f (x)] = f (x) −
m−1 ∑ k=1
3
k
f k (0) xk!
Homotopy Perturbation Method
The homotopy perturbation method was first proposed by He [11, 12] is applied to various problems [9, 11–14]. To illustrate the basic idea of this method, we consider the following nonlinear differential equation A(u) − f (r) = 0,
r∈Ω
(3.1)
with boundary conditions B(u,
∂u ) = 0, ∂n
r∈Γ
(3.2)
where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function, Γ is the boundary of the domain Ω. In general, the operator A can be divided into two parts L and N , where L is linear, while N is nonlinear. Eq. (3.1) therefore can be rewritten as follows L(u) + N (u) − f (r) = 0
(3.3)
By the homotopy technique [18,19]. We construct a homotopy v(r, p) : Ω × [0, 1] → R which satisfies H(v, p) = (1 − p)[L(v) − L(u0 )] + p[A(v) − f (r)] = 0 p ∈ [0, 1], r ∈ Ω
(3.4)
3
or H(v, p) = L(v) − L(u0 ) + pL(u0 ) + p[N (v) − f (r)] = 0
(3.5)
where p ∈ [0, 1]is an embedding parameter, u0 is an initial approximation of eq. (3.1) which satisfies the boundary conditions. From equations (3.4),(3.5) we have H(v, 0) = L(v) − L(u0 ) = 0,
(3.6)
H(v, 1) = A(v) − f (r) = 0.
(3.7)
The changing in the process of p from zero to unity is just that of v(r, p) from u0 (r) to u(r). In topology this is called deformation and L(v) − L(u0 ), and A(v) − f (r) are called homotopic. Now, assume that the solution of equations (3.4),(3.5) can be expressed as v = v0 + pv1 + p2 v2 + ...
(3.8)
The approximate solution of Eq. (3.1) can be obtained by Setting p = 1. u = lim v = v0 + v1 + v2 + ... p→1
4
(3.9)
The Variational Iteration Method
To illustrate the basic concepts of VIM, we consider the following differential equation L(u) + N (u) = g(x)
(4.1)
where L is a linear operator, N nonlinear operator, and g(x) is an non-homogeneous term. According to VIM, one construct a correction functional as follows ∫x λ [Lyn (s) − N yf n (s)]ds
yn+1 = yn +
(4.2)
0
where λ is a general Lagrange multiplier, and yf yn =0. rem n denotes restricted variation i.e δf For the analysis of HPM and VIM we refer the reader to [4, 16].
5
Numerical Examples
In this section, we have applied homotopy perturbation method and variational iteration method to linear and nonlinear Fredholm Integro-differential equations of fractional order with known exact solution Example 5.1. Consider the following linear Fredholm integro-differential equation: ∫x D u(x) = 2e − 1 − α
u(t)dt 0 ≤ x ≤ 1, 1 < α ≤ 2.
x
(5.1)
0
with initial boundary conditions u(0) = 1, the exact is u(x) = ex .
′
u (1) = e
(5.2)
4
1. HPM According to HPM, we construct the following homotopy: ∫x Dα u(x) = p 2ex − 1 − u(t)dt
(5.3)
0
Substitution of (3.8) in (5.3) and then equating the terms with same powers of p we get the series p0
:
Dα u0 (x) = 0
:
D u1 (x) = 2e − 1 −
∫x 1
p
α
x
u0 (t)dt
(5.4)
0
.. .
∫x
n
p
D un (x) = − α
:
et un−1 (t)dt 0
Now applying the operator Jα to equations (5.4) . In order to satisfy the boundary conditions and convenient calculation, we will chose u0 (x), u1 (x) and un (x)(n = 2, 3, 4, · · · ) as follows: u0 (x) = u1 (x) =
1
Ax + J α 2e2x − 1 −
∫x
u0 dt
(5.5)
0
.. . un (x) =
J α −
∫x
un−1 (t)dt
0
Then by solving Eqs.(5.5), we obtain u1 , u2 , · · · as u1 (x) u2 (x) =
= Ax + α+2 −A 2 x Γ(α+3)
−
xα Γ(α+1)
+
x2α+1 Γ(2α+2)
xα+1 Γ(α+2)
−
+
x2α+2 Γ(2α+3)
2xα+2 Γ(α+3)
−
+
2x2α+3 Γ(2α+4)
2xα+3 Γ(α+4)
−
2x2α+4 Γ(2α+5)
Now, we can form the 2-term approximation ϕ2 (x)
= 1 + Ax +
xα xα+1 2xα+2 2xα+3 + + + Γ(α + 1) Γ(α + 2) Γ(α + 3) Γ(α + 4)
(5.6)
− A2 xα+2 x2α+1 x2α+2 2x2α+3 2x2α+4 − − − − + ··· Γ(α + 3) Γ(2α + 2) Γ(2α + 3) Γ(2α + 4) Γ(2α + 5) where A can be determined by imposing initial-boundary conditions (5.2) on ϕ2 . Table1 shows the value of A for different values of α. 2. VIM According to the variational iteration method, Eq.(5.1) can be expressed in the following form: ∫x (α − 1) α α uk+1 (x) = uk (x) − J D u(x) − 2ex − 1 − uk (t)dt (5.7) 2 0
5 Table 1: Value of A for different values of α using (5.6) and (5.9)
A(HP M ) A(V IM )
α = 1.25 0.1452034490 1.432940029
α = 1.5 0.4549747740 1.241066017
α = 1.75 0.7244709330 1.138179159
α=2 0.9516151533 1.108032497
fig. 1: Absolute error of functions E1 (x), E2 (x) and E3 (x) obtained by 2-term HPM and E4 (x), E5 (x) and E6 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.1).
Then, in order to avoid the complex and difficult fractional integration, we can take the truncated Taylor expansions for exponential term in (5.7) for example, ex ∼ 1 + 2 3 x + x2 + x6 and further, suppose that an initial approximation has the following form which satisfies the initial-boundary conditions u0 (x) = 1 + Ax
(5.8)
Now by iteration formula (5.7), the first approximation takes the following form ∫x (α − 1) α α u1 (x) = u0 (x) − J D u0 (x) − 2ex − 1 − u0 (t)dt 2 0 [ ] (α − 1) α 1 x (2 − A)x2 2x3 = 1 + Ax + x + + + 2 Γ(α + 1) Γ(α + 2) Γ(α + 3) 3Γ(α + 4) (5.9) By imposing initial-boundary conditions (5.2) on u1 . Example 5.2. Consider the following nonlinear Fredholm integro-differential equation: ∫x D u(x) = x cos x − 2 sin x + α
tu(t)dt
0 ≤ x ≤ π/2, 1 < α ≤ 2.
(5.10)
0
with initial boundary conditions u(0) = 0, the exact is u(x) = sin x.
′
u (π/2) = 0
(5.11)
6 fig. 2: Comparison of approximate solutions obtained by 3-term HPM and with exact solution at α = 2 for example (5.1).
Table 2: Value of A for different values of α using (5.12) and (5.13)
A(HP M ) A(V IM )
α = 1.25 1.229102855 0.7967121064
α = 1.5 1.154176102 0.9093326779
α = 1.75 1.078262573 0.9808215282
α=2 1.005549222 1.017387171
1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= Ax −
xα+1 xα+3 3xα+5 5xα+7 + − + + ··· Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
(5.12)
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is ∫x (α − 1) α α u1 (x) = u0 (x) − J D u0 (x)x cos x − 2 sin x + tu0 (t)dt 2 0 [ ] (α − 1) α 2x (2A − 1)x3 3x5 5x7 = Ax − x − − + (5.13) 2 Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
Example 5.3. Consider the following linear Fredholm integro-differential equation: 1 1 D u(x) = ( + e2x ) + 2 2
∫x (1 − et )u(t)dt
α
0 ≤ x ≤ 1, 1 < α ≤ 2.
(5.14)
0
with initial boundary conditions u(0) = 1, the exact is u(x) = ex .
′
u (1) = e
(5.15)
7
fig. 3: Absolute error of functions E7 (x), E8 (x) and E9 (x) obtained by 1-term HPM and E10 (x), E11 (x) and E12 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.2).
fig. 4: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example5.2.
8 Table 3: Value of A for different values of α using (5.16) and (5.17)
A(HP M ) A(V IM )
α = 1.25 0.2375025470 1.411203960
α = 1.5 0.5218416880 1.208885080
α = 1.75 0.7720691390 1.100882101
α=2 0.9849484867 1.068880597
fig. 5: Absolute error of functions E13 (x), E14 (x) and E15 (x) obtained by 1-term HPM and E16 (x), E17 (x) and E18 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.3).
1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= 1 + Ax +
xα Γ(α+1)
+
xα+1 Γ(α+2)
+
xα+2 Γ(α+3)
+
3xα+3 2Γ(α+4)
+ ··· (5.16)
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is [ ] 3 4 ( 3+2A )x (α − 1) α 1 x (A − 1)x2 Ax u1 (x) = 1 + Ax + x + + + 12 − 2 Γ(α + 1) Γ(α + 2) 2Γ(α + 3) Γ(α + 4) 2Γ(α + 5) (5.17) Example 5.4. Consider the following nonlinear Fredholm integro-differential equation: ( ) ∫x x 1 α D u(x) + u2 (t)dt + − sinh x − sinh 2x 0 ≤ x ≤ 1, 1 < α ≤ 2. (5.18) 2 4 0
with initial boundary conditions u(0) = 0,
′
u (1) = cosh 1
(5.19)
the exact is u(x) = sinh x. 1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= Ax +
xα+1 3xα+3 9xα+5 33xα+7 + + + + ··· Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
(5.20)
9 fig. 6: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example (5.3).
Table 4: Value of A for different values of α using (5.20) and (5.21)
A(HP M ) A(V IM )
α = 1.25 0.6894636978 0.8410301049
α = 1.5 0.8118993456 0.8387819527
α = 1.75 0.9079287321 0.8648448808
α=2 0.9816578735 0.9053789914
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is [ ] (α − 1) α x (3 − 2A2 )x3 9x5 33x7 u1 (x) = Ax + x + + + (5.21) 2 Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
6
Results and Discussion
The FIDEs withe initial-boundary conditions have been solved by using HPM and VIM. Based on preliminary calculations, we decided to use 2-term in HPM and first-order in VIM solutions. We note that increasing the number of terms improves the accuracy of HPM solutions. Table (1) shows the values of A for different values of α using (5.6) and (5.9). In Figure (1) absolute error functions E1 (x), E2 (x), and E3 (x) by HPM, E4 (x), E5 (x), and E6 (x) by VIM for different values of α, where E(x) = |uexact − uapproximate | have been drawn . From this figure we can see that the absolute error for α = 1.75 is smaller than the absolute errors for α = 1.25, 1.5 in both methods. In addition, the absolute errors for all α slightly increased from 0 till reach to maximum value then slightly decrease till reach to zero at x = 1. It can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (2) displays the approximate solution when α = 2 for HPM and VIM with exact solution. Table (2) shows the values of A for different values of α using (5.12) and (5.13) . The
10
fig. 7: Absolute error of functions E19 (x), E20 (x) and E21 (x) obtained by 1-term HPM and E22 (x), E23 (x) and E24 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.4).
fig. 8: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example5.4.
11
absolute error functions E7 (x) , E8 (x) , E9 (x) by HPM and E10 (x),E11 (x) and E12 (x) by VIM of example (5.2) for different values of α, where have been drawn in Figure (3). From this figure we can see that the absolute error for all α by HPM smaller than the absolute errors for all α by VIM . In addition also it can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (4) displays the approximate solution α = 2 for HPM and VIM with exact solution. Table (3) shows the values of A for different values of α using (5.16) and (5.17). The absolute error functions E13 (x) , E14 (x) , E15 (x) by HPM and E16 (x),E17 (x) and E18 (x) by VIM of example (5.3) for different values of α, have been drawn in Figure (5). From this figure we can see that the absolute error for α = 1.75 smaller than the absolute errors for α = 1.25, 1.5 in both methods. In addition, the absolute errors for all α slightly increase from 0 till reach to maximum value then slightly decrease till reach to zero at x = 1. Also, we can see that the absolute error for all α by HPM smaller than the absolute errors for all α by VIM .It can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (6) displays the approximate solution α = 2 for HPM and VIM with exact solution. Table (4) shows the values of A for different values of α using (5.20) and (5.21). The absolute error functions E19 (x) , E20 (x) , E21 (x) by HPM and E22 (x),E23 (x) and E24 (x) by VIM of example (5.4) for different values of α, where have been drawn in Figure (7). From this figure we can see that the maximum absolute error for α = 1.75 by HPM smaller than the absolute errors for α = 1.75 by VIM, while the maximum absolute error for α = 1.25, 1.5 by HPM greater than the maximum absolute error for α = 1.25, 1.5 by VIM. Figure(8) displays the approximate solution α = 2 for HPM and VIM with exact solution. Overall, it can be seen that the approximate solution is in excellent agreement with exact solution. Also, it is to be noted that the accuracy can be improved by computing more terms of approximated solution and/ or by taking more terms in the Taylor expansion of the exponential term.
References [1] Abbasbandy, S. (2007). An approximation solution of a nonlinear equation with Riemann–Liouville’s fractional derivatives by He’s variational iteration method. Journal of Computational and Applied Mathematics, 207(1), 53–58. [2] Abdulaziz , O., Hashim, I. and Momani, S. (2008). Application of homotopy– perturbation method to fractional IVPs. Journal of Computational and Applied Mathematics, 216(2),574–584. [3] Baleanu, D., Diethelm, Scalas, K. E. and Trujillo, J. J. (2012). Fractional calculus, of Series on Complexity,Nonlinearity and Chaos, vol. 3 ,World Scientific, Hackensack, NJ, USA. [4] Elbeleze, A. Kilicman, A. Taib, M. B. (2012). Application of homotopy perturbation and variational iteration methods for Fredholm integro-differential equation of fractional order. Abstract and Applied Analysis, vol. 2012, 14 pages. [5] Gaul, L., Klein, P. and Kemple, S. (1991). Damping description involving fractional operators. Mechanicaln Systems and Signal Processing, 5(2), 81–88. [6] Glockle, W.G. and Nonnenmacher, T. F. A. (1995). A fractional calculus approach of self-similar protein dynamics. Biophysical Journal, 68, 46–53.
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[7] He, J.-H. (1999). Homotopy perturbation technique.Computer Methods in Applied Mechanics and Engineering,178(3-4), 257–262. [8] He, J.-H. (1999). Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics,34(4), 699–708. [9] He, J.-H. (2000). A Coupling Method of Homotopy Technique and Perturbation Technique for Nonlinear Problems. International Journal of Non-Linear Mechanics, 35, 37– 43. [10] He, J.-H. (1997). Non-Pertubation Methods for Strongly Nonlinear Problems[ Dissertation]. Shanghai University, 35, 37–43. [11] He, J.-H. (2006). Homotopy Perturbation Method for Solving Boundary Value Problems.Physics Letters A, 350(1-2), 87–88. [12] He, J.-H. (2005). Application of Homotopy Perturbation Method to Nonlinear Wave Equations. Chaos, Solitons & Fractals, 26(3), 695–700. [13] He, J.-H. (2005). Homotopy Perturbation Method for Bifurcation of Nonlinear Problems.International Journal of Nonlinear Sciences and Numerical Simulation, 6(2), 87–88. [14] He, J.-H. (2003). Homotopy Perturbation Method: A New Nonlinear Analytic Technique.Applied Mathematics and Computation, 135, 73–79. [15] Hilfert,R. (2000). Applications of fractional calculus in physics. World Scientific, River Edge, NJ, USA [16] Kadem, A. and Kilicman, A. (2012). The approximate solution of fractional Fredholm integrodifferential equations by variational iteration and homotopy perturbation methods. Abstract and Applied Analysis, vol. 2012, Article ID 486193, 10 pages. [17] Kythe, P. K. and Puri, P. (2002). Computational methodsfor linear integralequations, Birkhauser, Boston, Mass, USA. [18] Liao, S. J. (1995). An approximate solution technique not depending on small parameters: a special example. International Journal of Non-Linear Mechanics, 30( 3), 371–380. [19] Liao, S.-J. (1997). Boundary element method for general nonlinear differential operators. Engineering Analysis with Boundary Elements, 20( 2), 91–99. [20] Mainardi, F. (1997). Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, Vienna, Austria. [21] Podlubny, I. (1999). Fractional differential equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. [22] Yıldırım, A. (2008). Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Computers & Mathematics with Applications, 56(12), 3175–3180.
S1018-3647(15)00041-5 http://dx.doi.org/10.1016/j.jksus.2015.04.006 JKSUS 312
To appear in:
Journal of King Saud University - Science
Please cite this article as: A.A. Elbeleze, A. Kilicman, B.M. Taib, Approximate Solution of Integro-Differential Equation of Fractional (Arbitrary) Order, Journal of King Saud University - Science (2015), doi: http://dx.doi.org/ 10.1016/j.jksus.2015.04.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
APPROXIMATE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATION OF FRACTIONAL (ARBITRARY) ORDER Asma A. Elbeleze1, Adem Kilicman2* and Bachok M. Taib3 1,3
Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), 71800 Nilai, Malaysia 2
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia emails:[email protected], [email protected], [email protected] *Corresponding author
APPROXIMATE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATION OF FRACTIONAL (ARBITRARY) ORDER
Abstract In the present paper, we study the integro-differential equations which are combination of differential and Fredholm-Volterra equations that having the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented. Keywords: Fractional calculus, Homotopy perturbation method, The variational iteration method, Fractional integro-differential equations
1
Introduction
During the past decades, the topic of fractional calculus has attracted many scientists and researchers due to its applications in many areas, see [5, 6, 15, 21]. Thus several researchers have investigated existence results for solutions to fractional differential equations due to the fact that many mathematical formulation of physical phenomena lead to integro-differential equations, for instance, mostly these types of equations arise in continuum and statistical mechanics and chemical kinetics , fluid dynamics, and biological models, for more details see [3, 17, 20]. Integro-differential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. The homotopy perturbation method and variational iteration method which are proposed by He [7, 8] are of the methods which have received much concern. These methods have been successfully applied by many authors such as [1, 2, 22]. In this work, we study the Integro-differential equations which are combination of differential and Fredholm-Volterra equations that have the fractional order. In particular, we applied the HPM and VIM for fractional Fredholm Integro-differential equations with constant coefficients of the form ∞ ∑
∫a Pk D∗α u(t)
= g(t) + λ
k=0
H(x, t)u(t)dt,
a ≤ x,
,t ≤ b
(1.1)
0
under the initial-boundary conditions D∗α u(a)
= u(0)
D∗α u(0)
= u (a)
′
(1.2)
where a is constant and 1 < α ≤ 2 and D∗α is the fractional derivative in the Caputo sense.
2
Preliminaries
In this section, we give some basic definitions and properties of fractional calculus theory which used in this paper. Definition 2.1. A real function f (x), x > 0 is said to be in space Cµ, µ ∈ R if there exists a real number p > µ, such that f (x) = xp f1 (x) where f1 (x) ∈ C(0, ∞), and it is said to be in the space Cµn if f n ∈ Rµ , n ∈ N .
2 Definition 2.2. The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f ∈ Cµ, µ ≥ −1 is defined as: 1 J f (x) = (Γ(α)
∫x α−1
(x − t)
α
f (t)dt,
α>0,
t>0
(2.1)
0
in particular J 0 f (x) = f (x) For β ≥ 0 and γ ≥ −1, some properties of the operator J α 1. J α J β f (x) = J α+β f (x) 2. J α J β f (x) = J β J α f (x) 3. J α xγ =
Γ(γ+1) α+γ Γ(α+γ+1) x
m Definition 2.3. The Caputo fractional derivative of f ∈ C−1 , m ∈ N is defined as:
1 D f (x) = Γ(m − α)
∫x m−α−1 m
(x − t)
α
f (t)dt, m − 1 < α ≤ m
(2.2)
0
Lemma 2.4. if m − 1 < α ≤ m ,m ∈ N, f ∈ Cµm , µ > −1 then the following two properties hold 1. Dα [J α f (x)] = f (x) 2. J α [Dα f (x)] = f (x) −
m−1 ∑ k=1
3
k
f k (0) xk!
Homotopy Perturbation Method
The homotopy perturbation method was first proposed by He [11, 12] is applied to various problems [9, 11–14]. To illustrate the basic idea of this method, we consider the following nonlinear differential equation A(u) − f (r) = 0,
r∈Ω
(3.1)
with boundary conditions B(u,
∂u ) = 0, ∂n
r∈Γ
(3.2)
where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function, Γ is the boundary of the domain Ω. In general, the operator A can be divided into two parts L and N , where L is linear, while N is nonlinear. Eq. (3.1) therefore can be rewritten as follows L(u) + N (u) − f (r) = 0
(3.3)
By the homotopy technique [18,19]. We construct a homotopy v(r, p) : Ω × [0, 1] → R which satisfies H(v, p) = (1 − p)[L(v) − L(u0 )] + p[A(v) − f (r)] = 0 p ∈ [0, 1], r ∈ Ω
(3.4)
3
or H(v, p) = L(v) − L(u0 ) + pL(u0 ) + p[N (v) − f (r)] = 0
(3.5)
where p ∈ [0, 1]is an embedding parameter, u0 is an initial approximation of eq. (3.1) which satisfies the boundary conditions. From equations (3.4),(3.5) we have H(v, 0) = L(v) − L(u0 ) = 0,
(3.6)
H(v, 1) = A(v) − f (r) = 0.
(3.7)
The changing in the process of p from zero to unity is just that of v(r, p) from u0 (r) to u(r). In topology this is called deformation and L(v) − L(u0 ), and A(v) − f (r) are called homotopic. Now, assume that the solution of equations (3.4),(3.5) can be expressed as v = v0 + pv1 + p2 v2 + ...
(3.8)
The approximate solution of Eq. (3.1) can be obtained by Setting p = 1. u = lim v = v0 + v1 + v2 + ... p→1
4
(3.9)
The Variational Iteration Method
To illustrate the basic concepts of VIM, we consider the following differential equation L(u) + N (u) = g(x)
(4.1)
where L is a linear operator, N nonlinear operator, and g(x) is an non-homogeneous term. According to VIM, one construct a correction functional as follows ∫x λ [Lyn (s) − N yf n (s)]ds
yn+1 = yn +
(4.2)
0
where λ is a general Lagrange multiplier, and yf yn =0. rem n denotes restricted variation i.e δf For the analysis of HPM and VIM we refer the reader to [4, 16].
5
Numerical Examples
In this section, we have applied homotopy perturbation method and variational iteration method to linear and nonlinear Fredholm Integro-differential equations of fractional order with known exact solution Example 5.1. Consider the following linear Fredholm integro-differential equation: ∫x D u(x) = 2e − 1 − α
u(t)dt 0 ≤ x ≤ 1, 1 < α ≤ 2.
x
(5.1)
0
with initial boundary conditions u(0) = 1, the exact is u(x) = ex .
′
u (1) = e
(5.2)
4
1. HPM According to HPM, we construct the following homotopy: ∫x Dα u(x) = p 2ex − 1 − u(t)dt
(5.3)
0
Substitution of (3.8) in (5.3) and then equating the terms with same powers of p we get the series p0
:
Dα u0 (x) = 0
:
D u1 (x) = 2e − 1 −
∫x 1
p
α
x
u0 (t)dt
(5.4)
0
.. .
∫x
n
p
D un (x) = − α
:
et un−1 (t)dt 0
Now applying the operator Jα to equations (5.4) . In order to satisfy the boundary conditions and convenient calculation, we will chose u0 (x), u1 (x) and un (x)(n = 2, 3, 4, · · · ) as follows: u0 (x) = u1 (x) =
1
Ax + J α 2e2x − 1 −
∫x
u0 dt
(5.5)
0
.. . un (x) =
J α −
∫x
un−1 (t)dt
0
Then by solving Eqs.(5.5), we obtain u1 , u2 , · · · as u1 (x) u2 (x) =
= Ax + α+2 −A 2 x Γ(α+3)
−
xα Γ(α+1)
+
x2α+1 Γ(2α+2)
xα+1 Γ(α+2)
−
+
x2α+2 Γ(2α+3)
2xα+2 Γ(α+3)
−
+
2x2α+3 Γ(2α+4)
2xα+3 Γ(α+4)
−
2x2α+4 Γ(2α+5)
Now, we can form the 2-term approximation ϕ2 (x)
= 1 + Ax +
xα xα+1 2xα+2 2xα+3 + + + Γ(α + 1) Γ(α + 2) Γ(α + 3) Γ(α + 4)
(5.6)
− A2 xα+2 x2α+1 x2α+2 2x2α+3 2x2α+4 − − − − + ··· Γ(α + 3) Γ(2α + 2) Γ(2α + 3) Γ(2α + 4) Γ(2α + 5) where A can be determined by imposing initial-boundary conditions (5.2) on ϕ2 . Table1 shows the value of A for different values of α. 2. VIM According to the variational iteration method, Eq.(5.1) can be expressed in the following form: ∫x (α − 1) α α uk+1 (x) = uk (x) − J D u(x) − 2ex − 1 − uk (t)dt (5.7) 2 0
5 Table 1: Value of A for different values of α using (5.6) and (5.9)
A(HP M ) A(V IM )
α = 1.25 0.1452034490 1.432940029
α = 1.5 0.4549747740 1.241066017
α = 1.75 0.7244709330 1.138179159
α=2 0.9516151533 1.108032497
fig. 1: Absolute error of functions E1 (x), E2 (x) and E3 (x) obtained by 2-term HPM and E4 (x), E5 (x) and E6 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.1).
Then, in order to avoid the complex and difficult fractional integration, we can take the truncated Taylor expansions for exponential term in (5.7) for example, ex ∼ 1 + 2 3 x + x2 + x6 and further, suppose that an initial approximation has the following form which satisfies the initial-boundary conditions u0 (x) = 1 + Ax
(5.8)
Now by iteration formula (5.7), the first approximation takes the following form ∫x (α − 1) α α u1 (x) = u0 (x) − J D u0 (x) − 2ex − 1 − u0 (t)dt 2 0 [ ] (α − 1) α 1 x (2 − A)x2 2x3 = 1 + Ax + x + + + 2 Γ(α + 1) Γ(α + 2) Γ(α + 3) 3Γ(α + 4) (5.9) By imposing initial-boundary conditions (5.2) on u1 . Example 5.2. Consider the following nonlinear Fredholm integro-differential equation: ∫x D u(x) = x cos x − 2 sin x + α
tu(t)dt
0 ≤ x ≤ π/2, 1 < α ≤ 2.
(5.10)
0
with initial boundary conditions u(0) = 0, the exact is u(x) = sin x.
′
u (π/2) = 0
(5.11)
6 fig. 2: Comparison of approximate solutions obtained by 3-term HPM and with exact solution at α = 2 for example (5.1).
Table 2: Value of A for different values of α using (5.12) and (5.13)
A(HP M ) A(V IM )
α = 1.25 1.229102855 0.7967121064
α = 1.5 1.154176102 0.9093326779
α = 1.75 1.078262573 0.9808215282
α=2 1.005549222 1.017387171
1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= Ax −
xα+1 xα+3 3xα+5 5xα+7 + − + + ··· Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
(5.12)
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is ∫x (α − 1) α α u1 (x) = u0 (x) − J D u0 (x)x cos x − 2 sin x + tu0 (t)dt 2 0 [ ] (α − 1) α 2x (2A − 1)x3 3x5 5x7 = Ax − x − − + (5.13) 2 Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
Example 5.3. Consider the following linear Fredholm integro-differential equation: 1 1 D u(x) = ( + e2x ) + 2 2
∫x (1 − et )u(t)dt
α
0 ≤ x ≤ 1, 1 < α ≤ 2.
(5.14)
0
with initial boundary conditions u(0) = 1, the exact is u(x) = ex .
′
u (1) = e
(5.15)
7
fig. 3: Absolute error of functions E7 (x), E8 (x) and E9 (x) obtained by 1-term HPM and E10 (x), E11 (x) and E12 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.2).
fig. 4: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example5.2.
8 Table 3: Value of A for different values of α using (5.16) and (5.17)
A(HP M ) A(V IM )
α = 1.25 0.2375025470 1.411203960
α = 1.5 0.5218416880 1.208885080
α = 1.75 0.7720691390 1.100882101
α=2 0.9849484867 1.068880597
fig. 5: Absolute error of functions E13 (x), E14 (x) and E15 (x) obtained by 1-term HPM and E16 (x), E17 (x) and E18 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.3).
1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= 1 + Ax +
xα Γ(α+1)
+
xα+1 Γ(α+2)
+
xα+2 Γ(α+3)
+
3xα+3 2Γ(α+4)
+ ··· (5.16)
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is [ ] 3 4 ( 3+2A )x (α − 1) α 1 x (A − 1)x2 Ax u1 (x) = 1 + Ax + x + + + 12 − 2 Γ(α + 1) Γ(α + 2) 2Γ(α + 3) Γ(α + 4) 2Γ(α + 5) (5.17) Example 5.4. Consider the following nonlinear Fredholm integro-differential equation: ( ) ∫x x 1 α D u(x) + u2 (t)dt + − sinh x − sinh 2x 0 ≤ x ≤ 1, 1 < α ≤ 2. (5.18) 2 4 0
with initial boundary conditions u(0) = 0,
′
u (1) = cosh 1
(5.19)
the exact is u(x) = sinh x. 1. HPM We follow the same procedure in example (5.1) By HPM, so the approximate solution is ϕ2 (x)
= Ax +
xα+1 3xα+3 9xα+5 33xα+7 + + + + ··· Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
(5.20)
9 fig. 6: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example (5.3).
Table 4: Value of A for different values of α using (5.20) and (5.21)
A(HP M ) A(V IM )
α = 1.25 0.6894636978 0.8410301049
α = 1.5 0.8118993456 0.8387819527
α = 1.75 0.9079287321 0.8648448808
α=2 0.9816578735 0.9053789914
2. VIM we follow the same procedure in example (5.1) by VIM, so the approximate solution is [ ] (α − 1) α x (3 − 2A2 )x3 9x5 33x7 u1 (x) = Ax + x + + + (5.21) 2 Γ(α + 2) Γ(α + 4) Γ(α + 6) Γ(α + 8)
6
Results and Discussion
The FIDEs withe initial-boundary conditions have been solved by using HPM and VIM. Based on preliminary calculations, we decided to use 2-term in HPM and first-order in VIM solutions. We note that increasing the number of terms improves the accuracy of HPM solutions. Table (1) shows the values of A for different values of α using (5.6) and (5.9). In Figure (1) absolute error functions E1 (x), E2 (x), and E3 (x) by HPM, E4 (x), E5 (x), and E6 (x) by VIM for different values of α, where E(x) = |uexact − uapproximate | have been drawn . From this figure we can see that the absolute error for α = 1.75 is smaller than the absolute errors for α = 1.25, 1.5 in both methods. In addition, the absolute errors for all α slightly increased from 0 till reach to maximum value then slightly decrease till reach to zero at x = 1. It can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (2) displays the approximate solution when α = 2 for HPM and VIM with exact solution. Table (2) shows the values of A for different values of α using (5.12) and (5.13) . The
10
fig. 7: Absolute error of functions E19 (x), E20 (x) and E21 (x) obtained by 1-term HPM and E22 (x), E23 (x) and E24 (x) obtained by VIM with α = 1.25, α = 1.5, α = 1.75 for example (5.4).
fig. 8: Comparison of approximate solutions obtained by 2-term HPM with exact solution at α = 2 for example5.4.
11
absolute error functions E7 (x) , E8 (x) , E9 (x) by HPM and E10 (x),E11 (x) and E12 (x) by VIM of example (5.2) for different values of α, where have been drawn in Figure (3). From this figure we can see that the absolute error for all α by HPM smaller than the absolute errors for all α by VIM . In addition also it can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (4) displays the approximate solution α = 2 for HPM and VIM with exact solution. Table (3) shows the values of A for different values of α using (5.16) and (5.17). The absolute error functions E13 (x) , E14 (x) , E15 (x) by HPM and E16 (x),E17 (x) and E18 (x) by VIM of example (5.3) for different values of α, have been drawn in Figure (5). From this figure we can see that the absolute error for α = 1.75 smaller than the absolute errors for α = 1.25, 1.5 in both methods. In addition, the absolute errors for all α slightly increase from 0 till reach to maximum value then slightly decrease till reach to zero at x = 1. Also, we can see that the absolute error for all α by HPM smaller than the absolute errors for all α by VIM .It can be observed from this figure that increasing the accuracy of the HPM and VIM solution by increasing the fractional derivative. While, Figure (6) displays the approximate solution α = 2 for HPM and VIM with exact solution. Table (4) shows the values of A for different values of α using (5.20) and (5.21). The absolute error functions E19 (x) , E20 (x) , E21 (x) by HPM and E22 (x),E23 (x) and E24 (x) by VIM of example (5.4) for different values of α, where have been drawn in Figure (7). From this figure we can see that the maximum absolute error for α = 1.75 by HPM smaller than the absolute errors for α = 1.75 by VIM, while the maximum absolute error for α = 1.25, 1.5 by HPM greater than the maximum absolute error for α = 1.25, 1.5 by VIM. Figure(8) displays the approximate solution α = 2 for HPM and VIM with exact solution. Overall, it can be seen that the approximate solution is in excellent agreement with exact solution. Also, it is to be noted that the accuracy can be improved by computing more terms of approximated solution and/ or by taking more terms in the Taylor expansion of the exponential term.
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