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Open Physics

formerly Central European Journal of Physics

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Volume 14, Issue 1

Issues

Volume 13 (2015)

Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

A. Bolandtalat
  • Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran (Islamic Republic of)
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/ E. Babolian
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  • Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran (Islamic Republic of)
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/ H. Jafari
  • Department of Mathematical sciences, University of South Africa, UNISA003, South Africa\newline and Department of Mathematics, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran (Islamic Republic of)
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Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/phys-2016-0028

Abstract

In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

Keywords: Boubaker polynomials; Multi-order fractional differential equation; Operational matrix

PACS: 02.30.Mv; 02.60.-x

1 Introduction

During the last few decades, fractional differential equations (FDEs) have been applied to describe mathematical phenomenon in physics, chemistry, damping laws, rheology, control theory, signal processing, viscoelastic materials, polymers and so on [4, 10, 13, 17, 19]. Since most FDEs do not have exact analytic solutions, many researchers have tried to find solutions of FDEs using approximate and numerical techniques. For example see [3, 12, 16, 2024]. Our aim in this work is the following type of multi-order FDE:

Dαy(x)=i=1kyiDxβiy(x)+yk+1y(x)+g(x),(1)

y(p)(0)=dp,  p=0,1,,n1,(2)

where n − 1 < αn, the coefficients yi (i = 1, …, k + 1) are constant, 0 < β1 < β2 < ... < βk < α and g is a known function. Moreover, Dα y(x) denotes the Caputo fractional derivative of order α. It is defined as [4, 7, 17]:

Dxαy(x)={I(kα)y(k)(x),k1<α<k,kN,dkdtky(x),α=k.(3)

In (3) I(kα) denotes the Riemann-Liouville fractional. It is generally defined as follows:

Iαy(x)=1Γ(α)0xy(t)(xt)1αdt,α>0.(4)

Here we list the few properties of these two operators as follow:

(a)  Dxαxβ={0,βIN0,  β<α,Γ(β+1)Γ(1+βα)xβα,βIN0,  βα,or  βIN0,  β>α,

(b)  IαDxαy(x)=y(x)k=0n1y(k)(0+)xkk!,(5)

(c)  IαDxβy(x)=Iαβy(x)k=0m1y(k)(0)Γ(αβ+k+1)(xa)αβ+k,m1<βm.(6)

Many approximation and numerical techniques are utilized to determine the numerical solution of multi-order FDE [8, 11].

The Boubaker polynomials were established for the first time by Boubaker [1, 5, 6] as a guide for solving a one-dimensional heat transfer equation and second order differential equations. Kumar used these polynomials to solve Love’s equation in a particular physical system [14]. We will generalize the operational matrix for fractional integration using Boubaker polynomials [1, 5, 6, 14].

In this study, we want to solve the multi-order FDE using the operational matrix for fractional integration based on Boubaker Polynomials. The aim of this approach is converting the multi-order FDE into a set of algebraic equations by expanding the unspecified function within Boubaker polynomials.

This work is organized into six sections. Section 2 deals with some properties of Boubaker polynomials. The operational matrix is computed for fractional order integration in Section 3. We convert the multi-order FDE to system of algebraic equations in Section 4. Following this we solve a selection of numerical examples in Section 5 by using the proposed technique. A brief conclusion is presented in Section 6.

2 Boubaker polynomials

The Boubaker polynomials monomial definition is given by [5, 6, 12]:

Bn(x)=p=0ξ(n)[(n4p)(np)Cnpp](1)pxn2p,(7)

where

ξ(n)=n2=2n+((1)n1)4,Cnpp=(np)!p!(n2p)!.(8)

The symbol ⌊ ⌋ denotes the floor function.

The Boubaker polynomials could be calculated by following recursive formula:

{Bm(x)=xBm1(x)Bm2(x),form2,B0(x)=1,B1(x)=x.(9)

2.1 Approximation of function

The function f is approximated by Boubaker polynomials as following:

f(x)i=0NciBi=CTB(x),(10)

where B(x)T = [B0, B1, …, BN], Bi(x), i = 0, 1, 2, ···, N denote the Boubaker polynomials, CT = [c0, c1, …, cN] are unknown Boubaker cofficients and N is chosen as any positive integer.

Then CT can be obtained by

CTB(x),B(x)=f,B(x),(11)

where

f,B(x)=01f(x)B(x)Tdx=[f,B0,f,B1,,f,Bm],(12)

and 〈B(x), B(x)〉 is called the dual matrix of ϕ denoted by Q, and Q is obtained as:

Q=B(x),B(x)=01B(x)B(x)Tdx,(13)

and then

CT=(01f(x)B(x)Tdx)Q1.(14)

By using the expression (7) and taking n = 0, …, N, we find the corresponding matrix relation as we can express Boubaker polynomials in terms of power basis functions

B(x)=ZX(x),(15)

where

X(x)=[1  xxN]T,(16)

and if N is odd,

Z=[φ0,0000  000φ1,000  00φ2,10φ2,00  00  φN1,N120φN1,N320  φN1,000φN,N120φN,N32  0φN,0]

(default) and if N is even,

Z=[φ0,0000  000φ1,000  00φ2,10φ2,00  00  0φN1,N120φN1,N42  φN1,00φN,N20φN,N220  0φN,0]

where

Bn(x)=p=0ξ(n)φn,pxn2p,n=0,1,,N,p=0,1,,n2.(17)

φn,p=[(n4p)(np)Cnpp](1)p.(18)

3 Operational matrix for fractional order integration

For a vector B(Mx), we can approximate the operational matrices of fractional order integration as:

0IxαB(x)PαB(x),(19)

where Pα is the (N + 1) × (N + 1) Riemann-Liouville fractional operational matrix of integration for Boubaker polynomials. We compute Pα as follows:

0IxαB(x)=1Γ(α)0x(xτ)α1B(τ)dτ.(20)

By substituting B(x) = ZX(x), we get:

0IxαB(x)=1Γ(α)0x(xτ)α1ZX(τ)dτ=Z[Iα1,Iαx,,IαxN]T=Z[0!Γ(α+1)xα,,N!Γ(α+N+1)xα+N]T=ZDX¯(x),(21)

where the matrix D(N + 1) × (N + 1) is given by

D=[0!Γ(α+1)      0                    0      01!Γ(α+2)                    0                                    0      0      0        N!Γ(α+N+1)],

and

X¯(x)=[xαxα+1xα+N].

Now we approximate xk + α by N + 1 terms of the Boubaker basis

xα+iEiTB(x),(22)

where Ei = [Ei, 0, Ei, 1, …, Ei, N] and

Ei,j=Q101xα+iBi,j(x)dx=N!Γ(i+j+α+1)j!Γ(i+N+α+2),i,j=0,1,,N,(23)

where E is an (N + 1) × (N + 1) matrix with Ei as its column. Therefore, we can write

Iαϕ(x)=ZD[E0TB(x),E1TB(x),,EmTB(x)]T=ZDETB(x).(24)

Finally, we obtain

0Ixαϕ(x)PαB(x),(25)

where

Pα=ZDE,(26)

is called the operational matrix of fractional integration Boubaker polynomials.

4 Operational matrix for multi-order FDE

In this section, we employ the Boubaker polynomials for solving the multi-order FED (1). First we apply Iα on both sides of (1). It gives the following fractional order integral equation

y(x)k=0n1y(k)(0+)xkk!=i=1kγi(Iαβiy(x)j=0ni1y(j)(0)Γ(αβi+j+1)xαβi+k)+γk+1Iαy(x)+Iαg(x),y(i)(0)=di,    i=0,1,,n1,(27)

where ni − 1 < βini, niN. This implies that

y(x)=i=1kγi(Iαβiy(x))+γk+1Iαy(x)+h(x),            (28)

y(i)(0)=di,  i=0,1,,n1,(29)

where

h(x)=Ixαf(x)+k=0n1y(k)(0+)xkk!j=0ni1y(j)(0)Γ(αβi+j+1)xαβi+k.(30)

Now, we approximate y and g by Boubaker polynomials B(x) as follows:

y(x)i=0NcjBi(x)=CTB(x),(31)

h(x)i=0NgjBi(x)=GTB(x),(32)

such that

CT=[c0,,cN]T,  GT=[g0,,gN]T,(33)

where GT and CT are known and unknown vectors, respectively.

We also approximate the fractional order integrals by using (26) as follow:

Ixαy(x)CTPαB(x),  Ixαβiy(x)CTPαβiB(x).(34)

By substituting (31)-(34) in (28), we obtain:

(CTCTi=1kγiPαβiγk+1CTPα+GT)B(x)=0.(35)

Finally, we get:

CTCTi=1kγiIαβiγk+1CTIα+GT=0.(36)

Finally, by solving the above system of algebraic equations we find vector C can be obtained . Consequently y(x) can be approximated by (31).

5 Applications

Here, we use the presented numerical approach to solve several illustrative examples.

We solve the following FDE [3, 12]

D0.5y(x)+y(x)=x+π2,0<α1,y(0)=0,(37)

with the exact solution y(x)=x.

In Fig. 1, we plotted the exact solution and the approximate solutions of y for N = 3 and N = 5. Definitely, by increasing the value of N, the approximate value of y(x) will close to the exact values.

The exact solution (Red line) and approximation solutions for N = 3 (dotted line) and N = 5 (dashed line).
Figure 1

The exact solution (Red line) and approximation solutions for N = 3 (dotted line) and N = 5 (dashed line).

Consider the following FDE [12, 16]

Dαy(x)=y(x)+x2+2x2αΓ(3α),0<x<1,y(0)=0,(38)

the exact solution in this case:

y(x)=x2.(39)

We applied Boubaker polynomials approach to solve (38) with N = 3. In this case we obtain y(x) = x2

Consider the inhomogeneous Bagley-Torvik equation as a multi-order FED [2, 12, 20]

D2y(x)+D32y(x)+y(x)=1+x,0<x<1,y(0)=1,y(0)=1.(40)

The exact solution of (40) is:

y(x)=1+x.(41)

We solved this equation by Boubaker polynomials with N = 3 and obtained the exact solution 1 + x.

The exact solution (red line), approximation solutions for N = 6 (dotted line), N = 4 (dashed line) and N = 3 (long-dashed line).
Figure 2

The exact solution (red line), approximation solutions for N = 6 (dotted line), N = 4 (dashed line) and N = 3 (long-dashed line).

The last examined equation [3, 12] is

D2y(x)2Dy(x)+D0.5+y(x)=x7+2048429πx6.514x6+42x5x283πx1.5+4x2,y(0)=0,y(0)=0,  0<x<1,(42)

The exact solution y(x) = x7x2.

We applied this method for N = 3, 4, 6, and the result is plotted in Fig. 2. It can be seen that by increasing the value of N, the approximate solution will converge to the exact values.

6 Conclusion

In this work we applied Boubaker polynomials for solving multi-order FDE. The Boubaker polynomials operational matrices of fractional integration was used. Illustrative examples were presented to show the applicability and validity of the approach.

Mathematica was used for computation in this paper.

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About the article

Received: 2015-12-27

Accepted: 2016-02-24

Published Online: 2016-07-12

Published in Print: 2016-01-01


Citation Information: Open Physics, Volume 14, Issue 1, Pages 226–230, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2016-0028.

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© 2016 A. Bolandtalat et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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