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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

CiteScore 2018: 1.01

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Source Normalized Impact per Paper (SNIP) 2018: 0.541

ICV 2017: 162.45

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2391-5471
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Volume 15, Issue 1

# Numerical solution for fractional Bratu’s initial value problem

Fateme Ghomanjani
/ Stanford Shateyi
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/phys-2017-0131

## Abstract

In this paper, a method for solving fractional Bratu’s initial value problem (FBIVP) is presented. The main idea behind this work is the use of the Bezier curve method (BCM). To show the efficiency of the developed method, numerical results are presented.

PACS: 02.60.Jh; 02.60.Nm

## 1 Introduction

Fractional differential equations (FDEs) are modelled in different fields of science and engineering such as control engineering, electromagnetism, image processing, fluid flow, statistical mechanics, In general, most of FDEs do not have exact solutions, therefore some investigators studied various methods for finding approximate solutions. These techniques include, Adomian decomposition method (ADM) [1], homotopy perturbation method (HPM) [2], Bezier curve method (BCM) [3], finite difference method, etc. For example, He [1] proposed HPM when this method is an approach which searches for an analytical approximate solution of linear and nonlinear problems, also, the differential transform method (DTM) is applied to use for solving FBIVP [4].

BCM is used for solving dynamical systems, (see [5]). Also BCM is used for solving delay differential equations and switched systems (see [5]). Authors in [6] proposed the utilization of BCM on some linear optimal control systems with pantograph delays. Also, to solve the quadratic Riccati differential equation and the Riccati differential-difference equation, BCM is utilized (see [6]). In this study, BCM is extended for solving FBIVP as follows:

$D0+αx(t)+λex(t)=0,1<α<2,0<1,x(0)=x0,x′(0)=x0′,$(1)

where $\begin{array}{}{D}_{{0}_{+}}^{\alpha }\end{array}$ is derivative operator of fractional order α, x(t) is unknown function on [0,1], x′(0) is the derivative of x with respect to t at t = 0, and x0, $\begin{array}{}{x}_{0}^{\prime }\end{array}$ and λ are given constant.

Also, ADM was investigated for solving FBIVP in [7]. Babolian et al. [8] presented reproducing kernel method (RKM) for solving FBIVP.

The outline of this sequel is as follows: In Section 2, background materials are stated. In Section 3, problem statement is presented. Section 4 is devoted to numerical examples for the precision of the proposed technique. Finally, the conclusion is presented in Section 5.

## 2 Background materials

Several definitions of a fractional derivative of order α > 0 existed.

#### Definition 2.1

The caputo’s fractional derivative of order α is stated in [3]

$(D0+αx)(t)=1Γ(n1−α)∫0t(t−s)−α−1+n1x(n1)(s)ds,n1−1≤α≤n1,n1∈N,$

where α0 > 0 and n1 is the smallest integer greater than α.

## 2.1 Function approximation

Utilizing Bezier curves, this technique is to approximate the solutions x(t) where x(t) is given in Eq. (2). Define the Bezier polynomials of degree n that approximate over the interval t ∊ [t0, tf] as follows:

$x≈Pnx=∑i=0nciBi,nt−t0h=CTB(t),$(2)

where h = tft0, t0 = 0, tf = 1, CT = [c0,c1,…,cn]T, and

$BT(t)=[B0,n(t),B1,n(t),…,Bn,n(t)]T,Bi,n(t−t0h)=ni1hn(tf−t)n−i(t−t0)i,$(3)

is the Bernstein polynomial with degree n for t [t0,tf], and cr is the control point [3].

## 3 Problem statement

One may consider the following FBIVP:

$D0+αx(t)+λex(t)=0,1<α≤2,0(4)

By BCM, one may have

$xn(t)≃∑i=0nciBi,n(t),0≤t≤1,$(5)

by Eqs. (4) and (5), one may have

$D0+α∑i=0nciBi,n(t)=−λe∑i=0nciBi,n(t)$(6)

Hence

$R(x,c0,c1,…,cn)=∑i=0nciD0+αBi,n(x)+λe∑i=0nciBi,n(t)$

Suppose

$S(x,c0,c1,…,cn)=∫01R(x,c0,c1,…,n)2w1(x)dx,$

where w1(x) = 1,

$S(x,c0,c1,…,cn)==∫01∑i=0nciDαBi,n(x)+λe∑i=0nciBi,n(t)2dx,$

then, one may have

$∂S∂ci=0,0≤i≤n,$

and therefore

$∫01∑i=0nciDαBi,n(x)+λe∑i=0nciBi,n(t)×DαBi,n(x)+λe∑i=0nciBi,n(t)Bi,n(x)dx=0,$(7)

by Eq. (7), one can obtain a system of n+1 linear equations with n+1 unknown coefficients ci. Also by utilizing many subroutine algorithm for solving this linear equations, one can find the unknown coefficients ci, i = 0, 1, …, n.

## 4 Numerical application

In this section, some numerical examples are presented to illustrate the proposed method.

#### Example 1

The following FBIVP is considered (see [8]):

$D0+αx(t)−2ex(t)=0,1<α≤2,0

Using the described technique with n = 5, one may have the following xapprox(t), Figures 1, 2, Table 1 and Table 2. The computation takes 6 s of CPU time when it is performed by Maple 16. Also the value of approximate solution with the stated technique is more accurate than that with the stated technique in [8] (see Table 2).

$xapprox(t)=0.01076333172t(1−t)4+.9314121143t2(1−t)3+3.125786554t3(1−t)2+3.058176445t4(1−t)+1.231252941t5,forα=1.9xapprox(t)=0.01076333202t(1−t)4+.9314121129t2(1−t)3+3.125786554t3(1−t)2+3.058176446t4(1−t)−2t5ln(cos(1)),forα=1.5.$

Figure 1

The graphs of approximated (with α = 1.5) and exact (with α = 2) solution x(t) for Example 1

Figure 2

The graphs of approximated (with α = 1.9) and exact (with α = 2) solution x(t) for Example 1

Table 1

The comparison of approximation solution of the this method and RKM in [8] with α = 1.9 and n = 5 for Example 1

Table 2

The comparison of between the error solution of the this method and RKM in [8] with α = 2 for Example 1

#### Example 2

The following FBIVP is considered (see [8]):

$D0+αx(t)−e2x(t)=0,1<α≤2,0

Using the described technique with n = 5, one may have the following xapprox(t), Figures 3, 4, Table 4 and Table 3. Also the value of approximate solution with the stated technique is more accurate than that with the stated technique in [8] (see Table 3).

$xapprox(t)=0.7528184499t2(1−t)3+1.117894533t3(1−t)2+1.572350780t4(1−t)+0.6156264706t5,forα=1.9$

Figure 3

The graphs of approximated (with α = 1.5) and exact (with α = 2) solution x(t) for Example 2

Figure 4

The graphs of approximated (with α = 1.9) and exact (with α = 2) solution x(t) for Example 2

Table 3

The comparison of between the approximate solution of the this method and RKM in [8] with α = 1.9 and n = 5 for Example 2

Table 4

The comparison of between the error solution of the this method and RKM in [8] with α = 2 for Example 2

## 5 Conclusions

In this study, BCM is used to solve a class of FBIVP. The achieved results by the BCM are in good agreement with the given exact solutions. The study shows that the method is effective and is a simple technique to solve FBIVP.

One may extend this method for optimal control problems governed by fuzzy ordinary differential equations (FODEs), but it has a complicated manipulation. But the method is simple, by solving various numerical examples, accuracy can be found in comparison of other methods.

## Acknowledgement

The author wishes to acknowledge financial support from the University of Venda.

## References

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Daftardar-Gejji V., Jafari H., Solving a multi-order fractional differential equation using adomian decomposition, applied mathematics and computation, 2007, 189, 541-548. Google Scholar

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Abdulaziz O., Hashim I., Momani S., Solving systems of fractional differential equations by homotopy-perturbation method, Physics Letters A, 2008, 372, 451-459.

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Ghomanjani F., A new approach for solving fractional differential-algebraic equations, journal of taibah university for science, 2017, DOI:10.1016/j.jtusci.2017.03.006.

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Grover M., Tomer A.K., Numerical approach to differential equations of fractional order Bratu-type equations by differential transform method, Global Journal of Pure and Applied Mathematics, 2017, 13(9), 5813-5826. Google Scholar

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Ghomanjani F., Farahi M.H., Optimal control of switched systems based on bezier control points, International Journal of Intelligent Systems and Applications, 2012, 4(7), 16-22.

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Ghomanjani F., Farahi M.H., Kamyad AV., Numerical solution of some linear optimal control systems with pantograph delays. IMA J Math Control Inf, 2015, 32(2), 225-243.

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Ghazanfari B., Sepahvandzadeh A., Adomian decomposition method for solving fractional bratu-type equations, journal of mathematics and computer science, 2014, 8, 236-244. Google Scholar

• [8]

Babolian E., Javadi Sh., Moradi E., RKM for solving Bratu-type differential equations of fractional order, mathematical methods in the aplied sciences, 2015, 39(6), 1548–1557. Google Scholar

Accepted: 2017-10-31

Published Online: 2017-12-29

Conflict of Interest: The author declares that there is no conflict of interest in the use of the above mentioned software.

Citation Information: Open Physics, Volume 15, Issue 1, Pages 1045–1048, ISSN (Online) 2391-5471,

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