Startseite Numerical solution for fractional Bratu’s initial value problem
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Numerical solution for fractional Bratu’s initial value problem

  • Fateme Ghomanjani und Stanford Shateyi EMAIL logo
Veröffentlicht/Copyright: 29. Dezember 2017
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Open Physics
Aus der Zeitschrift Open Physics Band 15 Heft 1

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.

Keywords: Bezier curve; Fractional Bratu-type equation
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 D0+α 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, x0′ 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 = tf – t0, 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<t<1,x(0)=x0,x′(0)=x0′,(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<t<1,x(0)=0,x′(0)=0,xexact(t)=−2ln(cos(t)),forα=2.

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 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
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

tproposed methodapproximate solution of RKM
0.10.010315612031.3082 × 10−2
0.20.040269546185.1424 × 10−2
0.30.091213935951.1325 × 10−1
0.40.16434731631.9923 × 10−1
0.50.26116848083.1127 × 10−1
0.60.38393033884.5103 × 10−1
0.70.53609377246.2324 × 10−1
0.80.72278149388.3787 × 10−1
0.90.95123190221.1051
1.01.2312529411.4299

Table 2

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

terror of proposed methoderror of RKM
0.10.2989008300 × 10−31.6674 × 10−5
0.20.03.1 × 10−7
0.30.00016937595001.131 × 10−6
0.40.0001107219000212 × 10−6
0.50.02.9 × 10−6
0.60.04.1 × 10−6
0.70.7774260000 × 10−46.5 × 10−6
0.80.07.5 × 10−6
0.90.3470152000 × 10−33.35 × 10−5
1.00.04.37 × 10−5

Example 2

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

D0+αx(t)−e2x(t)=0,1<α≤2,0<t<1,x(0)=0,x′(0)=0,xexact(t)=ln(sec(t)),forα=2.

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 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
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

tproposed methodapproximate solution of RKM
0.10.6541208907 × 10−26.5411 × 10−3
0.20.2335095133 × 10−12.5712 × 10−2
0.30.048440451465.6625 × 10−2
0.40.082229018719.9615 × 10−2
0.50.12683406981.5563 × 10−1
0.60.18536115092.2552 × 10−1
0.70.26119395993.1162 × 10−1
0.80.35728436824.18893 × 10−1
0.90.47544244325.5255 × 10−1
1.00.61562647037.1494x × 10−1

Table 4

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

terror of proposed methoderror of RKM
0.10.1401819430 × 10−48.333 × 10−6
0.22.050000000 × 10−111.51 × 10−7
0.30.000026311483006.4 × 10−7
0.43.540000000 × 10−101.02 × 10−6
0.50.1081046000 × 10−31.35 × 10−6
0.60.2398590700 × 10−31.90 × 10−6
0.70.2507683000 × 10−33.2 × 10−6
0.81.000000000 × 10−103.5 × 10−6
0.90.3975147000 × 10−31.65 × 10−5
1.00.02.14 × 10−5

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.

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

Acknowledgement

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

References

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Received: 2017-9-12
Accepted: 2017-10-31
Published Online: 2017-12-29

© 2017 F. Ghomanjani and S. Shateyi

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  230. Modelling axial vibration in windings of power transformers
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  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
https://doi.org/10.1515/phys-2017-0131
Schlagwörter für diesen Artikel
Bezier curve; Fractional Bratu-type equation
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