Next Article in Journal
The Emergence of Fuzzy Sets in the Decade of the Perceptron—Lotfi A. Zadeh’s and Frank Rosenblatt’s Research Work on Pattern Classification
Next Article in Special Issue
The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense
Previous Article in Journal
Near Fixed Point Theorems in the Space of Fuzzy Numbers
Previous Article in Special Issue
Global Behavior of Certain Nonautonomous Linearizable Three Term Difference Equations
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem

Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, UAE
*
Author to whom correspondence should be addressed.
Mathematics 2018, 6(7), 109; https://doi.org/10.3390/math6070109
Submission received: 31 May 2018 / Revised: 13 June 2018 / Accepted: 15 June 2018 / Published: 25 June 2018
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)

Abstract

:
In this article, a modified implicit hybrid method for solving the fractional Bagley-Torvik boundary (BTB) value problem is investigated. This approach is of a higher order. We study the convergence, zero stability, consistency, and region of absolute stability of the modified implicit hybrid method. Three of our numerical examples are presented.

1. Introduction

Several engineering and physical phenomena are modeled mathematically using fractional derivatives. Fractional derivatives have many applications, such as diffusion problems, liquid crystals, proteins, mechanics structural control, and biosystems [1,2,3,4,5,6,7,8,9]. Several analytical and numerical methods are used to solve fractional boundary and initial value problems, such as generalized differential transform, the Adomian decomposition method, the homotopy perturbation technique, fractional multistep methods, the spline approximation method, and the collocation method [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].
In this article, we consider the Bagley-Torvik boundary (BTB) value problem in the form:
y ( x ) = a y ( 3 2 ) ( x ) + b y ( x ) + r ( x ) = f ( x , y , y ( 3 2 ) ) ,   x [ 0 , X ]
subject to
y ( 0 ) = α ,   y ( X ) = β ,
where a ,   b ,   α , and β are constants, and y , f L 1 [ 0 , X ] .
BTB was discussed analytically in [27]. Then, several numerical approaches were used to solve it such as the discrete spline method [28], the Hybridizable discontinuous Galerkin method [29], generalizing the Taylor collocation method [30], and the operational matrix of Haar wavelet method [30]. Special attention was given when y ( 0 ) = y ( 1 ) = 0 . In addition, analytical solutions for such cases are investigated using the modified spectral method and the Adomian decomposition method [31].
We use a local fractional derivative, which is presented in [32,33]. This definition of fraction derivative works efficiently with the proposed method since it has several properties such as the product rule, power rule, and chain rule. These properties are given in the next section.
We modify an implicit hybrid method to solve Equations (1) and (2). We find an explicit formula to solve such a problem. We investigate some analytical properties of the proposed method such as consistency, stability, convergence, and order of convergence.
We organize our paper as follow. In Section 2, we mention some definitions and results which we use in this article. A modified fractional implicit hybrid method and its analytical properties are presented in Section 3. Three of our numerical examples are presented in Section 4. We compare our results with the results in [19,29]. Finally, we draw some conclusions in Section 5.

2. Preliminaries

First, we mention the definition of fractional derivative which we will use.
Definition 1.
Let u ( x ) C α ( 0 , X ) , the fractional derivatives of order 1 α > 0 at x 0 is defined as [33,34]:
D α u ( x ) = lim x x 0 Δ α ( u ( x ) u ( x 0 ) ( x x 0 ) α ,
where Δ α ( u ( x ) u ( x 0 ) Γ ( α + 1 ) Δ ( u ( x ) u ( x 0 ) ) .
The power rule of this local fractional derivative is given in the following theorem.
Theorem 1.
This fractional derivative satisfies the following power rule:
D α x p = { Γ ( p + 1 ) Γ ( p α + 1 ) x p α , p α 0 o t h e r w i s e } .
In addition, it is easy to see that:
  • D α c = 0 for constant c .
  • D α ( l ( m ) ( x ) ) = ( d m d x ) α D α l ( m ( x ) ) .
  • D k α l ( x ) = D α D α D α l ( x ) .
  • D α ( l ( x ) m ( x ) ) = l ( x ) D α m ( x ) + m ( x ) D α l ( x ) .
In this paper, a modified fractional implicit hybrid multistep method will be presented. To the best of our knowledge, no work has been done to discuss this problem using the implicit hybrid multistep methods.
Two major techniques are used to solve
w ( m ) = f ( x , w , w , , w ( m 1 ) ) , w ( a ) = w 0 , w ( a ) = w 1 , , w ( m 1 ) ( a ) = w m 1 ,
which are one step methods, such as Taylor and Runge-Kutta methods, and multistep methods such as Adams-Bashforth and Adams-Moulton methods. One-step methods are suitable only for the first order since they have a very low order of accuracy. If the higher order Runge-Kutta method is used, more function evaluations per step are required. Hence, solving Equation (5) using one-step methods requires the transformation of the problem into a system of first order differential equations which makes the dimension of the problem high and its scale also high. As a result, it will be time consuming for large scale problems with a low accuracy.
On the other hand, multistep step methods do not need to transform Equation (5) into a system of first order differential equations. These methods give higher order accuracy. However, they are not efficient in terms of function evaluations as are the one step methods and require more than one value to start the integration process.
In this paper, we look for a method that is a continuous implicit hybrid one step method. This method is as efficient as the one step methods and has as high an accuracy as the multistep methods. Next, we define the k -step hybrid formula. Let { x 0 , x 1 , , x N } be a uniform partition of [ a , b ] with x i = a + i h , i = 0 , 1 , , N , and h = b a N .
Definition 2.
A k -step hybrid formula is defined by:
i = 0 k a i y n + i + i = 0 l a n + v i y n + v i = h i = 0 k b i f n + i + h i = 0 l b n + v i f n + v i ,
where a k = 1 , a 0 and b 0 are nonzero, v { 0 , 1 , , k } , y n + i = y ( x n + i h ) and f n + v i   = f ( x n + v i , y n + v i ) . For more details, see in [34].
Definition 3.
Let:
[ y [ x n ] ; h ] = i = 0 k a i y n + i + i = 0 l a n + v i y n + v i = h i = 0 k b i f n + i + h i = 0 l b n + v i f n + v i = c 0 y n + c 1 y n +
If c 0 = 0 , c 1 = 0 , , c p + 1 = 0 , c p + 2 0 , then the order of the method is p and the error constant is c p + 2 .
Definition 4.
If the first and second characteristic polynomials are ρ ( z ) = i = 0 k α i   z i and σ ( z ) = i = 0 l β i   z i with:
  • α 0 2 + β 0 2 0 ,
  • The order is greater than or equal 1,
  • i = 0 k α i = 0 ,
  • ρ ( 1 ) = ρ ( 1 ) = 0 ,
  • ρ ( 1 ) = 2 σ ( 1 ) ,
then, it and its block method are called consistent.
Definition 5.
If no zeros of the first characteristic polynomial have a modulus greater than one and every root of modulus one has multiplicity not greater than one, then it is called zero stable.
Definition 6.
If the method is consistent and zero stable, it is convergent.

3. Method of Solution

To derive the modified fractional implicit hybrid method, we approximate the solution of Equation (1) by:
y ( x ) = k = 0 4 a x   x k ,
with second derivative given by:
y ( x ) = k = 2 4 k ( k 1 ) a x   x k 2 .
Let { x 0 ,   x 1 , x 2 , , x m } be a uniform partition of [ 0 , X ] with h = X m . Collocate Equation (7) at x n ,   x n + 1 2 ,   x n + 1 and interpolate Equation (6) at x n ,   x n + 1 2 to get:
( 1 x n x n 2 x n 3 x n 4 1 x n + 1 2 x n + 1 2 2 x n + 1 2 3 x n + 1 2 4 0 0 0 0 0 0 2 6 x n 12 x n 2 2 6 x n + 1 2 12 x n + 1 2 2 2 6 x n + 1 12 x n + 1 2 ) ( a 0 a 1 a 2 a 3 a 4 ) = ( y n y n + 1 2 f n f n + 1 2 f n + 1 ) ,
where:
f n = f ( x n ,   y n , y n ( 3 2 ) ) ,   f n + 1 2 = f ( x n + 1 2 ,   y n + 1 2 , y n ( 3 2 ) ) ,   f n + 1 = f ( x n + 1 ,   y n + 1 , y n + 1 ( 3 2 ) ) .
Let x n = x h t 3 2 . Then, x n + 1 2 = x h t 3 2 + h 2 and x n + 1 = x h t 3 2 + h . Solving Equation (8), we get:
y ( x ) = α 0 ( t ) y n + α 1 ( t ) y n + 1 2 + β 0 ( t ) f n + β 1 ( t ) f n + 1 2 + β 2 ( t ) f n + 1 ,
where:
α 0 ( t ) = 1 2 t 3 2 ,
α 1 ( t ) = 2 t 3 2 ,
β 0 ( t ) = h 2 ( 7 48 t 3 2 + 1 2 t 3 1 2 t 9 2 + t 6 6 ) ,
β 1 ( t ) = h 2 ( 1 8 t 3 2 + 2 3 t 9 2 t 6 3 ) ,
β 2 ( t ) = h 2 ( 1 48 t 3 2 1 6 t 9 2 + t 6 6 ) .
For x = x n + 1 , t = 1 and
y n + 1 = y n + 2 y n + 1 2 + h 2 48 ( f n + 1 + 10 f n + 1 2 + f n ) .
Using the Taylor series about x = x n for Equation (9), we get:
y n + 1 + y n 2 y n + 1 2 h 2 48 ( f n + 1 + 10 f n + 1 2 + f n ) = h 6 15360 y ( 6 ) ( x n ) h 7 30360 y ( 7 ) ( x n )
which means that the order of Equation (10) is 4 and the error constant is O ( 0.000065 h 6 ) .
The first and the second characteristic functions are given by:
ρ ( z ) = z 2 z 1 2 + 1 = ( z 1 ) 2 ,
and
σ ( z ) = 1 48 ( z + 10 z 1 2 + 1 ) .
Simple calculation implies that:
  • The roots of ρ for which | z | = 1 are simple.
  • Sum of coefficients of ρ is zero.
  • ρ ( 1 ) = 0 = ρ ( 1 ) .
  • ρ ( 1 ) = 2 ! σ   ( 1 ) = 1 2 .
This means that Equation (10) is consistent and zero stable which means that it is convergent. To find the region of absolute stability, let:
g ( z ) = ρ ( z ) σ ( z ) = 48 ( z 2 z 1 2 + 1 ) z + 10 z 1 2 + 1 .
Let z = e i θ , then:
g ( θ ) = 96   sin 2 ( θ 4 ) 5 + cos ( θ 2 ) .
Thus, the interval of absolute stability is (−9.6, 0) and the region of absolute stability is given in Figure 1.
Differentiate α i and β i to get:
D 3 2 α 0 ( t ) = 3 2 π ,
D 3 2 α 1 ( t ) = 3 2 π ,
D 3 2 β 0 ( t ) = h 2 ( 7 π 64 + 4 π t 3 2 315 π 128 t 3 + 256 63 π t 9 2 ) ,
D 3 2 β 1 ( t ) = h 2 ( 3 π 32 + 105 π 32 t 3 512 63 π t 9 2 ) ,
D 3 2 β 2 ( t ) = h 2 ( π 64 105 π 128 t 3 + 256 63 π t 9 2 ) .
Similarly
D t 3 2 y ( x ) = 27 8 π 3 4 h 3 2 D t 3 2 y ( t ) ,
Thus
D t 3 2 y ( t ) = 8 27 π 3 4 h 3 2 ( D t 3 2 α 0 ( t ) y n + D t 3 2 α 1 ( t ) y n + 1 2 + D t 3 2 β 0 ( t ) f n + D t 3 2 β 1 ( t ) f n + 1 2 + D t 3 2 β 2 ( t ) f n + 1 ) .
Then, at x n ,   x n + 1 2   ,   x n + 1 , t = 0 , 1 4 3 ,   1 , which imply that:
D 3 2 y n = 4 3 π 4 h 3 2 ( y n + y n + 1 2 ) 8 h π 4 27 ( 7 f n + 6 f n + 1 2 f n + 1 ) ,
D 3 2 y n + 1 2 = 4 3 π 4 h 3 2 ( y n + y n + 1 2 )   + 8 h 27 π 5 4 ( 90896 23373 π   32256   f n + 9192 + 5859 π   8064 f n + 1 2 + 16384 6611 π   326   f n + 1 ) ,
D 3 2 y n + 1 = 4 3 π 4 h 3 2 ( y n + y n + 1 2 )   + 8 h 27 π 5 4 ( 65024 20727 π   8064 f n 8192 3213 π   1008   f n + 1 2 + 32768 6489 π   8064 f n + 1 ) ,
From Equation (12), we get:
y n + 1 2 = y n   3 π 4 h 3 2   4 D 3 2 y n 2 h 2 3 π ( 7 f n + 6 f n + 1 2 f n + 1 ) .
Let D 3 2 y 0 = θ . Then, y 1 , , y m are functions of θ . Using the shooting method, we find the value of θ .

4. Numerical Results

In this section, we present three of our examples. Comparison with References [19] and [29] will be presented.
Example 1.
Consider the following problem:
D 2 y ( t ) + D 3 2 y ( t ) + y ( t ) = 2 + 4 t π + t 2 + α = g ( t ) ,
subject to
y ( 0 ) = α , y ( 5 ) = 25 + α 2 ,
where the exact solution is y ( t ) = t 2 + α 2 . Let h = 0.01 and x i = i h ,   i = 0 ,   1 ,   ,   500 . Let D 3 2 y ( 0 ) = θ . Using the modified fractional implicit hybrid method, we get the following system:
y n + 1 = y n + 2 y n + 1 2 10 4 48 ( D 3 2 ( y n + 1 + 10 y n + 1 2 + y n ) + y n + 1 + 10 y n + 1 2 + y n ) + 10 4 4 g ( t n ) ,
y n + 1 2 = y n   3 π 4 h 3 2   4 D 3 2 y n 2 10 4 3 π ( D 3 2 ( 7 y n + 1 + 6 y n + 1 2 y n ( t ) ) + 7 y n + 1 + 6 y n + 1 2 y n ) + 24 10 4 3 π g ( t n ) ,
D 3 2 y n + 1 2 = 4 3 π 4 h 3 2 ( y n + y n + 1 2 ) 0.8 27 π 5 4 ( 90896 23373 π   32256 ( D 3 2 y n + y n g ( t n ) ) + 9192 + 5859 π   8064 ( D 3 2 y n + 1 2 + y n + 1 2 g ( t n ) + 16384 6611 π   326 ( D 3 2 y n + 1 + y n + 1 g ( t n ) ) ) ) ,
D 3 2 y n + 1 = 4 3 π 4 h 3 2 ( y n + y n + 1 2 )   + 0.8 27 π 5 4 ( 65024 20727 π   8064 ( D 3 2 y n + y n g ( t n ) ) 8192 3213 π 1008 ( D 3 2 y n + 1 2 + y n + 1 2 g ( t n ) ) + 32768 6489 π   8064 ( D 3 2 y n + 1 + y n + 1 g ( t n ) ) ) ,
where y 500 = 25 + α 2   0.0000790783 θ . To find θ , we set:
y 500 = 25 + α 2
to get θ = 0 . The effect of α on the solution is given in Table 1 where:
e ( α ) = m a x { | y ( t n ) y n | , n = 0 ,   1 , ,   100 } .
We compare our results with the results in Reference [19].
From Table 1, we see that there is no significant effect for the initial condition. In addition, the proposed method gives more accurate results than Reference [19]. The approximate and exact solutions are given in Figure 2.
Example 2.
Consider the following problem:
D 2 y ( t ) + D 3 2 y ( t ) + y ( t ) = 1 + 8 π t 3 2 + 6 t + t 3 = g ( t )
subject to
y ( 0 ) = 1 , y ( 1 ) = 2
then, the exact solution is y ( t ) = t 3 + 1 . Let h = 0.01 and x i = i h ,   i = 0 , 1 , , 100 . Let D 3 2 y ( 0 ) = θ . Following the procedure described in Example 1, we find that θ = 1.2 × 10 14 .
The errors for Example 2 are given in Table 2. The approximate and exact solutions are given in Figure 3.
Example 3.
Consider the following problem:
D 2 y ( t ) + D 3 2 y ( t ) + y ( t ) = 8 π t 3 2 + t 3 + 7 t + 1 = g ( t ) ,
subject to
y ( 0 ) = 1 , y ( 1 ) = 3 .
then, the exact solution is y ( t ) = t 3 + t + 1 . Let h = 0.01 and x i = i h ,   i = 0 ,   1 , ,   300 . Let D 3 2 y ( 0 ) = θ . Following the procedure described in Example 1, we find that θ = 2.6 × 10 15 .
We compare our results with the results in Reference [29]. Let
e r r o r = max { | y ( t n ) y n | : n = 0 , 1 , , 300 } .
In Table 3, we present the comparison between our results and the results in Reference [29] for Example 3. The approximate and exact solutions are given in Figure 4.

5. Conclusions

In this paper, the modified fractional implicit hybrid method is presented for solving a class of fractional BTB. The proposed method is based on an implicit hybrid multistep method. We study the convergence, zero stability, consistency, and region of absolute stability. Three numerical examples are presented. We notice the following:
  • The modified implicit hybrid method is consistent, zero stable, and the order of convergence is 4.
  • The interval of convergence is (−9.6, 0) and the region of absolute stability is given in Figure 1.
  • The order of convergence is high without the need to refer to more initial conditions.
  • As seen in Table 1 and Table 3, the modified fractional implicit hybrid method gives more accurate results than other methods.
  • From Table 1, Table 2 and Table 3 and Figure 2, Figure 3 and Figure 4, we see that the approximate solutions are very accurate and very close to the exact solutions.
  • The modified fractional implicit hybrid method can be applied to more physical and engineering applications.

Author Contributions

All authors equally contributed in this paper and approved the final version.

Acknowledgments

The authors would like to express their sincere appreciation to United Arab Emirates University for the financial support of Grant No. G00002758.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of the paper.

References

  1. Caponetto, R.; Dongola, G.; Fortuna, L.; Gallo, A. New results on the synthesis of FO-PID controllers. Commun. Nonlinear Sci. Numer. Simul. 2010, 15, 997–1007. [Google Scholar] [CrossRef]
  2. Di Matteo, A.; Pirrotta, A. Generalized Differential Transform Method for Nonlinear Boundary Value Problem of Fractional Order. Commun. Nonlinear Sci. Numer. Simul. 2015, 29, 88–101. [Google Scholar] [CrossRef]
  3. Sabatier, J.; Agrawal, O.P.; Tenreiro Machado, J.A. Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering; Springer: Dordrecht, The Netherlands, 2007. [Google Scholar]
  4. Hilfer, R. Applications of Fractional Calculus to Physics; World Scientific: Singapore, 2000. [Google Scholar]
  5. Atanackovic, T.M.; Pilipovic, S.; Stankovic, B.; Zorica, D. Fractional Calculus with Applications in Mechanics; Wiley: London, UK, 2014. [Google Scholar]
  6. Di Matteo, A.; Di Paola, M.; Pirrotta, A. Innovative modeling of tuned liquid column damper controlled structures. Smart Struct. Syst. 2016, 18, 117–138. [Google Scholar] [CrossRef]
  7. Gemant, A. On fractional differentials. Philos. Mag. Ser. 1938, 25, 540–549. [Google Scholar] [CrossRef]
  8. Bosworth, R.C.L. A definition of plasticity. Nature 1946, 157, 447. [Google Scholar] [CrossRef] [PubMed]
  9. Nutting, P.G. A new general law deformation. J. Frankl. Inst. 1921, 191, 678–685. [Google Scholar] [CrossRef]
  10. Obaidat, Z.; Momani, S. A generalized differential transform for linear partial differential equations of fractional order. Appl. Math. Lett. 2008, 21, 194–199. [Google Scholar]
  11. Ghorbani, A.; Alavi, A. Applications of He’s variational iteration method to solve semidifferential equations of nth order. Math. Prob. Eng. 2008, 2008, 627983. [Google Scholar] [CrossRef]
  12. Blank, L. Numerical Treatment of Differential Equations of Fractional Order. 1996. CiteSeerX—Numerical Treatment of Differential Equations of Fractional Order. Available online: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.1 (accessed on 1/12/2017).
  13. Rawashdeh, E.A. Numerical solution of semidifferential equation by collocation method. Appl. Math. Comput. 2006, 174, 869–876. [Google Scholar] [CrossRef]
  14. Syam, M.I. Analytical solution of the Fractional Fredholm integro-differntial equation using the modified residual power series method. Complexity 2017, 2017, 4573589. [Google Scholar] [CrossRef]
  15. Kashkari, B.; Syam, M. Evolutionary computational intelligence in solving a class of nonlinear Volterra–Fredholm integro-differential equations. J. Comput. Appl. Math. 2017, 311, 314–323. [Google Scholar] [CrossRef]
  16. Syam, M.; Jaradat, H.M.; Alquran, M. A study on the two-mode coupled modified Korteweg—De Vries using the simplified bilinear and the trigonometric-function methods. Nonlinear Dyn. 2017, 90, 1363–1371. [Google Scholar] [CrossRef]
  17. Syam, M.I. A computational method for solving a class of non-linear singularly perturbed Volterra Integro-differential boundary-value problems. J. Math. Comput. Sci. 2013, 3, 73–86. [Google Scholar]
  18. Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus Models and Numerical Models (Series on Complexity, Nonlinearity and Chaos); Word Scientific: Singapore, 2012. [Google Scholar]
  19. Al-Mdallal, Q.; Syam, M.; Anwar, M. A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2010, 15, 3814–3822. [Google Scholar] [CrossRef]
  20. Syam, M.I.; Siyyam, H.I. Numerical differentiation of implicitly defined curves. J. Comput. Appl. Math. 1999, 108, 131–144. [Google Scholar] [CrossRef]
  21. Syam, M.I. The modified Broyden-variational method for solving nonlinear elliptic differential equations. Chaos Solitions Fractals 2007, 32, 392–404. [Google Scholar] [CrossRef]
  22. El-Sayed, M.F.; Syam, M.I. Numerical study for the electrified instability of viscoelastic cylindrical dielectric fluid film surrounded by a conducting gas. Physica A 2007, 377, 381–400. [Google Scholar] [CrossRef]
  23. Syam, M. Cubic spline interpolation predictors over implicitly defined curves. J. Comput. Appl. Math. 2003, 157, 283–295. [Google Scholar] [CrossRef]
  24. El-Sayed, M.F.; Syam, M.I. Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas. Arch. Appl. Mech. 2007, 77, 613–626. [Google Scholar] [CrossRef]
  25. Syam, M.I.; Siyyam, H.I. An efficient technique for finding the eigenvalues of fourth-order Sturm-Liouville problems. Chaos Solitons Fractals 2009, 39, 659–665. [Google Scholar] [CrossRef]
  26. Syam, M.I.; Attili, B.S. Numerical solution of singularly perturbed fifth order two point boundary value problem. Appl. Math. Comput. 2005, 170, 1085–1094. [Google Scholar] [CrossRef]
  27. Beyer, H.; Kempfle, S. Definition of physically consistent damping laws with fractional derivatives. Z. Angew. Math. Mech. 1995, 75, 623–635. [Google Scholar] [CrossRef]
  28. Zahra, W.K.; Van Daele, M. Discrete spline methods for solving two point fractional Bagley–Torvik equation. Appl. Math. Comput. 2017, 296, 42–56. [Google Scholar] [CrossRef]
  29. Karaaslan, M.F.; Celiker, F.; Kurulay, M. Approximate solution of the Bagley–Torvik equation by hybridizable discontinuous Galerkin methods. Appl. Math. Comput. 2016, 285, 51–58. [Google Scholar] [CrossRef] [Green Version]
  30. Çenesiz, Y.; Keskin, Y.; Kurnaz, A. The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Frankl. Inst. 2010, 347, 452–466. [Google Scholar] [CrossRef]
  31. Saha Ray, S. On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Appl. Math. Comput. 2012, 218, 5239–5248. [Google Scholar] [CrossRef]
  32. Yang, X.J.; Machado, J.T.; Cattani, C.; Gao, F. On a fractal LC-electric circuit modelled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2017, 47, 200–206. [Google Scholar] [CrossRef]
  33. Yang, X.J. Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4, 1–225. [Google Scholar]
  34. Lambert, J.D. Numerical Methods for Ordinary Differential Systems; John Wiley: New York, NY, USA, 1991. [Google Scholar]
Figure 1. Region of absolute stability.
Figure 1. Region of absolute stability.
Mathematics 06 00109 g001
Figure 2. The exact and the approximate solutions for Example 1 when α = 0 .
Figure 2. The exact and the approximate solutions for Example 1 when α = 0 .
Mathematics 06 00109 g002
Figure 3. The exact and the approximate solutions for Example 2.
Figure 3. The exact and the approximate solutions for Example 2.
Mathematics 06 00109 g003
Figure 4. The exact and the approximate solutions for Example 3.
Figure 4. The exact and the approximate solutions for Example 3.
Mathematics 06 00109 g004
Table 1. Comparison between our results and the results in Reference [20] for Example 1.
Table 1. Comparison between our results and the results in Reference [20] for Example 1.
α e ( α ) in [19] e ( α ) Using Our Method
0 1.71 × 10 10 1.12 × 10 16
0.001 2.10 × 10 10 2.21 × 10 16
0.01 3.84 × 10 10 2.27 × 10 16
0.1 7.14 × 10 10 3.01 × 10 16
0.5 2.92 × 10 10 3.02 × 10 16
1 8.73 × 10 10 3.07 × 10 16
Table 2. The errors for Example 2.
Table 2. The errors for Example 2.
t | y ( t n ) y n |
0 1.01 × 10 16
0.1 1.21 × 10 16
0.2 1.35 × 10 16
0.3 1.39 × 10 16
0.4 2.62 × 10 16
0.5 3.53 × 10 16
0.6 2.89 × 10 16
0.7 2.21 × 10 16
0.8 1.98 × 10 16
0.9 1.45 × 10 16
1.0 1.11 × 10 16
Table 3. Comparison between our results and the results in Reference [29] for Example 3.
Table 3. Comparison between our results and the results in Reference [29] for Example 3.
MethodError
Our method 2.89 × 10 16
Results in Reference [29] 1.98 × 10 13

Share and Cite

MDPI and ACS Style

Syam, M.I.; Alsuwaidi, A.; Alneyadi, A.; Al Refai, S.; Al Khaldi, S. An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem. Mathematics 2018, 6, 109. https://doi.org/10.3390/math6070109

AMA Style

Syam MI, Alsuwaidi A, Alneyadi A, Al Refai S, Al Khaldi S. An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem. Mathematics. 2018; 6(7):109. https://doi.org/10.3390/math6070109

Chicago/Turabian Style

Syam, Muhammed I., Azza Alsuwaidi, Asia Alneyadi, Safa Al Refai, and Sondos Al Khaldi. 2018. "An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem" Mathematics 6, no. 7: 109. https://doi.org/10.3390/math6070109

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop