• 500 Views
  • 255 Downloads
Research in Applied Mathematics: Vol. 1
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101268, 13 pages
doi:10.11131/2017/101268

Sumudu Decomposition Method for Solving Fractional Delay Differential Equations

Hassan Eltayeb1 and Eltayeb Abdeldaim2

1Mathematics Department College of Science, King Saud University

2Omdurman Islamic University College of Science and technology, Sudan

Received 21 November 2016; Accepted 19 April 2017

Editor: Maria Alessandra Ragusa

Copyright © 2017 Hassan Eltayeb and Eltayeb Abdeldaim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, The Sumudu transform decomposition method is applied to solve the linear and nonlinear fractional delay differential equations (DDEs). Numerical examples are presented to support our method.

1. Introduction

In the literature there are a kind of integral transforms used in physics and engineering, the integral transforms were extensively used to solve the differential equations, several works on the theory and application of integral transforms such as Laplace, Fourier, Mellin and Hankel.

Watugala [1] introduce a new integral transform named the Sumudu transform and applied it to solution of ordinary differential equation in control engineering problems for properties of Sumudu transform see [2], [3], [4] and [5]. In [18] Maria Ragusa proved a sufficient condition for commutators of fractional integral operators. The Sumudu transform is defined over the set of the functions:

A = { f ( t ) : M , τ 1 , τ 2 > 0 , f ( t ) < M e t τ j , if t ( 1 ) j × [ 0 , ) } .

By the following formula:

F ( u ) = S [ f ( t ) ] = 1 u 0 e t u f ( t ) d t , u ( τ 1 , τ 2 ) .

Delay differential equations arise when the rate of change of a time dependent process in its mathematical modeling is not only determined by its present state but also at certain past estate known as its history. Introduction of delays in models enriches the dynamics of such models and allow a precise description of real life phenomena. DDEs arise frequently in single processing, digital images, control system [8], lasers, traffic models [6], metal cutting, population dynamic [9], chemical kinetics [7], and in many physical phenomena.

Theorem 1

If Fn(u) is the Sumudu transform of n-th order derivative of fn(t) then for n≥1,

F n ( u ) = F ( u ) u n k = 0 n 1 f k ( 0 ) u n k .

For more details see [4].

Analysis of the Method

In this paper we will consider a class of nonlinear delay differential equation of the form:

d n y d t n + R ( y ) + N ( t τ ) = f ( t ) ,

with the initial condition:

u k ( 0 ) = u 0 k ,

where y = y(t), R is a linear bounded operator and f(t) is a given continuous function N is a nonlinear bounded operator and d n y d t n is the term of the highest order derivative.

The Sumudu decomposition method consists of applying the Sumudu transform first on both side of 1 to give:

S [ d n y d t n ] + S [ R ( y ) ] + S [ N ( t τ ) ] = S [ f ( t ) ] .

By Theorem 1, we have

S ( y ( t ) ) u n C u n k + S [ R ( y ) ] + S [ N ( t τ ) ] = S [ f ( t ) ] ,

where C = ∑k=0n−1fk(0),

S ( y ( t ) ) = u k C u n S [ R ( y ) ] u n S [ N ( t τ ) ] + u n S [ f ( t ) ] .

The standard Sumudu decomposition method defines the solution y(t) by the series:

y ( t ) = n = 0 y n ( t ) ,

the nonlinear operator is decomposed as:

N ( t τ ) = n = 0 A n ,

where An is the a domain polynomial of y0, y1, y2, ⋯, yn that are given by:

A n = 1 n ! d n d λ n [ N ( n = 0 λ n y n ] λ = 0 , n = 0 , 1 , 2 ,

The first a domain polynomials are given by:

A 0 = f ( y 0 ) A 1 = y 1 f 1 ( y 0 ) A 2 = y 2 f 1 ( y 0 ) + 1 2 ! y 1 2 f 2 ( y 0 ) A 3 = y 3 f 1 ( y 0 ) + y 1 y 2 f 2 ( y 0 ) + 1 3 ! y 1 3 f 3 ( y 0 ) .

Apply 4 and 5 into 3 we have:

S [ n = 0 y n ] = u k C u n S [ R n = 0 y n ] u n S [ n = 0 A n ] + u n S [ f ( t ) ] ,

comparing both side of 8:

S [ y 0 ] = u k C + u n S [ f ( t ) ]

S [ y 1 ] = u n S [ R y 0 ] u n S [ A 0 ]

S [ y 2 ] = u n S [ R y 1 ] u n S [ A 1 ] .

In general the recursive relation is given by:

S [ y n ] = u n S [ R y n 1 ] u n S [ A n 1 ] , n 1

applying inverse Sumudu transform to 9–12 then:

y 0 = H ( t )

y n = S 1 [ u n S [ R y n 1 ] + u n S [ A n 1 ] ] , n 1

Where H(t) is a function that a rises from the source term and prescribed initial conditions.

Numerical Examples

Example 1

Consider the nonlinear delay differential equation of first order:

y ' ( t ) = 1 2 y 2 ( t 2 ) , 0 t 1 , y ( 0 ) = 0 .

Apply Sumudu transform to both side of equation 15:

S [ y ' ( t ) ] = S [ 1 2 y 2 ( t 2 ) ] .

Using Theorem 1 and initial condition we have:

Y ( u ) y ( 0 ) u = 1 S [ 2 y 2 ( t 2 ) ] ,

Y ( u ) u = 1 S [ 2 y 2 ( t 2 ) ] ,

S [ y ( t ) ] = u u S [ 2 y 2 ( t 2 ) ] .

Applying the inverse Sumudu transform to 16 we have:

y ( t ) = S 1 [ u ] S 1 [ u S ( 2 y 2 ( t 2 ) ) ] ,

y 0 ( t ) = S 1 [ u ] = t ,

y 0 ( t 2 ) = t 2 ,

y n + 1 ( t ) = S 1 [ u S ( 2 A n ) ] .

From equation 7

A 0 = y 0 2 ( t 2 ) A 1 = 2 y 0 ( t 2 ) y 1 ( t 2 ) A 2 = 2 y 2 ( t 2 ) y 0 ( t 2 ) + y 1 2 ( t 2 ) ,

At n = 0 in equation 18:

y 1 ( t ) = S 1 [ u S ( 2 A 0 ) ] ,

substituting equation 19 in 20 we get:

y 1 ( t ) = S 1 [ u S ( 2 y 0 2 ( t 2 ) ) ] ,

y 1 ( t ) = S 1 [ u S ( 2 ( t 4 ) ) ] = S 1 [ u S ( t 2 2 ) ] ,

y 1 ( t ) = S 1 [ u ( u 2 ) ] = S 1 [ u 3 ] = t 3 3 ! ,

y 1 ( t 2 ) = ( t 2 ) 3 3 ! = t 3 48 .

At n = 1 in equation 18 we have:

y 2 ( t ) = S 1 [ u S ( 2 A 1 ) ] ,

substituting equations 19 into 22:

y 2 ( t ) = S 1 [ u S ( 2 ( 2 y 0 ( t 2 ) y 1 ( t 2 ) ) ] ,

y 2 ( t ) = S 1 [ u S ( 4 t 2 ( t 3 48 ) ) ] = S 1 [ u S ( t 4 24 ) ] ,

y 2 ( t ) = S 1 [ u S ( 4 ! u 4 24 ) ] = S 1 [ u 5 ] = t 5 5 ! = t 5 120 ,

y 2 ( t 2 ) = t 5 3840 ,

At n = 2 in equation 18 we have:

y 3 ( t ) = S 1 [ u S ( 2 A 2 ) ]

Substituting equations 19 and 24:

y 3 ( t ) = S 1 [ u S ( 2 ( 2 y 2 ( t 2 ) y 0 ( t 2 ) + y 1 2 ( t 2 ) ) ] ,

y 3 ( t ) = S 1 [ 2 u S ( 2 t 5 3840 ( t 2 ) + ( t 3 48 ) 2 ) ] ,

y 3 ( t ) = S 1 [ 2 u S ( t 6 3840 + t 6 2304 ) ] ,

y 3 ( t ) = S 1 [ 2 u ( u 6 . 6 ! 3840 + u 6 . 6 ! 2304 ) ] ,

y 3 ( t ) = S 1 [ u 7 ] = t 7 7 ! = t 7 5040 .

The series solution is given by:

y ( t ) = y 0 ( t ) + y 1 ( t ) + y 2 ( t ) + ,

y ( t ) = t t 3 6 + t 5 120 t 7 5040 + .

The exact solution is

y ( t ) = sin ( t )

Example 2

Consider the linear delay differential equation of first order

y ' ( t ) y ( t 2 ) = 0 , 0 < α 1 , 0 < t 1 ,

with initial condition y(0) = 1, apply Sumudu transform to both side of 25

S [ y ' ( t ) ] = S [ y ( t 2 ) ] ,

using Theorem 1 and initial condition:

Y ( u ) y ( 0 ) u = S [ y ( t 2 ) ] ,

Y ( u ) 1 u = S [ y ( t 2 ) ] ,

Y ( u ) = 1 + u S [ y ( t 2 ) ] ,

S [ y ( t ) ] = 1 + u S [ y ( t 2 ) ] .

Applying the inverse Sumudu transform to 26:

y ( t ) = S 1 [ 1 ] + S 1 [ u S [ y ( t 2 ) ] ] ,

y 0 ( t ) = S 1 [ 1 ] = 1 ,

y 0 ( t 2 ) = 1 ,

y n + 1 ( t ) = S 1 [ u S [ y n ( t 2 ) ] ] ,

at n = 0 in equation 28:

y 1 ( t ) = S 1 [ u S [ y 0 ( t 2 ) ] ] ,

y 1 ( t ) = S 1 [ u S [ 1 ] ] ,

y 1 ( t ) = S 1 [ u ] = t ,

y 1 ( t 2 ) = t 2 ,

at n = 1 in equation 28:

y 2 ( t ) = S 1 [ u S [ y 1 ( t 2 ) ] ] ,

y 2 ( t ) = S 1 [ u S [ t 2 ] ] ,

y 2 ( t ) = S 1 [ u 2 2 ] = t 2 4 ,

at n = 1 in equation 28:

y 3 ( t ) = S 1 [ u S [ y 2 ( t 2 ) ] ] = S 1 [ u S [ t 2 16 ] ] = S 1 [ u 3 8 ] = t 3 48 .

The series solution is given by:

y ( t ) = y 0 ( t ) + y 1 ( t ) + y 2 ( t ) + ,

y ( t ) = 1 + t + t 2 4 + t 3 48 +

The exact solution is y ( t ) = k = 0 ( 1 2 ) 1 2 k ( k - 1 ) k ! t k .

Fractional Delay Differential Equation

In this section we apply the Sumudu decomposition method to solve linear and nonlinear fractional delay differential equation.

Definition 1

The Sumudu transform of the Caputo fractional derivative is defined as follows:

S [ D α f ( t ) ] = u α S [ f ( t ) ] k = 0 n 1 u α + k f k ( 0 ) , n 1 < α n ,

for more details see [16].

2. Analysis of the Method of Fractional Order

Here we will consider a class of nonlinear delay differential equation of the form:

D α y ( t ) + R ( y ) + N ( t τ ) = f ( t ) , τ R , t < τ , n 1 < α n ,

with the initial condition:

u k ( 0 ) = u 0 k ,

where R is a linear bounded operator and f(t) is a given continuous function N is a nonlinear bounded operator and Dαy(t) is the term of the fractional order derivative.

The Sumudu decomposition method consists of applying the Sumudu transform first on both side of 30 to give:

S [ D α y ( t ) ] + S [ R ( y ) ] + S [ N ( t τ ) ] = S [ f ( t ) ] ,

by Definition 1,

S ( y ( t ) ) u α C u α k + S [ R ( y ) ] + S [ N ( t τ ) ] = S [ f ( t ) ] .

Where C = ∑k=0n−1fk(0)

S ( y ( t ) ) = u k C + u α S [ f ( t ) ] u α S [ R ( y ) ] u α S [ N ( t τ ) ] .

The standard Sumudu decomposition method define the solution y(t) by the series:

y ( t ) = n = 0 y n ( t ) ,

the nonlinear operator is decomposed as:

N ( t τ ) = n = 0 A n

Where An as in 6. The first a domain polynomials are given as in 7. Apply 33 and 34 in 32 we have:

S [ n = 0 y n ] = u k C + u α S [ f ( t ) ] u α S [ R n = 0 y n ] u α S [ n = 0 A n ]

Comparing both side of 35:

S [ y 0 ] = u k C + u α S [ f ( t ) ] ,

S [ y 1 ] = u α S [ R y 0 ] u α S [ A 0 ] ,

S [ y 2 ] = u α S [ R y 1 ] u α S [ A 1 ] .

In general the recursive relation is given by:

S [ y n ] = u α S [ R y n 1 ] u α S [ A n 1 ] , n 1 ,

applying inverse Sumudu transform to 36–39 then:

y 0 = H ( t ) ,

y n = S 1 [ u α S [ R y n 1 ] + u α S [ A n 1 ] ] , n 1 ,

where H(t) is a function that a rises from the source term and prescribed initial conditions.

Example 3

Consider the nonlinear delay differential equation of first order:

D α y ( t ) = 1 2 y 2 ( t 2 ) , 0 t 1 , 0 < α 1 ,

y ( 0 ) = 0 ,

apply Sumudu transform to both side of equation 42:

S [ D α y ( t ) ] = S [ 1 2 y 2 ( t 2 ) ] ,

by using Definition 1 and initial condition 43 we have:

Y ( u ) y ( 0 ) u α = 1 S [ 2 y 2 ( t 2 ) ] ,

Y ( u ) u α = 1 S [ 2 y 2 ( t 2 ) ] ,

S [ y ( t ) ] = u α u α S [ 2 y 2 ( t 2 ) ] .

Applying the inverse Sumudu transform to 44 we have:

y ( t ) = S 1 [ u α ] S 1 [ u α S ( 2 y 2 ( t 2 ) ) ] ,

y 0 ( t ) = S 1 [ u α ] = t α Γ ( α + 1 ) ,

y 0 ( t 2 ) = ( t 2 ) α Γ ( α + 1 ) = t α 2 α Γ ( α + 1 ) ,

y n + 1 ( t ) = S 1 [ u α S ( 2 A n ) ] ,

From equation 7, we have

A 0 = y 0 2 ( t 2 ) A 1 = 2 y 0 ( t 2 ) y 1 ( t 2 ) A 2 = 2 y 2 ( t 2 ) y 0 ( t 2 ) + y 1 2 ( t 2 ) ,

at n = 0 in equation 46:

y 1 ( t ) = S 1 [ u α S ( 2 A 0 ) ] ,

substituting equation 47 in 48 we get:

y 1 ( t ) = S 1 [ u α S ( 2 y 0 2 ( t 2 ) ) ] ,

y 1 ( t ) = S 1 [ u α S ( 2 ( t α 2 α Γ ( α + 1 ) ) 2 ) ] = S 1 [ u α S ( t 2 α 2 2 α 1 ( Γ ( α + 1 ) ) 2 ) ] ,

y 1 ( t ) = S 1 [ u α ( u 2 α Γ ( 2 α + 1 ) 2 2 α 1 ( Γ ( α + 1 ) ) 2 ) ] ,

y 1 ( t ) = S 1 [ u 3 α Γ ( 2 α + 1 ) 2 2 α 1 ( Γ ( α + 1 ) ) 2 ] ,

y 1 ( t ) = t 3 α Γ ( 2 α + 1 ) 2 2 α 1 ( Γ ( α + 1 ) ) 2 Γ ( 3 α + 1 ) ,

y 1 ( t ) = - A t 3 α Γ ( 3 α + 1 ) , where A = Γ ( 2 α + 1 ) 2 2 α - 1 ( Γ ( α + 1 ) ) 2 ,

y 1 ( t 2 ) = A ( t 2 ) 3 α Γ ( 3 α + 1 ) = A t 3 α 2 3 α Γ ( 3 α + 1 ) ,

at n = 1 in equation 46 we have:

y 2 ( t ) = S 1 [ u α S ( 2 A 1 ) ] ,

substituting equations 47 in 50:

y 2 ( t ) = S 1 [ u α S ( 2 ( 2 y 0 ( t 2 ) y 1 ( t 2 ) ) ] ,

y 2 ( t ) = S 1 [ u α S ( 4 y 0 ( t 2 ) y 1 ( t 2 ) ) ] ,

y 2 ( t ) = S 1 [ u α S ( 4 ( t α 2 α Γ ( α + 1 ) ) ( A t 3 α 2 3 α Γ ( 3 α + 1 ) ) ) ] ,

y 2 ( t ) = S 1 [ u α S ( A t 4 α 2 4 α 2 Γ ( 3 α + 1 ) ) ] ,

y 2 ( t ) = S 1 [ u α ( A u 4 α Γ ( 4 α + 1 ) 2 4 α 2 Γ ( 3 α + 1 ) ) ] ,

y 2 ( t ) = S 1 [ A u 5 α Γ ( 4 α + 1 ) 2 4 α 2 Γ ( 3 α + 1 ) ] ,

y 2 ( t ) = A t 5 α Γ ( 4 α + 1 ) 2 4 α 2 Γ ( 3 α + 1 ) Γ ( 5 α + 1 ) .

The series solution is given by:

y ( t ) = y 0 ( t ) + y 1 ( t ) + y 2 ( t ) +

y ( t ) = t α Γ ( α + 1 ) A t 3 α Γ ( 3 α + 1 ) + A t 5 α Γ ( 4 α + 1 ) 2 4 α 2 Γ ( 3 α + 1 ) Γ ( 5 α + 1 ) + .

In particular caseα = 1then we have:

y ( t ) = t Γ ( 2 ) t 3 Γ ( 4 ) + t 5 Γ ( 5 ) 2 2 Γ ( 4 ) Γ ( 6 ) + ,

y ( t ) = t t 3 6 + t 5 120 + .

The exact solution when α = 1 is given by y(t) = sin(t)

Example 4

Consider the nonlinear delay differential equation of first order

D α y ( t ) y ( t 2 ) = 0 , 0 < α 1 , 0 < t 1 ,

with initial condition y(0) = 1.

Apply Sumudu transform to both side of 51

S [ D α y ( t ) ] = S [ y ( t 2 ) ] ,

Using Definition 1 and initial condition:

Y ( u ) y ( 0 ) u α = S [ y ( t 2 ) ] ,

Y ( u ) 1 u α = S [ y ( t 2 ) ] ,

Y ( u ) = 1 + u α S [ y ( t 2 ) ] ,

S [ y ( t ) ] = 1 + u α S [ y ( t 2 ) ] .

Applying the inverse Sumudu transform to 52:

y ( t ) = S 1 [ 1 ] + S 1 [ u α S [ y ( t 2 ) ] ] ,

y 0 ( t ) = S 1 [ 1 ] = 1 ,

y 0 ( t 2 ) = 1 ,

y n + 1 ( t ) = S 1 [ u α S [ y n ( t 2 ) ] ] ,

at n = 0 in equation 54:

y 1 ( t ) = S 1 [ u α S [ y 0 ( t 2 ) ] ] ,

y 1 ( t ) = S 1 [ u α S [ 1 ] ] ,

y 1 ( t ) = S 1 [ u α ] = t α Γ ( α + 1 ) ,

y 1 ( t 2 ) = t α 2 α Γ ( α + 1 ) ,

at n = 1 in equation 54:

y 2 ( t ) = S 1 [ u α S [ y 1 ( t 2 ) ] ] ,

y 2 ( t ) = S 1 [ u α S [ t α 2 α Γ ( α + 1 ) ] ] ,

y 2 ( t ) = S 1 [ u 2 α Γ ( α + 1 ) 2 α ] = t 2 α Γ ( α + 1 ) 2 α Γ ( 2 α + 1 ) .

The series solution is given by:

y ( t ) = y 0 ( t ) + y 1 ( t ) + y 2 ( t ) + ,

y ( t ) = 1 + t α Γ ( α + 1 ) + t 2 α Γ ( α + 1 ) 2 α Γ ( 2 α + 1 ) + .

In particular caseα = 1then we have:

y ( t ) = 1 + t + t 2 4 +

The exact solution is given by y ( t ) = k = 0 ( 1 2 ) 1 2 k ( k - 1 ) k ! t k .

Conclusion

In this paper the Sumudu decomposition method has been successfully applied to solve delay and fractional delay differential equations. The method is very powerful and efficient in finding the exact solution.

Competing Interests

The authors declare that they have no competing interests.

References

  1. G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35–43, 1993. Publisher Full Text | Google Scholar
  2. M. A. Asiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002. Publisher Full Text | Google Scholar
  3. S. Tuluce Demiray, H. Bulut, and F. B. Belgacem, “Sumudu transform method for analytical solutions of fractional type ordinary differential equations,” Mathematical Problems in Engineering, Article ID 131690, 6 pages, 2015. Publisher Full Text | Google Scholar
  4. F. B. Belgacem and A. Karaballi, “Sumudu transform fundamental properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis. JAMSA, Article ID 91083, 23 pages, 2006. Publisher Full Text | Google Scholar
  5. H. Eltayeb, A. K\i l\i \cCman, and B. Fisher, “A new integral transform and associated distributions,” Integral Transforms and Special Functions. An International Journal, vol. 21, no. 5-6, pp. 367–379, 2010. Publisher Full Text | Google Scholar
  6. L. C. Davis, “Modifications of the optimal velocity traffic model to include delay due to driver reaction time,” Physica A: Statistical Mechanics and Its Applications, vol. 319, pp. 557–567, 2003. Publisher Full Text | Google Scholar
  7. I. R. Epstein and Y. Luo, “Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator,” The Journal of Chemical Physics, vol. 95, no. 1, pp. 244–254, 1991. Publisher Full Text | Google Scholar
  8. E. Fridman, L. Fridman, and E. Shustin, “Steady modes in relay control systems with time delay and periodic disturbances,” Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 122, no. 4, pp. 732–737, 2000. Publisher Full Text | Google Scholar
  9. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, NY, USA, 2011.
  10. Y. Luchko and R. Gorenflo, “The initial value problem for some fractional differential equations with the Caputo derivative,” preprint series A 08-98, Fachbreich Mathematik und informatik Freic Universitat Berlin, 1998.
  11. B. P. Moghaddam and Z. S. Mostaghim, “A numerical method based on finite difference for solving fractional delay differential equations,” Journal of Taibah University for Science, vol. 7, no. 3, pp. 120–127, 2013. Publisher Full Text | Google Scholar
  12. M. L. Morgado, N. J. Ford, and P. M. Lima, “Analysis and numerical methods for fractional differential equations with delay,” Journal of Computational and Applied Mathematics, vol. 252, pp. 159–168, 2013. Publisher Full Text | Google Scholar
  13. A. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, China; Springer, Berlin, Germany, 2009.
  14. Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” Nonlinear Studies, vol. 18, no. 1, pp. 99–112, 2011.
  15. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, The Netherlands, 1994.
  16. H. Eltayeb, A. K\i l\i \cCman, and S. Mesloub, “Application of Sumudu decomposition method to solve nonlinear system Volterra integrodifferential equations,” Abstract and Applied Analysis, 6 pages, 2014. Publisher Full Text | Google Scholar
  17. P. Goswami and R. T. Alqahtani, “Solutions of fractional differential equations by Sumudu transform and variational iteration method,” Journal of Nonlinear Science and its Applications. JNSA, vol. 9, no. 4, pp. 1944–1951, 2016.
  18. M. A. Ragusa, “Commutators of fractional integral operators on vanishing-Morrey spaces,” Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, vol. 40, no. 1-3, pp. 361–368, 2008. Publisher Full Text | Google Scholar
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101268, 13 pages
doi:10.11131/2017/101268

Sumudu Decomposition Method for Solving Fractional Delay Differential Equations

Hassan Eltayeb1 and Eltayeb Abdeldaim2

1Mathematics Department College of Science, King Saud University

2Omdurman Islamic University College of Science and technology, Sudan

Received 21 November 2016; Accepted 19 April 2017

Editor: Maria Alessandra Ragusa

Copyright © 2017 Hassan Eltayeb and Eltayeb Abdeldaim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, The Sumudu transform decomposition method is applied to solve the linear and nonlinear fractional delay differential equations (DDEs). Numerical examples are presented to support our method.