International Journal of Systems Science and Applied Mathematics
Volume 1, Issue 4, November 2016, Pages: 63-68

A Coupling Method of Homotopy Perturbation and Aboodh Transform for Solving Nonlinear Fractional Heat - Like Equations

Mohand M. Abdelrahim Mahgoub1, 2

1Department of Mathematics, Faculty of Science & Technology, Omdurman Islamic University, Khartoum, Sudan

2Mathematics Department Faculty of Sciences and Arts, Almikwah-Albaha University, Albaha, Saudi Arabia

Mohand M. Abdelrahim Mahgoub. A Coupling Method of Homotopy Perturbation and Aboodh Transform for Solving Nonlinear Fractional Heat - Like Equations. International Journal of Systems Science and Applied Mathematics. Vol. 1, No. 4, 2016, pp. 63-68. doi: 10.11648/j.ijssam.20160104.15

Received: October 8, 2016; Accepted: October 18, 2016; Published: November 15, 2016

Abstract: In this paper, we present the solution of nonlinear fractional Heat - Like equations by using Aboodh transform homotopy perturbation method (ATHPM). The proposed method was derived by combining Aboodh transform and homotopy perturbation method. This method is seen as a better alternative method to some existing techniques for such realistic problems. The results showed the efficiency and accuracy of the combined Aboodh transform and homotopy perturbation method.

Keywords: Homotopy Decomposition Method, Nonlinear Fractional Heat - Like Equation, Aboodh Transform

1. Introduction

Nonlinear fractional partial differential equations (FPDEs) [1-3] are generalizations of classical differential equations of integer order. A great number of crucial phenomena in physics, chemistry, biology, biomedical sciences, signal processing, systems identification, control theory, viscoelastic materials and polymers are well described by fractional ordinary differential equations and nonlinear FPDEs. In the resent years many researchers mainly had paid attention to studying the solution of nonlinear fractional partial differential equations by using various methods. Among these are the Variational Iteration Method (VIM) [27-28], Adomian Decomposition Method (ADM) [16-17], projected differential transform method [25], and the Differential Transform Method (ADM) [26], are the most popular ones that are used to solve differential and integral equations of integer and fractional order. The Homotopy Perturbation Method (HPM) [4-6] is a universal approach which can be used to solve both fractional ordinary differential equations FODEs as well as fractional partial differential equations FPDEs. This method was originally proposed by He [7,8]. The HPM is a coupling of homotopy and the perturbation method. Recently, Khalid Aboodh, has introduced a new integral transform, named the Aboodh transform [18-24], and it has further applied to the solution of ordinary and partial differential equations. In this article, we use Aboodh transform and homotopy perturbation method together to solve Nonlinear Fractional Heat -Like Equations.

2. Fundamental Facts of the Fractional Calculus

In this section, some definitions and properties of the fractional calculus that will be used in this work are presented.

Definition 1:

The Gamma function is intrinsically tied in fractional calculus. The simplest interpretation of the gamma function is simply the generalization of the fraction for all real numbers. The definition of the gamma function is given by:

(1)

Definition 2:

A real function, is said to be in the space ,  if there exists a real number , such that , where  and it is said to be in space  if , .

Definition 3:

The Riemann-Liouville fractional integral operator of order , of a function , , is defined as

(2)

Some Properties of the operator:

For ,  and

,

Lemma 1:

If  and ,  then  and,

(3)

Definition 3: (Partial Derivatives of Fractional order)

Assume now that  is a function of  variables ,  also of class  on . As an extension of definition 2 we define partial derivative of order  for  respect to

(4)

If it exists, where  is the usual partial derivative of integer order .

3. Fundamental Facts of the Aboodh Transformation Method

A new transform called the Aboodh transform defined for function of exponential order we consider functions in the set A, defined by:

(5)

For a given function in the set  must be finite number,  may be finite or infinite. Aboodh transform which is defined by the integral equation

(6)

The following results can be obtained from the definition and simple calculations

1)

2)

3)

4)      .

Theorem 1:

If  is Aboodh transform of , we Knows that Aboodh transform of derivative with integral order is given as follows:

Proof:

Let us take the Aboodh transform, use integration by parts as follows:

(7)

Equation (7) gives us the proof of Theorem 1. When we continue in the same manner, we get the Aboodh transform of the second order derivative as follows

If we go on the same way, we get the Aboodh transform of the nth order derivative as follows:

for      (8)

or

(9)

Theorem 2:

If  is Aboodh transform of , one can take into consideration the Aboodh transform of the Riemann-Liouville derivative as follow:

(10)

Proof:

Therefore, we get the Aboodh transformation of fractional order of  as follows:

(11)

Definition 4:

The Aboodh transform of the Caputo fractional derivative by using Theorem 2 is defined as follows:

(12)

4. Basic Idea of Aboodh Transform Homotopy Perturbation Method (ATHPM)

To illustrate the basic idea of this method, we consider a general form of nonlinear non homogeneous partial differential equation as the follow:

(13)

with the following initial conditions

and

Where  denotes without loss of generality the Caputo fraction derivative operator,  is a known function,  is the general nonlinear fractional differential operator and  represents a linear fractional differential operator.

Taking Aboodh transform on both sides of equation (13), to get:

(14)

Using the differentiation property of Aboodh transform and above initial conditions, we have:

(15)

Operating with the Aboodh inverse on both sides of equation (15) gives:

(16)

Where  represents the term arising from the known function  and the initial condition.

Now, we apply the homotopy perturbation method

(17)

And the nonlinear term can be decomposed as:

(18)

Where  are He’s polynomial and given by:

(19)

Substituting equations. (18) and (17) in equation (16) we get:

(20)

Which is the coupling of the Aboodh transform and the homotopy perturbation method using He’s polynomials. Comparing the coefficient of like powers of , the following approximations are obtained:

(21)

Then the solution is;

(22)

The above series solution generally converges very rapidly.

5. Applications

Example 5.1:

Let consider the following one dimensional fractional heat- like equation:

(23)

with initial condition

(24)

Applying the Aboodh transform of both sides of Eq. (23),

(25)

Using the differential property of Aboodh transform Eq. (25) can be written as:

(26)

Using initial condition (24), Eq. (26) can be written as:

(27)

The inverse Aboodh transform implies that:

(28)

Now, we apply the homotopy perturbation method, we get:

(29)

Comparing the coefficient of like powers of , the following approximations are obtained;

,

,

Proceeding in a similar manner, we have:

,

,

Therefore the series solution  is given by:

(30)

This equivalent to the exact solution in closed form:

(31)

where  is the Mittag-Leffler function.

Example 5.2:

Consider the following tow - dimensional fractional heat like equation:

(32)

With the initial conditions

(33)

Applying the Aboodh transform of both sides of Eq. (32),

(34)

Using the differential property of Aboodh transform Eq. (34) can be written as:

(35)

Using initial condition (33), Eq. (35) can be written as:

(36)

The inverse Aboodh transform implies that:

(37)

Now, we apply the homotopy perturbation method, we get:

(38)

Comparing the coefficient of like powers of , the following approximations are obtained;

Proceeding in a similar manner, we have:

,

,

Therefore the series solution  is given by:

(39)

For the special case when , we can get the solution in a closed form

(40)

Example 5.3:

Consider the following three dimensional fractional heat-like equation:

(41)

With the initial condition;

(42)

Applying the Aboodh transform of both sides of Eq. (41),

(43)

Using the differential property of Aboodh transform Eq.(43), and using initial condition (42), Eq. (43) can be written as:

(44)

The inverse Aboodh transform implies that:

(45)

Now, we apply the homotopy perturbation method, we get:

(46)

Comparing the coefficient of like powers of , the following approximations are obtained;

,

Proceeding in a similar manner, we have:

,

,

Therefore the series solution  is given by:

(47)

Therefore the approximate solution of equation for the first  is given below as:

(48)

Now when  we obtained the follow solution

(49)

Where  is the generalized Mittag-Leffler function. Note that in the case

(50)

This is the exact solution for this case.

6. Conclusion

The main concern of this paper was to combine Aboodh transform and homotopy perturbation method (ATHPM). This method has been successfully employed to obtain an analytical solution for Nonlinear Fractional Heat -Like Equations. The results showed the efficiency and accuracy of the combined Aboodh transform and homotopy perturbation method.

References

1. K. B. Oldham and J. Spanier, "The Fractional Calculus", Academic Press, New York, NY, USA, (1974).
2. I.Podlubny."Fractional Differential Equations",Academic Press, NewYork, NY,USA, (1999).
3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations", Elsevier, Amsterdam, The Netherlands, (2006).
4. J. F. Cheng, Y. M. Chu, Solution to the linear fractional differential equation using Adomiandecomposition method, Mathematical Problems in Engineering, 2011, doi:10.1155 /2011/587068
5. J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinearproblems, International Journal of Non- Linear Mechanics, vol.35, 2000, pp. 37-43.
6. J. H. He, New interpretation of homotopy perturbation method, International Journal of Modern Physics B, vol.20, 2006b, pp. 2561- 2668
7. ] J.-H. He, "Homotopy perturbation technique," Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, (1999), pp.257–262.
8. J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, vol.167 (1-2), 1998, pp. 57– 68.
9. Esmail Hesameddini, Mohsen Riahi, Habibolla Latifizadeh,A coupling method of Homotopy technique and Laplace transform for nonlinear fractional differential equations,International Journal of Advances in Applied Sciences (IJAAS) Vol. 1, No. 4, December 2012, pp. 159~170.
10. Abdon Atangana and Adem Kihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis,Volume 2013
11. Rodrigue Batogna Gnitchogna, Abdon Atangana,Comparison of Homotopy Perturbation Sumudu Transform method and Homotopy Decomposition method for solving nonlinearFractional Partial Differential Equations,Advances in Applied and Pure Mathematics.
12. Abdolamir Karbalaie, Mohammad Mehdi Montazer, Hamed Hamid Muhammed, New Approach to Find the Exact Solution of Fractional Partial Differential Equation, WSEAS TRANSACTIONS on MATHEMATICS, Issue 10, Volume 11, October 2012.
13. M. Khalid, Mariam Sultana, Faheem Zaidi and Uroosa Arshad, Application of Elzaki Transform Method on Some Fractional Differential Equations, Mathematical Theory and Modeling, Vol.5, No.1, 2015.
14. Abdon Atangana and Adem Kihcman, The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations, Hindawi Publishing Corporation, Abstract and Applied Analysis,Volume 2013.
15. Eltayeb A. Yousif, Solution of Nonlinear Fractional Differential Equations Using the Homotopy Perturbation Sumudu Transform Method, Applied Mathematical Sciences, Vol. 8, 2014, no. 44, 2195 - 2210
16. G. Adomian, Solving frontier problems of physics: The decomposition method,Kluwer Academic Publishers, Boston and London, 1994.
17. J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, "A review of the Adomian decomposition method and its applications to fractional differential equations," Communications in FractionalCalculus, vol. 3, no. 2, (2012). pp. 73–99.
18. K. S. Aboodh, The New Integral Transform "Aboodh Transform" Global Journal of pure and Applied Mathematics, 9(1), 35-43(2013).
19. K. S. Aboodh, Application of New Transform "Aboodh transform" to Partial Differential Equations, Global Journal of pure and Applied Math, 10(2),249-254(2014).
20. Mohand M. Abdelrahim Mahgob " Homotopy Perturbation Method And Aboodh Transform For Solving Sine –Gorden And Klein – Gorden Equations" International Journal of Engineering Sciences & Research Technology,5(10): October, 2016
21. Mohand M. Abdelrahim Mahgob and Abdelilah K.Hassan Sedeeg "The Solution of Porous Medium Equation by Aboodh Homotopy Perturbation Method " American Journal of Applied Mathematics 2016; 4(5): 217-221.
22. Abdelilah K. Hassan Sedeeg and Mohand M. Abdelrahim Mahgoub, " Aboodh Transform Homotopy Perturbation Method For Solving System Of Nonlinear Partial Differential Equations," Mathematical Theory and Modeling Vol.6, No.8, 2016,
23. Abdelilah K. Hassan Sedeeg and Mohand M. Abdelrahim Mahgoub, "Combine Aboodh Transform And Homotopy Perturbation Method For Solving Linear And Nonlinear Schrodinger Equations," International Journal of Development Research Vol. 06, Issue, 08, pp. 9085-9089, August, 2016.
24. Abdelbagy A. Alshikhand Mohand M. Abdelrahim Mahgoub, "A Comparative Study Between Laplace Transform and Two New Integrals "ELzaki" Transform and "AboodhTransform,"Pure and Applied Mathematics Journal 2016; 5(5): 145-150.
25. T. M. Elzaki and E. M. A. Hilal, Solution of linear and nonlinear partial differential equations using mixture of Elzaki transform and the projected differential transform method, Math. Theo. & Model., 2 (2012), 50-59.
26. Sh. Chang, Il Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. Math. & Compu. 195 (2008), 799-808.
27. G. C. Wu, "New trends in the variational iteration method," Communications in Fractional Calculus, vol. 2, pp. 59–75, 2011.
28. G. C. Wu and D. Baleanu, "Variational iteration method for fractional calculus—a universal approach by Laplace transform," Advances in Difference Equations, vol. 2013, article 18, 2013.

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