 Research
 Open Access
 Published:
Approximate analytic solution of fractional heatlike and wavelike equations with variable coefficients using the differential transforms method
Advances in Difference Equations volume 2012, Article number: 198 (2012)
Abstract
This paper uses the differential transform method (DTM) to obtain analytical solutions of fractional heat and wavelike equations with variable coefficients. The time fractional heatlike and wavelike equations with variable coefficients were obtained by replacing a firstorder and a secondorder time derivative by a fractional derivative of order 0<\alpha <2. The approach mainly rests on the DTM which is one of the approximate methods. The method can easily be applied to many problems and is capable of reducing the size of computational work. Some examples are presented to show the efficiency and simplicity of the method.
1 Introduction
Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, have been used to model problems in fluid flow and other areas of application. Many phenomena in engineering physics, chemistry, and other sciences can be described very successfully by models using mathematical tools. Fractional derivatives provide an excellent instrument for the descriptive and hereditary properties of various materials and processes. In order to formulate certain electrochemical problems, halforder derivatives and integrals are more useful than the classical models [1]. Fractional differentiation and integration operators were also used for extensions of diffusion and wave equations [2].
Wazwaz and Gorguis [3] used the Adomian decomposition method for solving heatlike and wavelike models with variable coefficients. Momani [4] applied the method to the time fractional heatlike and wavelike equations with variable coefficients. The main disadvantage of the Adomian method is that the solution procedure for calculation of Adomian polynomials is complex and difficult as pointed out by many researchers [5–9]. Xu and Cang [10] solved the fractional heatlike and wavelike equations with variable coefficients using the homotopy analysis method (HAM). In 1998, the variational iteration method (VIM) was first proposed to solve fractional differential equations with great success [11]. Shou and He [12] used the VIM to solve various kinds of heatlike and wavelike equations, and it was claimed that by using the variational iteration method, demerits of complex calculation of Adomian polynomials were omitted [12]. Recently, Molliq, Noorani, and Hashim [13] applied the VIM to solve various kinds of fractional heatlike and wavelike equations. The differential transform method (DTM) was first applied to engineering problems [14].
The DTM is a numerical method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional highorder Taylor series method requires symbolic computation. However, the DTM obtains a polynomial series solution by means of an iterative procedure. Recently, the application of DTM is successfully extended to obtain analytical approximate solutions to ordinary differential equations of fractional order [15]. Application of fractional calculus in physics was presented by Hilfer [16]. A comparison between the VIM and the Adomian decomposition method for solving fractional differential equations is given by Odibat [17]. Recently, Kurulay [18] demonstrated the application of DTM for solving fmKdV. In this letter, we will apply the DTM [19] to fractional heatlike and wavelike equations to show the simplicity and straightforwardness of the method [20, 21].
In this paper, we will consider the fractional heatlike and wavelike equations of the form [4]
subject to boundary conditions
and the initial conditions
where α is a parameter describing the fractional derivative, {u}_{t} is the rate of change of temperature at a point over time. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of 0<\alpha \le 1, Eq. (1.1) reduces to the fractional heatlike equation with variable coefficients, and it does to a wavelike equation with variable coefficients for 1<\alpha \le 2.
2 Fractional calculus
We have wellknown definitions of a fractional derivative of order \alpha >0 such as RiemannLiouville, GrunwaldLetnikow, Caputo and generalized functions approach [1, 22]. The most commonly used definitions are the RiemannLiouville and Caputo. We give some basic definitions and properties of the fractional calculus theory which are used throughout the paper.
Definition 2.1 A real function f(x), x>0 is said to be in the space {C}_{\mu}, \mu \in R if there exists a real number (p>\mu) such that f(x)={x}^{p}{f}_{1}(x), where {f}_{1}(x)\in C[0,\mathrm{\infty}), and it is said to be in the space {C}_{\mu}^{m} iff {f}^{m}\in {C}_{\mu}, m\in N.
Definition 2.2 The RiemannLiouville fractional integral operator of order \alpha \ge 0 of a function f\in {C}_{\mu}, \mu \ge 1, is defined as
It has the following properties:
For f\in {C}_{\mu}, \mu \ge 1, \alpha ,\beta \ge 0, and \gamma >1:

1.
{J}^{\alpha}{J}^{\beta}f(x)={J}^{\alpha +\beta}f(x),

2.
{J}^{\alpha}{J}^{\beta}f(x)={J}^{\beta}{J}^{\alpha}f(x),

3.
{J}^{\alpha}{x}^{\gamma}=\frac{\mathrm{\Gamma}(\gamma +1)}{\mathrm{\Gamma}(\alpha +\gamma +1)}{x}^{\alpha +\gamma}.
The RiemannLiouville fractional derivative is mostly used by mathematicians, but this approach is not suitable for physical problems of the real world since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet. Caputo introduced an alternative definition, which has the advantage of defining integer order initial conditions for fractional order differential equations.
Definition 2.3 The fractional derivative of f(x) in the Caputo sense is defined as
for m1<v<m, m\in N, x>0, f\in {C}_{1}^{m}.
Lemma 2.1 If m1<\alpha <m, m\in N and f\in {C}_{\mu}^{m}, \mu \ge 1, then
The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem.
Definition 2.4 For m to be the smallest integer that exceeds α, the Caputo timefractional derivative operator of order \alpha >0 is defined as
3 Differential transform method
The DTM is applied to the solution of electric circuit problems. The DTM is a numerical method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional highorder Taylor series method requires symbolic computation. However, the DTM obtains a polynomial series solution by means of an iterative procedure. The method is well addressed in [19].
Consider a function of two variables u(x,y), and suppose that it can be represented as a product of two singlevariable functions, i.e., u(x,y)=f(x)g(y). Based on the properties of the generalized twodimensional differential transform [6, 23], the function u(x,y) can be represented as
where 0<\alpha, \beta \le 1, {U}_{\alpha \beta}(k,h)={F}_{\alpha}(k){G}_{\beta}(h) is called the spectrum of u(x,y). The generalized twodimensional differential transform of the function u(x,y) is given by
where {({D}_{{x}_{0}}^{\alpha})}^{k}={D}_{{x}_{0}}^{\alpha}{D}_{{x}_{0}}^{\alpha}\cdots {D}_{{x}_{0}}^{\alpha}, ktimes. In the case of \alpha =1 and \beta =1, the generalized twodimensional differential transform (3.1) reduces to the classical twodimensional differential transform [24].
The operators in twodimensional differential transformation method [24].
Let {U}_{\alpha ,\beta}(k,h), {V}_{\alpha ,\beta}(k,h) and {W}_{\alpha ,\beta}(k,h) be the differential transformations of the functions u(x,y), v(x,y) and w(x,y):

(a)
If u(x,y)=v(x,y)\pm w(x,y), then {U}_{\alpha ,\beta}(k,h)={V}_{\alpha ,\beta}(k,h)\pm {W}_{\alpha ,\beta}(k,h).

(b)
If u(x,y)=av(x,y), a\in R, then {U}_{\alpha ,\beta}(k,h)=a{V}_{\alpha ,\beta}(k,h).

(c)
If u(x,y)=v(x,y)w(x,y), then {U}_{\alpha ,\beta}(k,h)={\sum}_{r=0}^{k}{\sum}_{s=0}^{h}{V}_{\alpha ,\beta}(r,hs){W}_{\alpha ,\beta}(kr,s).

(d)
If u(x,y)={(x{x}_{0})}^{n\alpha}{(y{y}_{0})}^{m\beta}, then {U}_{\alpha ,\beta}(k,h)=\delta (kn)\delta (hm).

(e)
If u(x,y)=v(x,y)w(x,y)q(x,y), then
{U}_{\alpha ,\beta}(k,h)=\sum _{r=0}^{k}\sum _{t=0}^{kr}\sum _{t=0}^{h}{V}_{\alpha ,\beta}(r,hsp){W}_{\alpha ,\beta}(t,s){Q}_{\alpha ,\beta}(krt,p). 
(f)
If u(x,y)={D}_{{x}_{0}}^{\alpha}v(x,y), 0<\alpha \le 1, then {U}_{\alpha ,\beta}(k,h)=\frac{\mathrm{\Gamma}(\alpha (k+1)+1)}{\mathrm{\Gamma}(\alpha k+1)}{V}_{\alpha ,\beta}(k+1,h).

(g)
If u(x,y)=f(x)g(y) and the function f(x)={x}^{\lambda}h(x), where \lambda >1, h(x) has the generalized Taylor series expansion h(x)={\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{(x{x}_{0})}^{\alpha k}, and [24],

(i)
\beta <\lambda +1 and α arbitrary or

(ii)
\beta \ge \lambda +1, α arbitrary and {a}_{n}=0 for n=0,1,\dots ,m1, where m1<\beta \le m.
Then the generalized differential transform (3.2) becomes

(h)
If u(x,y)={D}_{{x}_{0}}^{\gamma}v(x,y), m1<\gamma \le m and v(x,y)=f(x)g(y), then
{U}_{\alpha ,\beta}(k,h)=\frac{\mathrm{\Gamma}(\alpha k+\gamma +1)}{\mathrm{\Gamma}(\alpha k+1)}{V}_{\alpha ,\beta}(k+\gamma /\alpha ,h). 
(i)
If u(x,y,t)={D}_{\ast {x}_{0}}^{\alpha}v(x,y,t), 0<\alpha \le 1, then
{U}_{\alpha ,\beta ,\gamma}(k,h,m)=\frac{\mathrm{\Gamma}(\alpha (k+1)+1)}{\mathrm{\Gamma}(\alpha k+1)}{V}_{\alpha ,\beta ,m}(k+1,h,m). 
(j)
If u(x,y)=a(x,y)\frac{{\partial}^{2}v(x,y)}{\partial {v}^{2}(x,y)}, then
U(k,h)=\sum _{i=0}^{k}\sum _{j=0}^{h}(ki+2)(ki+1)A(i,j)U(ki+2,hj).
The proofs of some properties can be found in [24].
4 Examples
Example 4.1 We consider the following onedimensional fractional heatlike problem:
subject to the boundary conditions
with the initial condition
The exact solution (\alpha =1) was found to be [4]
Taking the differential transform of Eq. (4.1), by using the property, we have
We start with the initial condition that was given by Eq. (4.2). By using the above formula (4.4), we can obtain the other components by using mathematical tools MAPLE package as follows:
So, the solution for the standard heatlike equation (\alpha =1) is given by [4]
Example 4.2 In this example, we consider the twodimensional fractional heatlike equation
subject to the boundary conditions
and the initial condition
The exact solution (\alpha =1) was found to be [4]
Taking the differential transform of Eq. (4.5), by using the related property, we have
We start with the initial condition that was given by Eq. (4.6). The solution for the fractional heatlike Eq. (4.5) in a series form is given by
For the special case (\alpha =1), we can reproduce the series solution of [4], and the solution in a closed form
Example 4.3 Consider the following threedimensional fractional heatlike equation:
subject to the boundary conditions
and the initial condition
The exact solution (\alpha =1) was found to be [4]
Taking the differential transform of Eq. (4.7), by using the related property, we have
We start with the initial condition. The solution for the fractional heatlike Eq. (4.7) in a series form is given by
For the special case (\alpha =1), we can reproduce the series solution of [4], and the solution in a closed form
We also conclude that our approximate solutions are in good agreement with the exact values. Both DTM and ADM have highly accurate solutions, but DTM has an easier way than ADM. We can solve the equation directly without calculating the Adomian polynomials.
Example 4.4 Next, we consider the onedimensional fractional wavelike equation
subject to the boundary conditions
and the initial conditions
Taking the differential transform of Eq. (4.8), by using the related property, we have
In the case of \alpha =2 in Eq. (4.8), we start with the initial conditions that were given by Eq. (4.9).
The solution for the fractional wavelike Eq. (4.8) in a series form is given by
and the exact [3] solution for this special case is
Example 4.5 We consider the threedimensional fractional wavelike equation
subject to the boundary conditions
and the initial conditions
The exact solution (\alpha =2) was found to be [3]
Taking the differential transform of Eq. (4.10), by using the related property, we have
In case of \alpha =2 in Eq. (4.10), we start with the initial conditions that were given by Eq. (4.11). The solution for the fractional wavelike Eq. (4.10) in a series form is given by
We can reproduce the series solution of [3], and the solution in a closed form
5 Conclusions
The application of the differential transform method (DTM) has been successfully employed to obtain the approximate solution of the fractional heatlike and wavelike equations with variable coefficients. The method was used in a direct way without using linearization, perturbation or restrictive assumptions. The procedure presented to solve the fractional heatlike and wavelike equations is the same as that for standard heatlike and wavelike equations, and in special cases of \alpha =1 and \alpha =2, the general solution reduces to the heatlike and wavelike solutions.
References
 1.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
 2.
Schneider WR, Wyss W: Fractional diffusion and wave equations. J. Math. Phys. 1989, 30: 134–144. 10.1063/1.528578
 3.
Wazwaz AM, Gorguis A: Exact solutions of heatlike and wavelike equations with variable coefficients. Appl. Math. Comput. 2004, 149(1):15–29. 10.1016/S00963003(02)009463
 4.
Momani S: Analytical approximate solution for fractional heatlike and wavelike equations with variable coefficients using the decomposition method. Appl. Math. Comput. 2005, 165(2):459–472. 10.1016/j.amc.2004.06.025
 5.
Odibat ZM, Momani S: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 2006, 7(1):27–34.
 6.
Bildik N, Konuralp A: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 2006, 7: 65–70.
 7.
Sweilam NH, Khader MM: Variational iteration method for one dimensional nonlinear thermoelasticity. Chaos Solitons Fractals 2007, 32: 145–149. 10.1016/j.chaos.2005.11.028
 8.
Momani S, Odibat Z: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 2007, 31(5):1248–1255. 10.1016/j.chaos.2005.10.068
 9.
Soliman AA: A numerical simulation and explicit solutions of KdVBurgers’ and Lax’s seventhorder KdV equations. Chaos Solitons Fractals 2006, 29: 294–302. 10.1016/j.chaos.2005.08.054
 10.
Xu H, Cang J: Analysis of a time fractional wavelike equation with the homotopy analysis method. Phys. Lett. A 2008, 372: 1250–1255. 10.1016/j.physleta.2007.09.039
 11.
He JH: Approximate analytical solution for seepage flow with fractional derivatives porous media. Comput. Methods Appl. Mech. Eng. 1998, 167: 57–68. 10.1016/S00457825(98)00108X
 12.
Shou DH, He JH: Beyond Adomian method: the variational iteration method for solving heatlike equations with variable coefficients. Phys. Lett. A 2008, 372: 233–237. 10.1016/j.physleta.2007.07.011
 13.
Molliq RY, Noorani MSM, Hashim I: Variational iteration method for fractional heat and wavelike equations. Nonlinear Anal., Real World Appl. 2009, 10(3):1854–1869. 10.1016/j.nonrwa.2008.02.026
 14.
Zhou JK: Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan; 1986.
 15.
Arikoglu A, Ozkol I: Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals 2007, 34: 1473–1481. 10.1016/j.chaos.2006.09.004
 16.
Hilfer R (Ed): Applications of Fractional Calculus in Physics. Academic Press, Orlando; 1999.
 17.
Odibat Z, Momani S: Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 2008, 32: 28–39. 10.1016/j.apm.2006.10.025
 18.
Kurulay M, Bayram M: Approximate analytical solution for the fractional modified KdV by differential transform method. Commun. Nonlinear Sci. Numer. Simul. 2010, 15(7):1777–1782. 10.1016/j.cnsns.2009.07.014
 19.
Odibat Z, Momani S, Erturk VS: Generalized differential transform method: for solving a space and timefractional diffusionwave equation. Phys. Lett. A 2007, 370: 379–387. 10.1016/j.physleta.2007.05.083
 20.
Hristov J: Heatbalance integral to fractional (halftime) heat diffusion submodel. Therm. Sci. 2010, 14(2):291–316. 10.2298/TSCI1002291H
 21.
Histov J: Starting radial subdiffusion from a central point through a diverging medium (a sphere): heatbalance integral method. Therm. Sci. 2011, 15(1):S5S20. 10.2298/TSCI11S1005H
 22.
Caputo M: Linear models of dissipation whose Q is almost frequency independent. Part II. Geophys. J. R. Astron. Soc. 1967, 13: 529–539. 10.1111/j.1365246X.1967.tb02303.x
 23.
AbdelHalim Hassan IH: Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos Solitons Fractals 2008, 36: 53–65. 10.1016/j.chaos.2006.06.040
 24.
Momani S, Odibat Z: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula. J. Comput. Appl. Math. 2008, 220: 85–95. 10.1016/j.cam.2007.07.033
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Secer, A. Approximate analytic solution of fractional heatlike and wavelike equations with variable coefficients using the differential transforms method. Adv Differ Equ 2012, 198 (2012). https://doi.org/10.1186/168718472012198
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718472012198
Keywords
 wavelike equations
 heatlike equations
 differential transform method
 fractional calculus