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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.
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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
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Keywords
 wavelike equations
 heatlike equations
 differential transform method
 fractional calculus