versão On-line ISSN 1807-0302
Comput. Appl. Math. vol.30 no.3 São Carlos 2011
Differential transformation method for solving one-space-dimensional telegraph equation
Young Researchers Club, Sarab Branch, Islamic Azad University, Sarab, Iran. E-mail: firstname.lastname@example.org
In this research, the Differential Transformation Method (DTM) has been utilized to solve the hyperbolic Telegraph equation. This method can be used to obtain the exact solutions of this equation. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.
Mathematical subject classification: 35Lxx, 35Qxx.
Key words: differential transformation method, spectral method, telegraph equation, Taylor expansion.
In this paper we focus on the following one-space-dimensional Telegraphequation:
where α, β ∈ R, ψ : R × R → R are known and u : R × R → R is the unknown function. This equation with initial and boundary conditions is considered by some authors. Numerical solution of Telegraph equation with variable coefficient, was discussed in . Mohanty and his coworkers [17, 18], also developed new three-level alternative direction implicit schemes for the two and three-space-dimensional linear hyperbolic equations that these are unconditionally stable. Saadatmandi and Dehghan , applied a shifted Chebyshev Tau method. Gao and Chi  solved Eq. (1) by a semi-discretization method. They transformed Eq. (1) into a system consisting of ordinary differential equations with respect to time t and then found an exact solution containing an infinite matrix series. Authors of , presented two unconditionally stable numerical schemes based on C3 quartic splines with accuracy orders of O(k5 + h4) and O(k7 + h4).
The concept of DTM was first proposed by Zhou , who solved linear and nonlinear problems in electrical circuit problems. Chen and Ho  developed this method for partial differential equations. Ayaz  applied it to the system of differential equations. During recent years, this method has been used for solving various types of equations by many authors. For example, this method has been used for differential algebraic equations , partial differential equations [5, 6, 7], fractional differential equations [10, 11], Volterra integral equations  and Difference equations . Shahmorad et al. [19, 20] developed DTM to fractional-order integro-differential equations with nonlocal boundary conditions and a class of two-dimensional Volterra integral equations. Borhanifar and Abazari applied this method for Burgers and Schrödinger equations [4, 21, 23]. In , this method has been utilized for the Kuramoto-Sivashinsky equation with an initial condition. There exist similar problem. For example, authors of [12, 13, 14] presented several matrix formulation method for solving some equation with a boundary integral condition.
The aim of this paper is to extend the differential transformation method to solve the hyperbolic Telegraph equation. The method can be used to evaluate the approximating solution by the finite Taylor series and by an iteration procedure described by the transformed equations obtained from the original equation using the operations of differential transformation.
2 The definitions and operations of DT
2.1 The one-dimensional differential transform
The basic definitions and operations of one-dimensional DTM are introducedin [1, 2, 3] as follows:
Definition 2.1. If u(t) is analytic in the time domain T then
For t = ti, φ(t, k) = φ(ti, k), where k belong to the non-negative integer, denoted as the K domain. Therefore, Eq. (2) can be rewritten as
where Ui(k) is called the spectrum of u(t) at t = ti in the K domain.
Definition 2.2. If u(t) is analytic, then it can be shown as
Equation (4) is known as the inverse transformation of U(k). If U(k) isdefined as
then the function u(t) can be described as
where M(k) ≠ 0, q(t) ≠ 0. M(k) is called the weighting factor and q(t) is regarded as a kernel corresponding to u(t). If M(k) = 1 and q(t) = 1 then Eqs. (4) and (6) are equivalent. In this paper, the transformation with M(k) = 1/k! and q(t) = 1 is applied. Thus from Eq. (7), we have
Using the differential transform, a differential equation in the domain of interest can be transformed to be an algebraic equation in the K domain and u(t) can be obtained by finite-term Taylor series plus a remainder, as
In order to speed up the convergence rate and improve the accuracy of calculation, the entire domain of t needs to be split into sub-domains [9, 10].
2.2 The two-dimensional differential transform
Consider a function of two variables w(x, t): R × R → R, and supposethat it can be represented as a product of two single-variable functions, i.e., w(x, t) = u(x)v(t). Based on the properties of one-dimensional differential transform, function w(x, t) can be represented as
where W(i, j) is called the spectrum of w(x, t). Now we introduce the basic definitions and operations of two-dimensional DT as follows .
Definition 2.3. If w(x, t) is analytic and continuously differentiable with respect to time t in the domain of interest, then
where the spectrum function W(k, h) is the transformed function, which is also called the T-function. Let w(x, t) as the original function while the uppercase W(k, h) stands for the transformed function. Now we define the differential inverse transform of W(k, h) as following:
Using Eq. (10) in (11), we have
where x0 = 0 and t0 = 0.
Now from the above definitions and Eqs. (11) and (12), we can obtain someof the fundamental mathematical operations performed by two-dimensionaldifferential transform in Table 1.
3 Application of the DTM
In this section, we apply the DTM to solve the presented Telegraph equation. Consider the equation (1),
with the initial conditions
and boundary conditions
Let U(k, h) as the differential transform of u(x, t). Applying Table 1, Eq. (2) and Definition 2.3 when x0 = t0 = 0, we get the differential transform version of Eq. (13) as following:
By the first initial condition we get
and from the second initial condition and Table 1, we have
Similarly, from the boundary conditions, we have
Therefore, the values of U(k, 0), U(k, 1), and for U(0, h) can be obtained from Eqs. (20), (22) and (23). By using Eqs. (18) and (24), the remainder values of U can be found as follows:
Example 3.1. Consider the Eqs. (13)-(15) with the
Applying Eqs. (20), (22) and (23) in initial and boundary conditions of this problem, we get
From Eqs. (24) and (25), we have
By continuing this process, we obtain
which implies that
which is the Taylor expansion of the
u(x, t) = x × exp(t) 0 < x < L,
which is the exact solution of the Example 3.1. The computational results of Example 3.1 are presented in Table 3, and the plot of corresponding exact and approximate functions are shown in Figs. 1 and 2.
Example 3.2 () Consider the Eqs. (13)-(15) with the
From the above initial and boundary conditions and Eqs. (20), (22) and (23), we obtain the corresponding spectra as follows:
From Eqs. (24) and (25), we have
Thus we get
which implies that
which is the Taylor expansion of the
u(x, t) = exp(t) sin(x),
which is the exact solution of the Example 3.2. The computational results of Example 4.2 are presented in Table 5, and the plot of corresponding exact and approximate functions are shown in Figs. 3 and 4.
In this paper, we solved the Telegraph equation with initial conditions by theDifferential Transformation method. By using this method, Numerical/analytical results obtained by a simple iterative process. The numerical results prove that this method is a powerful techniques for this case of problems. Consequently, it is seen that this method can be an alternative way for the solution of partial differential equations that have no analytic solutions.
Acknowledgments. The author would like to thank the reviewers for carefully reading the paper and for their constructive comments and suggestions that have improved the paper. The author is deeply grateful to the Young Researchers Club, Sarab Branch, Islamic Azad University, for the financial supports.
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