Research Article | Open Access
M. Asgari, R. Ezzati, T. Allahviranloo, "Numerical Solution of Time-Fractional Order Telegraph Equation by Bernstein Polynomials Operational Matrices", Mathematical Problems in Engineering, vol. 2016, Article ID 1683849, 6 pages, 2016. https://doi.org/10.1155/2016/1683849
Numerical Solution of Time-Fractional Order Telegraph Equation by Bernstein Polynomials Operational Matrices
We present a new method to solve time-fractional order telegraph equation (TFOTE) by using Bernstein polynomials. By implementation of Bernstein polynomials operational matrices of fractional differential on TFOTE, we reduce the original problem to a linear system of algebraic equations. Also, we prove the convergence analysis. In order to show the efficiency of the proposed method, we present two numerical examples.
Telegraph equations are hyperbolic partial differential equations that are applicable in modeling the reaction diffusion processes in various branches of engineering sciences and biological sciences. Those equations frequently arise in the study of wave propagation of electrical signal in a cable of transmission line and wave phenomena [1–4].
Many authors have used various numerical and analytical methods to solve the TFOTE. Chen and coworkers derived the analytical solution of the nonhomogeneous TFOTE by method of separation of variables . Huang presented a new analytical solution for three basic problems of time-fractional telegraph equation. He solved Cauchy and signaling problems by Laplace and Fourier transforms and the boundary problem by spatial Sine transform . Dehghan and Shokri developed a numerical method to solve the one-dimensional hyperbolic telegraph equation using the collocation points and approximated the solution by using thin plate spline radial basic functions . Saadatmandi and Dehghan developed a numerical solution based on Chebyshev tau method . Yousefi in  used Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation. In , Das and Gupta used homotopy analysis method for solving fractional hyperbolic partial differential equation. In , Mollahasani et al. applied hybrid functions of Legendre polynomials and block pulse functions to obtain the solution of telegraph equation of fractional order.
In this paper, our study focuses on the time-fractional telegraph equation of order :with the initial and boundary conditions: The right-hand-side function is given, , and also , , and are arbitrary positive constants. If , we have one-dimensional hyperbolic telegraph equation.
The rest of this paper is organized as follows: First, we present some preliminaries in fractional calculus. In Section 3, we briefly review some general concepts concerning Bernstein polynomials and the Bernstein polynomials operational matrix for fractional derivative. In Section 4, the method is applied to solve linear TFOTE. Section 5 exhibits an error estimation for the presented method. Section 6 illustrates two numerical examples to show the convergence and accuracy of the proposed method.
2. Basic Definitions
In this section, we present some basic definitions and properties of the fractional calculus which are going to be used in this paper.
Definition 1 (see ). The Riemann-Liouville fractional integral operator of order on the Lebesgue space is given by where denotes Euler Gamma function.
Definition 2 (see ). The Caputo fractional derivative of order is defined by where and is the smallest integer greater than . For the Caputo derivative, we have We use the ceiling function to denote the smallest integer greater than or equal to and . If , the Caputo differential operator coincides with the usual differential operator of an integer order.
For , , we have the following properties:
3. Bernstein Polynomials and Their Properties
3.1. The Definition of Bernstein Polynomials Basis
The Bernstein polynomials (BPs) of degree on the interval are defined by These polynomials have the following properties on :(1), , ,(2),(3), ,(4), .
Theorem 3 (see ). Suppose that is a Hilbert space with the inner product and is a finite dimensional and closed subspace; therefore, is a complete subspace of . So, if is an arbitrary element in , it has a unique best approximation out of such as ; that is, where and . So, there exist unique coefficients such that where and
Lemma 4 (see ). If is a complete basis, then , where is an upper triangular matrix with for and .
3.2. Function Approximation
A function can be expressed in terms of the Bernstein polynomials basis as where with
3.3. Operational Matrix for Fractional Derivative
Theorem 5. Let be vector defined in (10); then, where , , and are matrices that is defined in (11); is a diagonal matrix with is called the Bernstein polynomials operational matrix of fractional derivative .
4. Description of the Method
Clearly, can be approximated by using Bernstein polynomials as where and are vectors defined in (10) and the unknown is matrix that can be shown as Now, we can write Substituting (19) and (21) into (1), we have Now, we collocate (22) in Newton-Cotes nodes as So, we have equations as Applying (17) and (20) in the initial and boundary conditions (2), we getBy collocating (25) in Newton-Cotes nodes and for and , we get equations. These equations together with (24) give equations, which can be solved for , . So, the unknown function can be approximated.
5. Convergence Analysis
Theorem 6. Suppose that is a continuous function and all partial derivatives of exist and are continuous. Let , . If is the best approximation for out of and also all th-order partial derivatives of are bounded in magnitude by , then the error bound is presented as follows: where and .
Proof. By applying the Taylor expansion in two variables for , we have Hence, from Taylor expansion, we have Since is the best approximation for out of , we conclude that Clearly, we obtain the following result:
Remark 7. Equation (26) shows that if , then
6. Numerical Examples
To demonstrate the validity and applicability of the numerical scheme, we apply the present method for the following illustrative examples.
Example 1. Consider the time-fractional telegraph equation of order : The exact solution of this example is . In Table 1, we compare the obtained numerical results with the method of interpolating scaling functions operational matrix  (ISF), for . Also, in this table, we present the maximum absolute errors in for and various values of .
Example 2. Consider the time-fractional telegraph equation of order :Table 2 shows the absolute errors using the proposed method in the previous section for and different values of . The exact solution of this equation is .
This paper proposed a numerical approach for solving TFOTE by using the operational matrices of Bernstein polynomials. The operational matrix of fractional derivative, , as well as collocation method was used to transform the TFOTE to a linear system of algebraic equations. The numerical results show that the proposed method in this paper can be a suitable method for solving these equations. To obtain the numerical results, we applied Mathematica 9 software.
The authors declare that they have no competing interests.
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