A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method
Introduction
In this paper, we consider the following second-order linear two-space dimensional hyperbolic telegraph equationwhere R is the region in and is the time interval. are the constants. For , Eq. (1.1) represents a damped wave equation and for , it is called telegraph equation.
The initial conditions are given by,
The Dirichlet boundary conditions areor Neumann boundary conditions are given bywhere denotes the boundary of R.
The hyperbolic partial differential equations have significant role in formulating fundamental equations in atomic physics [25] and are also very useful in understanding various phenomena in applied sciences like engineering industry aerospace as well as in chemistry and biology too. On one hand vibrations of structures (e.g. buildings, machines and beams) can be easily analyzed and studied and on the other hand it is more convenient than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences [7].
On delving through the literatures, we found that much effort has been taken for the numerical solution of 1D and 2D hyperbolic telegraph equations. Various numerical schemes were developed for one dimensional telegraph equation such as Taylor matrix method [5], dual reciprocity boundary integral method [7], unconditionally stable finite difference scheme [15], modified B-spline collocation method [26], Chebyshev tau method [32], interpolating scaling function method [25] etc. Numerical solution of linear hyperbolic telegraph equation in three space dimension has been proposed by Mohanty et al. [29] by constructing an unconditionally stable alternating direction implicit scheme. In [27], Mohanty propounded a unconditionally stable implicit difference scheme for one, two and three space dimension telegraphic equations. Dehghan et al. [13], used He’s variational iteration method for solving linear, variable coefficient, fractional derivative and multi space telegraph equations.
In the recent past much emphasis have been given in the literature for numerical solution of two dimensional hyperbolic telegraph equation (1.1). Bülbül and Sezer [4], proposed Taylor matrix method that converts the telegraph equation into the matrix equation. Dehghan and Ghesmati [6] have explored two meshless methods namely meshless local weak-strong (MLWS) and meshless local Petrov–Galerkin (MLPG) method for Eqs. (1.1), (1.2), (1.3), (1.4). Also in [9], Eq. (1.1) is solved using higher order implicit collocation method. Numerical solution of 2D telegraph equation with variable coefficients has been tackled by Dehghan and Shorki [12]. Ding and Zhang [14] have discussed compact finite difference scheme which is of fourth order in both space and time. Mohanty and Jain [28] derived an unconditionally stable alternating direction implicit scheme for Eq. (1.1). A combination of boundary knot method (BKM) and analog equation method (AEM) for Eq. (1.1) has been proposed by Dehghan and Salehi [11]. A differential quadrature method, which approximates the solution of the problem on a finite dimensional space by using polynomials as the basis of the space is applied to the two dimensional telegraph equation with both Dirichlet and Neumann boundary conditions by Jiwari et al. [16].
Differential quadrature method (DQM) is a numerical discretization technique for the approximation of derivatives and has been successfully applied to solve various problem in biosciences, fluid dynamics, chemical engineering, etc [33]. DQM was introduced for the first time by Bellman et al. [3], in the framework to approximate the solution of differential equations. It follows that the partial derivatives of a function with respect to coordinate variable can be approximated by weighted linear combination of functional values at all grid points in the whole computational domain. The key of the DQM is to determine the weighting coefficient in the weighting sum by using some test functions. Various test functions have been used in the literature to calculate these weighting coefficients such as Legendre polynomials and spline functions by Bellman et al. [2], [3] respectively. To improve Bellman’s approach, Quan and Chang [31], [30], have proposed an explicit formulation using Lagrange interpolation polynomial as the test functions. Shu and Xue [35] used the Lagrange interpolated trigonometric functions. In [34], Shu and Wu presented an implicit approach using radial basis functions as the test functions. After that Shu [33] proposed a general approach to determine the weighting coefficients for first order derivatives and higher order derivatives. A number of differential quadrature methods have been developed in the literature to approximate the solution of differential equations such as quartic B-spline differential quadrature method [19], sinc differential quadrature method [18], modified cubic B-spline differential quadrature method [1], local radial basis functions based differential quadrature method [10], polynomial based differential quadrature method [17], harmonic differential quadrature method [37] etc.
In this paper we present a modified cubic B-spline differential quadrature method (MCB-DQM) to solve two dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. B-splines are extensively used in the literature to develop various numerical methods, for detail see [1], [8], [20], [21], [22], [23], [24], [26]. We use modified cubic B-spline basis functions [26], to compute the weighting coefficients. First the telegraph equation is converted into system of partial differential equations and then equations are discretized spatially by MCB-DQM. The obtained systems of ODEs in time are solved using SSP-RK43 [36] scheme and consequently the approximate solution is computed.
The outline of the paper is as follows. In Section 2, modified cubic B-spline differential quadrature method is introduced. In Section 3, numerical scheme using MCB-DQM for telegraph equation is presented. In Section 4, we discuss the stability of scheme. In Section 5, computational results for some test problems are illustrated and compared with some previous results and finally conclusions are included in Section 6.
Section snippets
Modified cubic B-spline differential quadrature method
To represent the mathematical formulation of two dimensional (DQM), first the region is discretized by taking N and M grid points in x and y direction respectively, such that and . Then the nth order partial derivatives of the function with respect to x at a point , at any line that is parallel to the x axis can be approximated as follows:
The mth order partial derivatives of the
Numerical solution of two dimensional hyperbolic telegraph equation
The telegraph equation (1.1) is converted into the coupled system using the following transformationThen transformed form of Eq. (1.1) is
Now the first and second order spatial derivatives of u are discretized by modified B-spline differential quadrature method, which reduces the system of Eq. (3.1) into the following system of first order ordinary differential equations,
Stability analysis
We consider the system (3.2) and rewrite it in compact form as:orwhere,
O = null matrix,
I = identity matrix of order ,
B is a square matrix of weighting coefficients of order ,
is solution vector at the interior grid points, given by.
.
is a vector containing nonhomogeneous part and boundary conditions, where is null
Numerical results
In this section Eqs. (1.1), (1.2), (1.3), (1.4) is solved numerically for different values of and at different time levels. The effectiveness of the approach is demonstrated by performing experiments on the problems available in the literature. To assess the performance of the method we compute norms and relative error. Example 1 We consider (1.1) in the region with and with the following initial conditions
The Dirichlet
Conclusion
In this paper a differential quadrature method based on modified cubic B-spline basis functions has been developed to solve two dimensional hyperbolic telegraph equation. The telegraph equation is first converted into a system of partial differential equations and then using MCB-DQM we get a system of first order ordinary differential equations. Finally we use SSP-RK43 scheme to solve this system. The SSP-RK43 scheme needs low storage space that causes less accumulation of numerical errors. The
Acknowledgments
The authors would like to thank the anonymous referees for their time, efforts and extensive comments on the revision of the manuscript.
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