ISRN Computational Mathematics

VolumeΒ 2012Β (2012), Article IDΒ 302923, 6 pages

http://dx.doi.org/10.5402/2012/302923

## Cubic Spline Method for 1D Wave Equation in Polar Coordinates

^{1}Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India^{2}Department of Mathematics, Deenbandhu Chhotu Ram University of Science & Technology, Murthal 131039, India

Received 10 September 2011; Accepted 28 September 2011

Academic Editors: P. B.Β Vasconcelos and J. G.Β Zhou

Copyright Β© 2012 R. K. Mohanty et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of for the numerical solution of 1D wave equations in polar coordinates, where and are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

#### 1. Introduction

We consider the 1D linear singular hyperbolic equation of the form subject to the initial conditions and with boundary conditions at of the form where and are time and distance variable, respectively. For and 2, the equation above represent, 1D wave equation in cylindrical and spherical polar coordinates, respectively. We shall assume that the initial and boundary conditions are given with sufficient smoothness to maintain the order of accuracy of the difference scheme and spline functions under consideration.

In this paper, we are interested to discuss a new approximation based on cubic spline polynomial for the solution to singular hyperbolic equation (1). During last three decades, several numerical schemes for the solution of two-point boundary value problems and partial differential equations have been developed by many researchers. First Bickley [1] and Fyfe [2] have discussed the second-order accurate spline method for the solution of linear two-point boundary value problems. Jain and Aziz [3] have derived fourth-order cubic spline method for solving the nonlinear two-point boundary value problems with significant first derivative terms. Further, Khan and Aziz [4] have studied parametric cubic spline method to the solution of a system of second-order boundary value problem. In 1974, Raggett and Wilson [5] have used a cubic spline technique of lower order accuracy to solve the wave equation. Later, Fleck Jr [6] has proposed a cubic spline method for solving the wave equation of linear optics. In recent years, Mohanty [7], Gao and Chi [8], Rashidinia et al. [9, 10], Mohebbi and Dehghan [11], Ding and Zhang [12, 13], and Liu and Liu [14] have derived various numerical methods for solution of nonsingular 2D hyperbolic equations. A difference method of accuracy two in time and four in space for the solution of differential equation (1) has been studied by Mohanty [15] using finite difference method. A new discretization method of order four for the numerical solution of one-space dimensional second-order nonlinear hyperbolic equations has been studied by Mohanty et al. [16, 17]. Recently, Mohanty and Dahiya [18] have derived highly accurate cubic spline method for the solution of parabolic equations. In this paper, using nine grid points, we discuss a new three-level implicit cubic spline finite difference method of accuracy two in time and four in space for the solution of differential equation (1). To the authorsβ knowledge, no cubic spline method of accuracy two in time and four in space for the solution of (1) has been discussed in the literature so far. In next section, we discuss the derivation of the proposed cubic spline method. It has been experienced in the past that the cubic spline solutions for the wave equation in polar coordinates usually deteriorate in the vicinity of the singularity. We overcome this difficulty by modifying the method in such a way that the solutions retain its order and accuracy everywhere in the vicinity of the singularity. In Section 3, we discuss the linear stability analysis of the proposed cubic spline method. In Section 4, we compare the computed results with the one obtained by the method discussed in [17]. Final remarks are given in Section 5.

#### 2. The Method Based on Cubic Spline Approximation

The solution domain is divided into mesh with the spatial step size in -direction and the time step size in -direction, respectively, where and are positive integers. The mesh ratio parameter is given by . Grid points are defined by . The notations are used for the discrete approximation and the exact value of at the grid point , respectively.

At the grid point , we denote

Let be the cubic spline interpolating polynomial of the function between the grid points and and be given by which satisfies at th level the following properties:(i) coincides with a polynomial of degree three on each , , ,(ii), and(iii), , .

The derivatives of cubic spline function are given by where and replacing by β,β we get Combining (8) and (9), we obtain Further, from (8), we have and, from (9), we have We consider the following approximations: Since the derivative values of defined by (7), (10), (11), and (12) are not known at each grid point , we use the following approximations for the derivatives of .

Let where and are approximations to and , respectively.

We may rewrite the differential equation (1) as

Let us denote .

A fourth-order method (see [3]) based on cubic spline approximations for the differential equation (15) may be written as where

Multiplying throughout by 6 and by the help of the approximations (17), from (16), we obtain a cubic spline finite difference method with accuracy of for the solution of differential equation (1) as

Note that the method (18) is of for the numerical solution of (1). However, the method fails to compute at . We modify the method in such a manner that it retains its order and accuracy in the vicinity of the singularity.

We use the following approximations: where and . Substituting the approximations (19) into (18) and neglecting higher order terms, we get

Note that the cubic spline method (20) is of for the numerical solution of differential equation (1), which is free from the terms hence can be computed for .

#### 3. Stability Analysis

Now we discuss the stability analysis for the scheme (20). For stability, we consider the homogeneous part of the scheme (20), which can be written as where

It is difficult to find the stability region for the scheme (21). In order to obtain a valid stability region, we may modify the scheme (21) (see [19]) as

The additional terms added in (23) are of high order and do not affect the accuracy of the scheme.

Put (where such that ) in the modified scheme (23), we getwhere is a nonzero real parameter to be determined. Left-hand side of (24) is a real quantity. Thus, the imaginary part of right-hand side of (24) must be zero.

Thus, we obtain provided.

Hence, .

Substituting the values of in (24) and (25), we obtain

Since , the method (23) is stable as long as , which is true if .

Hence, the required stability condition is which is the required stability interval for the scheme (23).

#### 4. Numerical Results

A difference method of , for the differential equation (1) may be written as

Note that the proposed cubic spline method (23) and the difference method (28) for second-order hyperbolic equation (1) are three-level schemes. The value of *u* at is known from the initial condition. To start any computation, it is necessary to know the numerical value of *u* of required accuracy at . In this section, we discuss an explicit scheme of for *u* at first time level, that is, at , in order to solve the differential equation (1) using the proposed method (23) and second-order method (28).

Since the values of and are known explicitly at , this implies all their successive tangential derivatives are known at , that is, the values of , and so forth, are known at .

An approximation for of at may be written as

From (1), we have

Thus, using the initial values and their successive tangential derivative values, from (30), we can obtain the value of , and, then ultimately, from (29), we can compute the value of at first time level, that is, at .

We solve the differential equation (1) in the region , whose exact solution is given by . The maximum absolute errors at are tabulated in Table 1 for various values of and in Table 2 for . We have compared the numerical results of the proposed method with the results obtained by using the method discussed in [17] in terms of accuracy. All computations were performed using double precision arithmetic.

A relation between the exact solution and the approximate numerical solution as given in the following equation: where is the measure of the mesh discretization, is a constant, and is the order (rate) of convergence. If the meshes to be considered are sufficiently refined, the higher-order terms can be neglected. Then, the maximum absolute errors can be approximated as

Taking the logarithm of both sides of (32), we obtain

For two different refined mesh spacing and , we have the following two relations

Subtracting (34b) from (34a), we obtain the order (rate) of convergence where and are maximum absolute errors for two uniform mesh widths and , respectively. For computation of order of convergence of the proposed method, we have considered and in Table 2, and we found the order of convergence of the proposed method for is 4.08 and for is 4.01.

#### 5. Final Remarks

Available numerical methods based on spline approximations for the numerical solution of 1D wave equation in polar coordinates are of accurate, which require nine grid points. In this paper, using the same number of grid points, we have discussed a new stable three-level implicit cubic spline finite difference method of accuracy for the solution of wave equation in polar coordinates. For a fixed parameter , the proposed method behaves like a fourth-order method, which is exhibited from the computed results.

#### Acknowledgment

The authors thank the anonymous reviewers for their constructive suggestions, which substantially improved the standard of the paper.

#### References

- W. G. Bickley, βPiecewise cubic interpolation and two-point boundary value problems,β
*The Computer Journal*, vol. 11, pp. 206β208, 1968. View at Google Scholar - D. J. Fyfe, βThe use of cubic splines in the solution of two-point boundary value problems,β
*The Computer Journal*, vol. 12, pp. 188β192, 1969. View at Publisher Β· View at Google Scholar - M. K. Jain and T. Aziz, βCubic spline solution of two-point boundary value problems with significant first derivatives,β
*Computer Methods in Applied Mechanics and Engineering*, vol. 39, no. 1, pp. 83β91, 1983. View at Publisher Β· View at Google Scholar - A. Khan and T. Aziz, βParametric cubic spline approach to the solution of a system of second-order boundary-value problems,β
*Journal of Optimization Theory and Applications*, vol. 118, no. 1, pp. 45β54, 2003. View at Publisher Β· View at Google Scholar - G. F. Raggett and P. D. Wilson, βA fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique,β
*Journal of the Institute of Mathematics and its Applications*, vol. 14, pp. 75β77, 1974. View at Publisher Β· View at Google Scholar - J. A. Fleck Jr., βA cubic spline method for solving the wave equation of nonlinear optics,β
*Journal of Computational Physics*, vol. 16, pp. 324β341, 1974. View at Publisher Β· View at Google Scholar - R. K. Mohanty, βAn unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation,β
*Applied Mathematics Letters*, vol. 17, no. 1, pp. 101β105, 2004. View at Publisher Β· View at Google Scholar - F. Gao and C. Chi, βUnconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation,β
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 1272β1276, 2007. View at Publisher Β· View at Google Scholar - J. Rashidinia, R. Jalilian, and V. Kazemi, βSpline methods for the solutions of hyperbolic equations,β
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 882β886, 2007. View at Publisher Β· View at Google Scholar - J. Rashidinia, R. Mohammadi, and R. Jalilian, βSpline methods for the solution of hyperbolic equation with variable coefficients,β
*Numerical Methods for Partial Differential Equations*, vol. 23, no. 6, pp. 1411β1419, 2007. View at Publisher Β· View at Google Scholar - A. Mohebbi and M. Dehghan, βHigh order compact solution of the one-space-dimensional linear hyperbolic equation,β
*Numerical Methods for Partial Differential Equations*, vol. 24, no. 5, pp. 1222β1235, 2008. View at Publisher Β· View at Google Scholar - H. Ding and Y. Zhang, βParameters spline methods for the solution of hyperbolic equations,β
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 938β941, 2008. View at Publisher Β· View at Google Scholar - H. Ding and Y. Zhang, βA new unconditionally stable compact difference scheme of $O({\mathrm{{\rm O}\x84}}^{2}+{h}^{4})$ for the 1D linear hyperbolic equation,β
*Applied Mathematics and Computation*, vol. 207, no. 1, pp. 236β241, 2009. View at Publisher Β· View at Google Scholar - H.-W. Liu and L.-B. Liu, βAn unconditionally stable spline difference scheme of $O({k}^{2}+{h}^{4})$ for solving the second-order 1D linear hyperbolic equation,β
*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1985β1993, 2009. View at Publisher Β· View at Google Scholar - R. K. Mohanty, βNew unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations,β
*International Journal of Computer Mathematics*, vol. 86, no. 12, pp. 2061β2071, 2009. View at Publisher Β· View at Google Scholar - R. K. Mohanty, M. K. Jain, and K. George, βOn the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients,β
*Journal of Computational and Applied Mathematics*, vol. 72, no. 2, pp. 421β431, 1996. View at Publisher Β· View at Google Scholar - R. K. Mohanty and U. Arora, βA new discretization method of order four for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equation,β
*International Journal of Mathematical Education in Science and Technology*, vol. 33, no. 6, pp. 829β838, 2002. View at Publisher Β· View at Google Scholar - R. K. Mohanty and V. Dahiya, βAn $O({k}^{2}+k{h}^{2}+{h}^{4})$ accurate two-level implicit cubic spline method for one space dimensional Quasi-linear parabolic equations,β
*American Journal of Computational Mathematics*, vol. 1, pp. 11β17, 2011. View at Google Scholar - R. K. Mohanty, βStability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms,β
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1683β1690, 2007. View at Publisher Β· View at Google Scholar