ISRN Computational Mathematics

Volume 2013 (2013), Article ID 710529, 8 pages

http://dx.doi.org/10.1155/2013/710529

Research Article

Applying Cubic B-Spline Quasi-Interpolation to Solve 1D Wave Equations in Polar Coordinates

Hossein Aminikhah and Javad Alavi

Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 27 August 2013; Accepted 14 October 2013

Academic Editors: Y. Peng and J. G. Zhou

Copyright © 2013 Hossein Aminikhah and Javad Alavi. 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.

We provide numerical solution to the one-dimensional wave equations in polar coordinates, based on the cubic B-spline quasi-interpolation. The numerical scheme is obtained by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors -norm and -norm. The numerical results are found to be in good agreement with the exact solutions. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement.

1. Introduction

The term “spline” in the spline function arises from the prefabricated wood or plastic curve board, which is called spline, and is used by the draftsman to plot smooth curves through connecting the known point. The use of spline function and its approximation play an important role in the formation of stable numerical methods. As the piecewise polynomial, spline, especially B-spline, have become a fundamental tool for numerical methods to get the solution of the differential equations. In the past, several numerical schemes for the solution of boundary value problems and partial differential equations based on the spline function have been developed by many researchers. As early in 1968 Bickley [1] has discussed the second-order accurate spline method for the solution of linear two-point boundary value problems. Raggett and Wilson [2] have used a cubic spline technique of lower order accuracy to solve the wave equation. Chawla et al. [3] solved the one-dimensional transient nonlinear heat conduction problems using the cubic spline collocation method in 1975. Rubin and Khosla [4] first proposed the spline alternating direction implicit method to solve the partial differential equation using the cubic spline and enhanced accuracy of the approximate solution of the second derivative to the same as that of the first derivative. Jain and Aziz [5] have derived fourth-order cubic spline method for solving the nonlinear two-point boundary value problems with significant first derivative terms. In recent years, El-Hawary and Mahmoud [6], Mohanty [7], Mohebbi and Dehghan [8], Zhu and Wang [9], Ma et al. [10], Dosti and Nazemi [11], Wang et al. [12], and other researchers [13–16] have derived various numerical methods for solution of partial differential equations based on the spline function.

The hyperbolic partial differential equations model the vibrations of structures (e.g., buildings, beams, and machines) and are the basis for fundamental equations of atomic physics.

The one-dimensional linear singular hyperbolic equation is given by

subject to the initial conditions

and Dirichlet boundary conditions at and of the form

where , is time variable and is distance variable, and subscripts and denote differentiation. For and , the equation above represents one-dimensional wave equation in cylindrical and spherical polar coordinates, respectively. We assume that , , and , and their derivatives are continuous functions of and , respectively.

Mohanty et al. [17] have a numerical solution equation (1). In this paper, we provide a numerical scheme to solve singular hyperbolic equation (1) using the derivative of the cubic B-spline quasi-interpolation to approximate the spatial derivative of the differential equations and utilize a forward difference to approximate the time derivative such as [9, 11] shown.

This paper is organized as follows. In Section 2, the univariate spline quasi-interpolants were introduced and we obtain the numerical schemes using cubic B-spline interpolation to solve singular hyperbolic equation (1). The stability of this method is studied in Section 3. Numerical experiments for various test problems are solved to assess the accuracy of the technique and the maximum absolute errors will be presented in Section 4. Finally, we give some concluded remarks in Section 5.

2. Univariate Spline Quasi-Interpolants Applied to Singular Hyperbolic Equation

According to recurrence relation of B-spline [18] the th B-spline of degree for the knot sequence is denoted by or and is obtain by the rule

with

Now assume that is a uniform partition of interval , where , and with meshlength and consider that subject to and , , and . Moreover suppose that and that is the B-spline of degree for the knot sequence . We denote by the space of splines of degree and on the uniform partition . Let the B-spline basis of be . With these notations, the support of is . Figure 1 shows the thirteen B-splines for the knot sequence . Note that in Figure 1  , .

Figure 1: The B-splines for the knot sequence , , .

In [19] univariate spline quasi-interpolants (abbreviation QIs) can be defined as operators of the form . We denote by the space of polynomials of total degree at most . In general we impose that for all . As a consequence of this property, the approximation order is on smooth functions. According to [19], we assume that the coefficient is a linear combination of discrete values of at some points in the neighborhood of .

The main advantage of QIs is that they have a direct construction without solving any system of linear equations. Moreover, they are local, in the sense that the value of depends only on values of in a neighborhood of . Finally, they have a rather small infinity norm, so they are nearly optimal approximants [19]. For any subinterval , , and for any function ,

where the distance of to polynomials is defined by

Here . Therefore, for, this implies that [19]

Since the cubic spline has become the most commonly used spline, we use cubic B-spline quasi-interpolation in this paper.

For cubic QI,

and (4) implies that

Let ; the coefficient functional are, respectively,

For approximate derivatives of by derivatives of up to the order , we can evaluate the value of at by and . We set , , and where , , . Using (4), we can compute , , and , ; see [11]. By solution of the linear systems

we obtain the differential formulas for cubic B-spline QI as

where and obtain as follows:

Now, we present the numerical scheme for solving one-dimensional linear singular hyperbolic equation (1) with initial conditions (2) and Dirichlet boundary conditions (3) based on the cubic B-spline quasi-interpolant.

Discretizing (1),

in time, we get

where is the approximation of the value at , , and is the time step. Then, we use the derivatives of the cubic B-spline quasi-interpolant to approximate and .

Assume that is known for the nonnegative integer . We set unknown vectors as

Then

From the initial conditions (2) and Dirichlet boundary conditions (3), we can compute the numerical solution of (1) step by step using the scheme and formulas (16).

3. Stability Analysis

Sharma and Singh provided a method to study the ability of the nonlinear partial equation in [20], which we used to study the stability of our scheme. According to (18) and , , the scheme (16) can be rewritten as

If we set , , then the scheme is

Therefore we obtained

Taking the norm of (21), we have

Since and , thus from (22) we have

where and . We set , and ; then from (23) we have

It implies that the method is stable if .

4. Numerical Experiments

In this section, some numerical solutions of the one-dimensional linear singular hyperbolic equation in the form (1) with the initial conditions (2) and boundary conditions (3) with the scheme (16) are presented.

The versatility and the accuracy of the proposed method is measured using the and error norms for the test problems. The error norms are defined as

where and are the exact and approximate solution of in and arbitrary value of , respectively.

Example 1. In this example, we consider (1) with , and , , . The initial condition are given by

and the boundary conditions

The exact solution of this example is . The root-mean-square error and maximum error are presented in Table 1. The space-time graph of the exact and numerical solution up to are shown in Figures 2 and 3. Absolute error between the numerical and analytical solution is also depicted at all mesh points in Figure 4.

Table 1: Error norms and invariants ().

Time
error
error

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 2: Exact solution of Example 1 for and .

Figure 3: Numerical solution of Example 1 using and for and .

Figure 4: Absolute error of Example 1 using and .

Example 2. We consider the initial and boundary conditions for (1) as follows:

In Figures 5 and 6 exact and numerical solutions corresponding to and are depicted. In our computations, we consider that and . The exact solution of this example is . The maximum absolute error and the norm error, at some time levels, are presented in Table 2. Absolute error between the numerical and analytical solution is also depicted at all mesh points in Figure 7.

Table 2: Error norms and invariants ().

Time
error
error

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 5: Exact solution of Example 2 for and .

Figure 6: Numerical solution of Example 2 using and for and .

Figure 7: Absolute error of Example 2 using and for and .

Example 3. As a third test problem, we consider (1) with and . The initial condition is given by

and the boundary conditions

The space-time graph of the exact and estimated solution up to is presented in Figures 8 and 9. Absolute error between the numerical and analytical solution is also depicted at all mesh points in Figure 10. The root-mean-square error and maximum error are presented in Table 3.

Table 3: Error norms and invariants ().

Time
error
error

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 8: Exact solution of Example 3 for and .

Figure 9: Numerical solution of Example 3 using and for and .

Figure 10: Absolute error of Example 3 using and for and .

5. Conclusions

In this paper, a numerical scheme for the one-dimensional linear singular hyperbolic equation is proposed using cubic B-spline quasi-interpolation. The numerical solutions are compared with the exact solution by finding and errors. From the test examples, we can say that the BSQI scheme is feasible and the error is acceptable. The implementation of the present method is a very easy, acceptable, and valid scheme.

Conflict of Interests

The authors of the paper do not have a direct financial relation that might lead to a conflict of interests for any of the authors.

References

W. G. Bickley, “Piecewise cubic interpolation and two-point boundary problems,” Computer Journal, vol. 11, no. 2, pp. 206–208, 1968.

G. F. Raggett and P. D. Wilson, “A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique,” IMA Journal of Applied Mathematics, vol. 14, no. 1, pp. 75–78, 1974.

T. C. Chawla, G. Leaf, W. L. Chen, and M. A. Grolmes, “The application of the collocation method using hermite cubic spline to nonlinear transient one-dimensional heat conduction problem,” Journal of Heat Transfer, vol. 97, no. 4, pp. 562–569, 1975.

S. G. Rubin and P. K. Khosla, “Higher-order numerical solution using cubic splines,” AIAA Journal, vol. 14, no. 7, pp. 851–858, 1976.

M. K. Jain and T. Aziz, “Cubic spline solution of two-point boundary value problems with significant first derivatives,” Computer Methods in Applied Mechanics and Engineering, vol. 39, no. 1, pp. 83–91, 1983.

H. M. El-Hawary and S. M. Mahmoud, “Spline collocation methods for solving delay-differential equations,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 359–372, 2003.

R. K. Mohanty, “An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation,” Applied Mathematics Letters, vol. 17, no. 1, pp. 101–105, 2004.

A. Mohebbi and M. Dehghan, “High order compact solution of the one-space-dimensional linear hyperbolic equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 5, pp. 1222–1235, 2008.

C.-G. Zhu and R.-H. Wang, “Numerical solution of Burgers' equation by cubic B-spline quasi-interpolation,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 260–272, 2009.

L. Ma, Z. Mo, and X. Xu, “Quasi-interpolation operators based on a cubic spline and applications in SAMR simulations,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3853–3868, 2010.

M. Dosti and A. Nazemi, “Solving one-dimensional hyperbolic telegraph equation using cubic B-spline quasi-interpolation,” International Journal of Mathematical & Computer Sciences, vol. 7, no. 2, p. 57, 2011.

C.-C. Wang, J.-H. Huang, and D.-J. Yang, “Cubic spline difference method for heat conduction,” International Communications in Heat and Mass Transfer, vol. 39, no. 2, pp. 224–230, 2012.

M. K. Kadalbajoo, L. P. Tripathi, and A. Kumar, “A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1483–1505, 2012.

S. A. Khuri and A. Sayfy, “A spline collocation approach for a generalized parabolic problem subject to non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9187–9196, 2012.

R. K. Mohanty, R. Kumar, and V. Dahiya, “Cubic spline iterative method for Poisson’s equation in cylindrical polar coordinates,” International Scholarly Research Network ISRN Mathematical Physics, vol. 2012, Article ID 234516, 11 pages, 2012.

R. C. Mittal and R. K. Jain, “Numerical solutions of nonlinear Burgers' equation with modified cubic B-splines collocation method,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7839–7855, 2012.

R. K. Mohanty, R. Kumar, and V. Dahiya, “Cubic spline method for 1D wave equation in polar coordinates,” International Scholarly Research Network ISRN Computational Mathematics, vol. 2012, Article ID 302923, 6 pages, 2012.

C. De Boor, A Practical Guide to Splines, Springer, New York, NY, USA, 1978.

P. Sablonnière, “Univariate spline quasi-interpolants and applications to numerical analysis,” Rendiconti del Seminario Matematico, vol. 63, no. 3, pp. 211–222, 2005.

K. K. Sharma and P. Singh, “Hyperbolic partial differential-difference equation in the mathematical modeling of neuronal firing and its numerical solution,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 229–238, 2008.