15.1 Introduction

We have already discussed Akbari–Ganji's method (AGM), exp‐function method, Adomian decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), and homotopy analysis method (HAM) in Chapters 9, 10, 11, 12, 13, and 14, respectively. This chapter presents another effective numerical method that approximates the solution of the PDEs by functional values at certain discretized points. This method can be applied with considerably less number of grid points, whereas the methods like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) as given in Chapters 5, 6, and 7, respectively, may need more number of grid points to obtain the solution. However, FDM, FEM, and FVM are versatile methods which may certainly be used to handle regular as well as irregular domains.

Approximating partial derivatives by means of weighted sum of function values is known as Differential Quadrature (DQ). The differential quadrature method (DQM) was proposed by Bellman et al. [1–4]. Determination of weighted coefficients plays an important and crucial role in DQM. As such, an effective procedure for finding weighted coefficients suggested by Bellman is to use a simple algebraic formulation of weighted coefficient with the help of coordinates of grid points. Here, the coordinate of the grid points are the roots of the base functions like Legendre polynomials, Chebyshev polynomials, etc. Interested readers may see Refs. [5–10] and references therein for more details and application of DQM.

15.2 DQM Procedure

In this section, DQM has been illustrated [ 1 ,2] by considering a second‐order nonlinear partial differential equation (PDE) of the form

(15.1)

subject to initial condition

(15.2)

The DQ of the first‐ and second‐order spatial derivatives at the grid points x_i are given by

(15.3)

(15.4)

By plugging Eqs. (15.3) and (15.4) in Eq. (15.1), we obtain a system of N ordinary differential equations (ODEs) as follows:

(15.5)

subject to initial conditions

(15.6)

The weighted coefficients a_ij and b_ij are to be found using N grid points x_i and these grid points depend on the considered polynomials like Legendre, Chebyshev, etc. In this chapter, shifted Legendre polynomials in [0, 1] have been used to find weighted coefficients.

The shifted Legendre polynomials [11,12] of order N are given as

First, five shifted Legendre polynomials are

The weighted coefficients in terms of shifted Legendre polynomial [2] are taken as follows:

(15.7)

and

(15.8)

Once all a_ii are obtained, it is then easy to find b_ii following the below procedure.

Let us denote Eq. ( 15.3 ) as

(15.9)

where A = [a_ij]_N × N.

The second‐order derivative can be approximated as

(15.10)

where [b_ij] = A².

Hence, weighted coefficients b_ij can be obtained by squaring A as

Substituting the obtained weighted coefficients a_ij and b_ij in Eq. (15.5) and then using a suitable method to solve the system of ODEs, the solution of the model problem may be found at selected grid points.

Next, we solve two test problems to demonstrate the DQM.

15.3 Numerical Examples

In this section, two nonlinear PDEs are solved by considering different N for better understanding of the DQM.

References

1. 1 Bellman, R. and Casti, J. (1971). Differential quadrature and long‐term integration. Journal of Mathematical Analysis and Applications 34: 235–238.
2. 2 Bellman, R., Kashef, B.G., and Casti, J. (1972). Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics 10: 40–52.
3. 3 Bellman, R., Kashef, B.G., Lee, E.S., and Vasudevan, R. (1975a). Solving hard problems by easy methods: differential and integral quadrature. Computers and Mathematics with Applications 1: 133–143.
4. 4 Bellman, R., Kashef, B.G., Lee, E.S., and Vasudevan, R. (1975b). Differential quadrature and splines. Computers and Mathematics with Applications 1: 371–376.
5. 5 Bert, C.W., Jang, S.K., and Striz, A.G. (1989). Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. Computational Mechanics 5: 217–226.
6. 6 Bert, C.W. and Kang, K.J. (1996). Stress analysis of closely‐coiled helical springs using differential quadrature. Mechanics Research Communications 23 (6): 589–598.
7. 7 Bert, C.W. and Malik, M. (1996). Differential quadrature method in computational mechanics. Applied Mechanics Reviews 49 (1): 1–28.
8. 8 Shu, C. and Chen, W. (1999). On optimal selection of interior points for applying discretized boundary conditions in DQ vibration analysis of beams and plates. Journal of Sound and Vibration 222 (2): 239–257.
9. 9 Shu, C. (1999). Application of differential quadrature method to simulate natural convection in a concentric annulus. International Journal for Numerical Methods in Fluids 30: 977–993.
10. 10 Wazwaz, A.M. (2010). Partial Differential Equations and Solitary Waves Theory. Beijing: Springer Science & Business Media, Higher Education Press.
11. 11 Ezz‐Eldien, S.S., Doha, E.H., Baleanu, D., and Bhrawy, A.H. (2017). A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. Journal of Vibration and Control 23 (1): 16–30.
12. 12 Bhrawy, A.H., Alofi, A.S., and Ezz‐Eldien, S.S. (2011). A quadrature tau method for fractional differential equations with variable coefficients. Applied Mathematics Letters 24 (12): 2146–2152.
13. 13 Debnath, L. (2012). Nonlinear Partial Differential Equations for Scientists and Engineers. New York: Springer Science & Business Media.