4.1 Introduction

Boundary characteristic orthogonal polynomials (BCOPs) proposed by Bhat [1,2] in 1985 have been used in various science and engineering problems. Further, many authors like Bhat and Chakraverty [3], Singh and Chakraverty [4] have also used BCOPs in different problems. BCOPs are found to be advantageous in well‐known methods like Rayleigh–Ritz, Galerkin, collocation, etc.

BCOPs may be generated by using the Gram–Schmidt orthogonalization procedure [5]. The generated BCOPs have to satisfy the boundary conditions of the considered problem [ 3 ,6]. Initially, the general approximated solution of the considered problem is assumed as linear combination of BCOPs. By substituting the approximated solution in the boundary value problem, one may get the residual [ 4 ,7]. Further, by using the residual and following the algorithm of a particular method, one may develop a linear system of equations. In the final step, one can handle the obtained linear system by using any known analytical/numerical procedure. The orthogonal nature of the BCOPs makes the analysis simple and straightforward.

The following section presents the Gram–Schmidt orthogonalization process for generating orthogonal polynomials.

4.2 Gram–Schmidt Orthogonalization Process

Let us suppose a set of functions (g_i(x), i = 1, 2, …) in [a, b]. From these set of functions, one can construct appropriate orthogonal functions by using the well‐known procedure known as the Gram–Schmidt orthogonalization [ 3 , 5 ] process as follows:

where , etc. and 〈   〉 defines the inner product of the respective polynomials.

In general, one can write the above procedure as

where and w(x) is a weight function.

Next, we address the generation procedure for BCOPs.

4.3 Generation of BCOPs

The first member of BCOPs set viz. φ₀(x) is chosen as the simplest polynomial of the least order which satisfies the boundary conditions of the considered problem. The other members of the orthogonal set in the interval a ≤ x ≤ b are generated by using the Gram–Schmidt process [ 3 , 4 ] as follows:

where and .

Here, we consider w(x) = 1. The polynomials φ_k(x) satisfy the orthogonality condition:

Next, we present Galerkin's method with BCOPs to handle various problems.

4.4 Galerkin's Method with BCOPs

Let us consider a second‐order boundary value problem [4] on [a, b] as

(4.1)

An approximate solution of Eq. (4.1) is considered as

(4.2)

where φ_i(x) are BCOPs which satisfy the given boundary conditions and c₀, c₁, …, c_n are real constants.

By substituting Eq. (4.2) in Eq. ( 4.1 ), one may obtain the residual R as

(4.3)

The residual R is orthogonalized to the (n + 1) BCOPs φ₀, φ₁, …, φ_n as

(4.4)

where .

From Eq. (4.4), one may obtain (n + 1) system of linear equations with (n + 1) unknowns which can be solved by any standard analytical method. Further, by substituting the evaluated constants c₀, c₁, …, c_n in Eq. ( 4.2 ) we get the approximate solution of the boundary value problem ( 4.1 ). One may note that the terms containing φ₀, φ₁, …, φ_n will vanish due to the orthogonal property. This makes the method efficient.

Let us now solve a boundary value problem by using BCOPs with Galerkin's method.

4.5 Rayleigh–Ritz Method with BCOPs

Let us consider a second‐order boundary value problem [ 2 , 4 ] as

(4.10)

Then, one can find a functional (function of functions) S(x, y, y′) as

(4.11)

where

(4.12)

Now, from Eq. (4.12), one can have the differential equation 4.10) by applying the well‐known Euler–Lagrange equation [8] as

(4.13)

(4.14)

Further, assume an approximate solution with BCOPs to Eq. ( 4.10 ) as

(4.15)

where φ_i's are linearly independent BCOPs.

Let us substitute Eq. (4.15) in Eq. (4.11) and thus the integral “I” is evaluated as a function of c_i. The necessary conditions for the extremum value of “I” from ordinary calculus gives

(4.16)

From Eq. (4.16), one may obtain (n + 1) system of linear equations with (n + 1) unknowns. The obtained system of equations 4.16) may be solved to evaluate c_i by using the standard analytical/numerical method to get the approximate solution of Eq. ( 4.10 ).

Next, we solve a boundary value problem by using BCOPs with the Rayleigh–Ritz method.

Exercise

1. 1 (i) (ii) Use BCOPs in Galerkin and Rayleigh–Ritz methods to approximate the solutions of the following differential equations:
   1. (i) 2y″ = 5x + 3 subject to y(0) = 1,  y(1) = 3.
   2. (ii) 2y″ − xy′ + 3 = 0 subject to y(0) = 0,  y′(2) = 4.

References

1. 1 Bhat, R.B. (1985). Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh‐Ritz method. Journal of Sound and Vibration 102 (4): 493–499.
2. 2 Bhat, R.B. (1986). Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh‐Ritz method. Journal of Sound and Vibration 105 (2): 199–210.
3. 3 Bhat, R.B. and Chakraverty, S. (2004). Numerical Analysis in Engineering. Oxford: Alpha Science Int'l Ltd.
4. 4 Singh, B. and Chakraverty, S. (1994). Boundary characteristic orthogonal polynomials in numerical approximation. Communications in Numerical Methods in Engineering 10 (12): 1027–1043.
5. 5 Johnson, P.S. (2014). Gram‐Schmidt Orthogonalization Process.
6. 6 Singh, B. and Chakraverty, S. (1994). Flexural vibration of skew plates using boundary characteristic orthogonal polynomials in two variables. Journal of Sound and Vibration 173 (2): 157–178.
7. 7 Chakraverty, S., Saini, H., and Panigrahi, S.K. (2008). Prediction of product parameters of fly ash cement bricks using two dimensional orthogonal polynomials in the regression analysis. Computers and Concrete 5 (5): 449–459.
8. 8 Agrawal, O.P. (2002). Formulation of Euler–Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications 272 (1): 368–379.