3.1 Introduction

Weighted residual is treated as another powerful method for computation of solution to differential equations subject to boundary conditions referred to as boundary value problems (BVPs). Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in error or residual. Finally, in the WRM the residual is forced to vanish at average points or made as small as possible depending on the weight function in order to find the unknown coefficients. WRMs, viz. collocation and Galerkin methods, have been discussed by Gerald and Wheatley [1]. Further discussion of various WRMs may be found in standard books viz. [2–4]. As regards, least‐square method for solving BVPs has been given by Locker [5]. Weighted residual‐based finite‐element methods are discussed in Refs. 1 –3, 6 , 7 . In Chapter 6, finite‐element discretization approach using Galerkin WRM has been introduced. Moreover, sometimes trial or shape functions taken as boundary characteristic orthogonal polynomials are advantageous. So, Chapter 4 is dedicated in solving BVP using boundary characteristic orthogonal polynomials incorporated into Galerkin and Rayleigh–Ritz methods.

In this regard, this chapter is organized such that various WRMs, viz. collocation, subdomain, least‐square, and Galerkin methods applied for solving ordinary differential equations subject to boundary conditions have been illustrated in Sections 3.2, 3.3, 3.4, and 3.5 respectively. For better understanding of the methods, comparative results for specific differential equations with respect to WRMs are also included. Lastly, few exercise problems are also given at the end for self‐validation.

Let us consider an ordinary differential equation,

(3.1)

subject to boundary conditions (BCs) in the domain Ω. Here, L is the linear differential operator acting on u and f is the applied force. In WRM, the solution u(x) of Eq. (3.1) is approximately considered as satisfying the BCs. Here, c_i are the unknown coefficients yet to be determined for the trial functions φ_i(x) for i = 0, 1, …, n, where φ_i(x) are linearly independent to each other. The assumed solution is substituted in the governing differential equation 3.1) resulting in error or residual. This residue is then minimized or forced to vanish in the domain Ω, resulting in system of algebraic equations in terms of unknown coefficients c_i. As such, WRMs mainly consist of the following steps:

• Step (i): Assume an approximate solution
  (3.2)

  involving linearly independent trial functions φ_i(x) such that satisfies the boundary conditions. Alternatively,

  (3.3)

  may also be considered where u₀(x) is the function satisfying the BCs.

  Generally, the choice of trial functions are considered such that the shape functions interpolate the desired solution subject to boundary conditions as mentioned in Eq. (3.2) or Eq. (3.3). Various cases studied for assumption of trial functions may be found in Ref. 2 . One possible assumption for the shape functions may be φ₀ = 1, φ₁ = x, and φ_i = x^i − 1(x − X) for i = 2, 3, …, n − 1 over the domain Ω = [0, X].

• Step (ii): Substitute in the given differential equation that results in error.
• Step (iii): The measure of error is considered as residual,
  (3.4)
• Step (iv): An arbitrary weight function w_i(x) is then multiplied in Eq. ( 3.3 ) and integrated over Ω resulting in
  

  for i = 0, 1, …, n. It is cumbersome to identically make the residue zero in the entire domain. So, the integral is either set to vanish at finite points or made as small as possible depending on the weight function.

• Step (v): Forcing the integral to vanish over the entire domain Ω using
  (3.5)

  n + 1 independent algebraic equations are obtained using Eq. (3.5) for computing unknowns c_i, where i = 0, 1, …, n.

It is worth mentioning that in Eq. ( 3.2 ), as n → ∞. Moreover, depending on weight function, there exist different types of WRMs, viz. collocation, least‐square, Galerkin, and Rayleigh–Ritz methods. In this regard, WRMs are discussed in Sections 3.2 – 3.5 .

3.2 Collocation Method

In the collocation method [ 1 – 3 , 7 ], the weight function is taken in terms of Dirac delta function δ as

(3.6)

where k = 0, 1, …, n. Often in literature studies, this approach is also referred to as the point collocation method. Now, using Eq. (3.6), the integrand

Generally, the residual is set to zero at n + 1 distinct points x_k within the domain Ω,

(3.7)

for computing the unknown coefficients c_i for n − 1 collocating points. For clear understanding, a BVP is solved using the collocation method in Example 3.1.

3.3 Subdomain Method

In the subdomain method [2] , the domain Ω is finitely divided into nonoverlapping n subdomains or subintervals Ω_k + 1 = [x_k, x_k + 1], where k = 0, 1, …, n − 1. Here, the weight function is selected as unity in the subdomains, otherwise zero,

(3.10)

Then, Eq. (3.10) reduces to

(3.11)

We rewrite Eq. (3.11) as , which in turn is expressed in terms of system of equations:

(3.12)

The unknown coefficients in the approximate solution are then obtained by solving Eq. (3.12). An example illustrating the usage of the subdomain method for solving BVPs is considered in Example 3.2.

3.4 Least‐square Method

In the least‐square method [ 2 , 3 , 5 ], the residue given in Eq. (3.4) is squared and integrated over the entire domain Ω,

(3.14)

The integrand I is then minimized using , where c_j are unknown coefficients of approximate solution , which further reduces to

3.15

From Eq. (3.15), the weight function for the least‐square method is considered as

(3.16)

3.5 Galerkin Method

In the Galerkin method [ 1 – 3 , 7 ], the weight function is considered in terms of trial functions,

(3.18)

such that φ_k are (n + 1) basis functions of x satisfying the boundary conditions. As such, Eq. ( 3.5 ) reduces to

(3.19)

Then, the system of equations in terms of unknown coefficients are obtained using Eq. (3.19). To illustrate Galerkin's method, we consider an example problem given below.

3.6 Comparison of WRMs

In this section, we check the efficiency of various WRMs discussed in Sections 3.2 – 3.5 by comparing the solution obtained using collocation, subdomain, least‐square, and Galerkin methods.

Exercise

1. 1 Apply the subdomain method to solve differential equation y″ + y′ + y = t³ subject to boundary conditions y(0) = 1 and y(1) = 0.
2. 2 Use collocation and Galerkin methods to compute the approximate solution of following BVPs subject to boundary conditions y(0) = 0 and y(1) = 0 in the domain Ω = [0, 1].
   1. y″ + 2y = x²
   2. y″ + y = sin(x).
3. 3 Find the approximate solution of BVP y″ − 2y′ + x = 0 subject to y(0) = 1 and y(1) = 0 using the least‐square method.

References

1. 1 Gerald, C.F. and Wheatley, P.O. (2004). Applied Numerical Analysis, 7e. New Delhi: Pearson Education Inc.
2. 2 Finlayson, B.A. (2013). The Method of Weighted Residuals and Variational Principles, vol. 73. Philadelphia: SIAM.
3. 3 Hatami, M. (2017). Weighted Residual Methods: Principles, Modifications and Applications. London: Academic Press.
4. 4 Baluch, M.H., Mohsen, M.F.N., and Ali, A.I. (1983). Method of weighted residuals as applied to nonlinear differential equations. Applied Mathematical Modelling 7 (5): 362–365.
5. 5 Locker, J. (1971). The method of least squares for boundary value problems. Transactions of the American Mathematical Society 154: 57–68.
6. 6 Lindgren, L.E., 2009. From Weighted Residual Methods to Finite Element Methods. Technical report.
7. 7 Logan, D.L. (2011). A First Course in the Finite Element Method, 5e. Stamford: Cengage Learning.