Computational Fractional Dynamical Systems

10Weighted Residual Methods

10.1 Introduction

The weighted residual is another effective approach for solving fractional differential equations (FDEs) with boundary conditions (BCs), often known as boundary value problems (BVPs). The weighted residual method (WRM) is an approximation approach that uses a linear combination of trial or shape functions with unknown coefficients to estimate the solution of FDEs. The approximate solution is then replaced in the governing FDE, yielding error or residual. Finally, in order to determine the unknown coefficients, the residual is made to zero at average points or made as minimal as possible based on the weight function. Gerald and Wheatley (2004) addressed WRMs, including collocation and Galerkin approaches. More information on various WRMs may be found in classic books (Baluch et al. 1983; Finlayson 2013; Hatami 2017). Locker (1971) has provided a least‐square method for solving BVPs. Gerald and Wheatley (2004), Lindgren (2009), Finlayson (2013), Hatami (2017), and Logan (2011) address weighted residual‐based finite‐element algorithms. Boundary characteristic orthogonal polynomials (Bhat and Chakraverty 2004) utilized as trial or shape functions are sometimes advantageous. As a result, Chapter 11 focuses on solving BVP using boundary characteristic orthogonal polynomials integrated into Galerkin and least‐square techniques.

In this regard, this chapter includes several WRMs, namely collocation, Galerkin method, and least‐square techniques ...