Quadrature rules for numerical integration based on Haar wavelets and hybrid functions

In this paper Haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration. This approach is the generalization and improvement of the methods (Siraj-ul-Islam et al. (2010) [9]) where the numerical methods are only applicable to the integrals with constant limits. Apart from generalization of the methods [9], the new approach has two major advantages over the classical methods based on quadrature rule: (i) No need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically (ii) Mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n. The new methods are more efficient. The novel methods are compared with existing methods and applied to a number of benchmark problems. Accuracy of the methods are measured in terms of absolute errors.