In this paper an efficient numerical scheme to approximate the solutions of fifth-order boundary value problems in a finite domain with two different types of boundary conditions has been prsented, by taking basis functions with quartic Bsplines and weight functions with quintic B-splines in PetrovGalerkin method. In this method, the quartic B-splines and quintic B-splines are redefined into new sets of functions which contain the equal number of functions. The analysis is accompanied by numerical examples. The obtained results demonstrate the reliability and efficiency of the proposed scheme.

In this paper, a finite element method involving Petrov-Galerkin method with quintic B-splines as basis functions and sextic B-splines as weight functions has been developed to solve a general tenth order boundary value problem with a particular case of boundary conditions. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the Dirichlet, the Neumann, the second order derivative and the third order derivative type of boundary conditions are prescribed. The weight functions are also redefined into a new set of weight functions which in number match with the number of redefined basis functions. The proposed method was applied to solve several examples of tenth order linear and nonlinear boundary value problems. The obtained numerical results were found to be in good agreement with the exact solutions available in the literature.