A finite element method involving collocation method with quintic B-splines as basis functions has been developed to solve tenth order boundary value problems. The fifth order, sixth order, seventh order, eighth order, ninth order and tenth order derivatives for the dependent variable are approximated by the central differences of fourth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of selected collocated points in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

A finite element method involving collocation method with quartic B-splines as basis functions have been developed to solve fifth order boundary value problems. The fifth order and fourth order derivatives for the dependent variable are approximated by the central differences of third order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on four linear and two non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

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