Spline-based DQM for multi-dimensional PDEs: Application to biharmonic and Poisson equations in 2D and 3D

The idea of differential quadrature is used to construct a new algorithm for the solution of differential equations. To determine the weighting coefficients of DQM, B-spline basis functions of degree r are used as test functions. The method is constructed on a set of points mixed from grid points and mid points of a uniform partition. Using the definition of B-splines interpolation as alternative, some error bounds are obtained for DQM. The method is successfully implemented on nonlinear boundary value problems of order m. Also the application of the proposed method to approximate the solution of multi-dimensional elliptic PDEs is included in the paper. As test problem, some examples of biharmonic and Poisson equations are solved in 2D and 3D. The results are compared with some existing methods to show the efficiency and performance of the proposed algorithm. Also some examples of time dependent PDEs are solved to compare the results with other existing spline based DQ methods.