Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models

A numerical method based on sextic B-spline is developed to solve the fourth-order time-dependent partial differential equations subjected to fixed and cantilever boundary conditions. We use finite difference approximation to discretize the temporal variable and the spatial variable by means of a σσ-method, σ∈[0,1]σ∈[0,1] (σ=12 corresponds to the Crank–Nicolson method), and a sextic B-spline collocation method on uniform meshes, respectively. Using Von Neumann method, the proposed method is also shown to be conditionally stable if σ<0.25σ<0.25 and unconditionally stable if σ⩾0.25σ⩾0.25. The convergence analysis of the proposed sextic B-spline approximation for the Euler–Bernoulli problem is discussed in details and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.