Numerical Solutions For Fourth-order Ordinary Differential Equations With Polynomial Nonlinearity

The main goal of the present thesis is to study the numerical techniques for finding mul-tiple solutions of nonlinear fourth-order ordinary differential equations(ODEs)with constant and variable-coefficients.These techniques mainly include the eigenfunction expansion method(EEM)for the differential equation and homotopy continuation method for solving the resulting system of polynomial equations.In the first part of this thesis,the fourth-order ODEs are discretized by using the eigenfunc-tion expansion method,and then the error estimates for the discretization method are derived in H1-norm ana L2-norm,respectively.A numerical example is given to show the rate of conver-gence for the eigenfunction expansion method.In the second part of the thesis,the multiple solutions of fourth-order ODEs with cubic polynomial nonlinearity are searched.We constructed an efficient polynomial homotopy to find all solutions for the system of polynomial equations on the coarse level.The idea of the homotopy is that the polynomial system can be solved by a recursive process,in which the solution set of the system in step k can be used to obtain the solution set of the system in step k+1.Finite difference filter and Newton filter are suggested to remove possible spurious solutions appearing in the solution set of the discretized system of BVPs.These filters are based on the error estimates.Some numerical experiments are included to verify the efficiency of the presented homotopy.Finally,a symmetric homotopy method for solving variable-coefficient fourth-order ODEs with cubic and quintic nonlinearities is given.The ODE problem is discretized by using EEM and then,we analyze the symmetry in the solution set for the resulting system of polynomial equation.Based on the analysis of symmetry,a simple fourth-order ODE is constructed as a starting system and then discretized in eigensubspaces,where the subsystems can be solved easily.Then,we put together the resulting systems of polynomial equations in a block-wise manner in order to construct the starting system to the proposed fourth-order ODEs.Utilizing the symmetry reduces the number of computations,because we only need to follow the representative solution paths of the homotopy.We apply Newton filter to remove the spurious solutions.Several numerical results are stated to demonstrate the efficiency of the symmetric homotopy.