A sixth-order wavelet integral collocation method for solving nonlinear boundary value problems in three dimensions

A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions. In order to realize the establishment of this method, an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension. Through applying such an integral technique, during the solution of nonlinear partial differential equations, the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes. A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method. The validation and convergence of the proposed method are examined through several benchmark problems, including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates. Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods. Most importantly, the convergence rate of the proposed method seems to be independent of the order of the differential equations, because it is always sixth order for second-, fourth-. sixth-, and even eighth-order problems.