In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem. 相似文献
In this concept review, the fundamental and polymerization chemistry of inverse vulcanization for the preparation of statistical and segmented sulfur copolymers, which have been actively developed and advanced in various applications over the past decade is discussed. This concept review delves into a discussion of step-growth polymerization constructs to describe the inverse vulcanization process and discuss prepolymer approaches for the synthesis of segmented sulfur polyurethanes. Furthermore, this concept review discusses the advantages of inverse vulcanization in conjunction with dynamic covalent polymerization and post-polymerization modifications to prepare segmented block copolymers with enhanced thermomechanical and flame retardant properties of these materials. 相似文献
The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions.
A new fast algorithm based on the augmented immersed interface method
and a fast Poisson solver is proposed to solve three dimensional elliptic interface
problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal
derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable
which should be chosen such that the original flux jump condition is satisfied. The
discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing
PDEs in a neighborhood of control points on the interface. The interpolation scheme
is the key to the success of the augmented IIM particularly. In this paper, the key
new idea is to select interpolation points along the normal direction in line with the
flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The
number of the GMRES iterations is independent of the mesh size. 相似文献