A fourth-order boundary value problem for a Sturm-Liouville type equation

An existence result of multiple solutions for a fourth-order Sturm-Liouville boundary value problem with variable parameters is established. As a consequence, three solutions for a boundary value problem with a fourth-order equation in a complete form are obtained. Our approach is based on variational methods.     

一类四阶非线性椭圆问题变号解的存在性

通过建立一个新空间,在新空间中讨论了一个含Hardy位势的四阶非线性椭圆问题变号解的存在性.在一个环绕定理下,得到了问题变号解的存在性.     

On a double degenerate fourth-order parabolic equation

The existence of solutions to a fourth-order p-Laplacian equation with boundary degeneracy is studied. For the purpose of solving the corresponding non-degenerate (with respect to the coefficient of fourth-order term) regularized problem, a fourth-order semi-discrete elliptic problem with homogeneous boundary conditions is established and its existence and uniqueness are obtained by the functional minimization method. It follows that the approximate solutions of the non-degenerate parabolic problem are constructed and the corresponding existence and uniqueness are discovered by a limit procedure from the energy estimation method and a compactness argument. Finally, the existence and regularity of solutions for the problem with boundary degeneracy is obtained by using a regularization parameter vanishing limit.     

Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.     

Uniqueness of a high-order accurate bicompact scheme for quasilinear hyperbolic equations

The possibility of constructing new third- and fourth-order accurate differential-difference bicompact schemes is explored. The schemes are constructed for the one-dimensional quasilinear advection equation on a symmetric three-point spatial stencil. It is proved that this family of schemes consists of a single fourth-order accurate bicompact scheme. The result is extended to the case of an asymmetric three-point stencil.     

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.     

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.     

一类含参数半正四阶边值问题的正解存在性与多解性

考察了一类含有两个参数的非线性四阶边值问题的n个解和/或正解的存在性,其中允许非线性项有一个非正的函数型下界.在力学和工程上,这类四阶边值问题描述了两端简单支撑的弹性梁的变形.主要工具是锥上的Krasnosel’skii不动点定理和局部化方法.     

On the solvability of a boundary value problem for a fourth-order equation on a graph

We consider a boundary value problem for a fourth-order equation on a graph modeling elastic deformations of a plane rod system with conditions of rigid connection at the vertices. Conditions for the unique solvability are stated. We also present sufficient conditions for the problem to be degenerate.     

Existence and blow-up of solutions of a fourth-order nonlinear diffusion equation

In this paper, we are concerned with the existence, uniqueness and blow up of solutions for a fourth-order nonlinear diffusion equation.     

On the Positive Solutions of a Nonlinear Fourth-order Boundary Value Problem with Two Parameters

The existence of m positive solutions is proven for a nonlinear fourth-order boundary value problem with two parameters, where m is an arbitrary natural number. This kind of fourth-order boundary value problems usually describes the equilibrium state of elastic beam where both ends are simply supported. The main ingredient is Krasnosel'skii fixed point theorem of cone expansion–compression type.     

一类四阶混合型微分方程的周期解

运用Fourier级数理论和实分析不等式方法,研究了一类四阶混合型微分方程的周期解问题,得到了若干周期解存在性和唯一性的判定定理.     

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

This paper is devoted to the testing and comparison of numerical solutions obtained from higher-order accurate finite difference schemes for the two-dimensional Burgers' equation having moderate to severe internal gradients. The fourth-order accurate two-point compact scheme, and the fourth-order accurate Du Fort Frankel scheme are derived. The numerical stability and convergence are presented. The cases of shock waves of severe gradient are solved and checked with the fourth-order accurate Du Fort Frankel scheme solutions. The present study shows that the fourth-order two-point compact scheme is highly stable and efficient in comparison with the fourth-order accurate Du Fort Frankel scheme.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

四阶不完全对称张量M-特征值的新包含域及应用

四阶不完全对称张量的M-特征值在非线性弹性材料分析中有着广泛的应用.本文的目的是给出四阶不完全对称张量M-特征值的新包含域,得到最大M-特征值上界更精确的估计,并将得到的上界估计值应用到计算最大M-特征值的WQZ算法中,数值例子验证了结果的有效性.最后,基于得到的包含域,给出了四阶不完全对称张量正定性判定的充分条件.     

Fully implicit finite differences methods for two-dimensional diffusion with a non-local boundary condition

Three new fully implicit methods which are based on the (5,5) Crank-Nicolson method, the (5,5) N-H (Noye-Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on the standard implicit backward time centered space (BTCS) method, use a large amount of central processor (CPU) time, the high accuracy of the new fourth-order fully implicit scheme is significant. Like the BTCS method, the new methods are also unconditionally stable.     

On the solvability of a boundary value problem for a fourth-order ordinary differential equation

We study the existence and multiplicity of nontrivial periodic solutions for a semilinear fourth-order ordinary differential equation arising in the study of spatial patterns for bistable systems. Variational tools such as the Brezis–Nirenberg theorem and Clark theorem are used in the proofs of the main results.     

Normal solvability of boundary value problems for a class of fourth-order operator-differential equations in a weighted space

In a weighted space, we find sufficient conditions for the normal solvability of a boundary value problem for a class of fourth-order operator-differential equations whose leading part has multiple characteristics. Note that these conditions are expressed in terms of properties of operator coefficients of the equation considered.     

两个基于不同张量乘法的四阶张量分解

提出了两个基于不同张量乘法的四阶张量分解. 首先, 在矩阵乘法的基础上, 定义第一种四阶张量乘法(F-乘), 基于F-乘提出了第一种四阶张量分解(F-TD). 其次, 基于三阶张量t-product给出了第二种四阶张量乘法(B-乘)和分解(FT-SVD). 同时, 利用两种分解方法, 分别给出两个张量逼近定理. 最后, 三个数值算例阐明提出的两种分解方法的准确性和可行性.     

Space/time convergence analysis of a class of conservative schemes for linear wave equations

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.