Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

Positive solutions for fourth-order singular p -Laplacian boundary value problems

This article investigates fourth-order singular p-Laplacian boundary value problems (BVPs), and obtains the necessary and sufficient conditions for existence of positive solutions for fourth-order singular p-Laplacian BVPs on closed interval.     

An O(K2 + h4) finite difference method for one-space Burger's equation in polar coordinates

A two-level implicit difference method of O(k² + h⁴) for a class of singular initial boundary value problem, where α, β, γ, and ν are constants, is discussed using three spatial grid points. The method is shown to be unconditionally stable when applied to linearized equations. The fourth-order convergence for a fixed mesh ratio parameter is illustrated with the help of two examples. © 1996 John Wiley & Sons, Inc.     

Compact block boundary value methods for semi-linear delay-reaction–diffusion equations with algebraic constraints

In the present paper, we study a class of linear approximation methods for solving semi-linear delay-reaction–diffusion equations with algebraic constraint (SDEACs). By combining a fourth-order compact difference scheme with block boundary value methods (BBVMs), a class of compact block boundary value methods (CBBVMs) for SDEACs are suggested. It is proved under some suitable conditions that the CBBVMs are convergent of order 4 in space and order p in time, where p is the local order of the used BBVMs, and are globally stable. With several numerical experiments for Fisher equation with delay and algebraic constraint, the computational effectiveness and theoretical results of CBBVMs are further illustrated.     

Positive Solutions of One-Dimensional <Emphasis Type="Italic">p</Emphasis>-Laplacian Boundary Value Problems for Fourth-Order Differential Equations with Deviating Arguments

This paper considers the existence of positive solutions of four-point boundary value problems for fourth-order ordinary differential equations with deviating arguments and p-Laplacian. We discuss such problems in the cases when the deviating arguments are delayed or advanced, what may concern optimization issues related to some technical problems. To obtain the existence results, a fixed point theorem for cones due to Avery and Peterson is applied. According to the Author’s knowledge, the results are new. It is a first paper where a fixed point theorem for cones is applied to fourth-order differential equations with deviating arguments and p-Laplacian. An example is included to verify the theoretical results.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

On a new method for the study of the spatial behavior in a homogeneous elastic arch-like region

We consider a two-dimensional homogeneous elastic state in the arch-like region a?≤?r?≤?b, 0?≤?θ?≤?α, where (r,θ) denotes plane polar coordinates. We assume that three of the edges are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load. The Airy stress function ‘?’ satisfies a fourth-order differential equation in the plane polar coordinates with appropriate boundary conditions. We develop a method which allows us to treat in a unitary way the two problems corresponding to the self-equilibrated loads distributed on the straight and curved edges of the region. In fact, we introduce an appropriate change for the variable r and for the Airy stress functions to reduce the corresponding boundary value problem to a simpler one which allows us to indicate an appropriate measure of the solution valuable for both the types of boundary value problems. In terms of such measures we are able to establish some spatial estimates describing the spatial behavior of the Airy stress function. In particular, our spatial decay estimates prove a clear relationship with the Saint-Venant's principle on such regions.     

Improving performance when solving high-order and mixed-order boundary value problems in ODEs

Solving high-order or mixed-order boundary value problems by general purpose software often requires the system to be first converted to a larger equivalent first-order system. The cost of solving such problems is generally O(m ³), where m is the dimension of the equivalent first-order system. In this paper, we show how to reduce this cost by exploiting the special structure the “equivalent” first-order system inherits from the original associated mixed-order system. This technique applies to a broad class of boundary value methods. We illustrate the potential benefits by considering in detail a general purpose Runge–Kutta method and a multiple shooting method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

The condition number of the Schur complement in domain decomposition

Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order , where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements. Received: January 15, 1998     

Extinction and Positivity for a Doubly Nonlinear Degenerate Parabolic Equation

The aims of this paper are to discuss the extinction and positivity for the solution of the initial boundary value problem and Cauchy problem of ut = div（[↓△u^m｜p-2↓△u^m）. It is proved that the weak solution will be extinct for 1 〈 p ≤ 1 ＋ 1/m and will be positive for p 〉 1 ＋ 1/m for large t, where m 〉 0.     

The perturbed Sparre Andersen model with a threshold dividend strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu functions. The special case where the inter-claim times are Erlang(2) distributed and the claim size distribution is exponential is considered in some details.     

Regularity of weak solutions of semilinear parabolic systems of arbitrary order

Letu be a weak solution of the initial boundary value problem for the semilinear parabolic system of order 2m:u′(t)+Au(t)+f(t,.,u,..., ▽ ^m u)=0. Letf satisfy controllable growth conditions. Thenu is smooth. This result is proved by a kind of continuity method, where the timet is the parameter of continuity.     

An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

The fourth-order nonlinear partial differential equation forsurface diffusion is approximated by a new integrable nonlinearevolution equation. Exact solutions are obtained for thermalgrooving, subject to boundary conditions representing a sectionof a grain boundary. When the slope m of the groove centre islarge, the linear model grossly overestimates the groove depth.In the linear model dimensionless groove depth increases linearlywith m, but in the nonlinear model it approaches an upper limitA nontrivial similarity solution is found for the limiting caseof a thermal groove whose central slope is vertical.     

Near-separating,self-similar turbulent boundary layer

The self-similar flow in a turbulent boundary layer, which is in a state close to separation as a result of the effect of adverse pressure gradient, is investigated. Such a boundary layer has a triple-deck asymptotic structure. Between outer and near-wall regions above the logarithmic sublayer, i. e. the constant-stress layer, an intermediate region — the gradient sublayer — is formed, where the shear stress varies linearly due to adverse longitudinal pressure gradient. In the external part of the gradient sublayer, the velocity profile obeys the square-root law. The velocity profile obtained from the solution for the outer region satisfies a slip condition on the wall. The slip value decreases as the similarity parameter increases and vanishes at the value of Ω = 0.0911, which corresponds to separation, here δ^* is the displacement thickness, and U and U′ are the free-stream velocity and its derivative with respect to the longitudinal coordinate. In this case, the exponent m in the law specifying the free-stream self-similar velocity distribution increases, with separation occurring not at the minimal value of m = −1/3, which corresponds to the strongest adverse pressure gradient, but at the value m = 0.228. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fredholm property of boundary value problems for a fourth-order elliptic differential-operator equation with operator boundary conditions

In a Hilbert space H, we study the Fredholm property of a boundary value problem for a fourth-order differential-operator equation of elliptic type with unbounded operators in the boundary conditions. We find sufficient conditions on the operators in the boundary conditions for the problem to be Fredholm. We give applications of the abstract results to boundary value problems for fourth-order elliptic partial differential equations in nonsmooth domains.     

Existence and asymptotic behaviour of solutions for a class of quasi‐linear evolution equations with non‐linear damping and source terms

We consider a class of quasi‐linear evolution equations with non‐linear damping and source terms arising from the models of non‐linear viscoelasticity. By a Galerkin approximation scheme combined with the potential well method we prove that when m<p, where m(?0) and p are, respectively, the growth orders of the non‐linear strain terms and the source term, under appropriate conditions, the initial boundary value problem of the above‐mentioned equations admits global weak solutions and the solutions decay to zero as t→∞. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Fourth-order finite difference method for three-dimensional elliptic equations with nonlinear first-derivative terms

We present a 19-point fourth-order finite difference method for the nonlinear second-order system of three-dimensional elliptic equations Au _xx + Bu _yy + Cu _zz = f , where A , B , C , are M × M diagonal matrices, on a cubic region R subject to the Dirichlet boundary conditions u (x, y, z) = u ⁽⁰⁾(x, y, z) on ?R. We establish, under appropriate conditions, O(h⁴) convergence of the difference method. Numerical examples are given to illustrate the method and its fourth-order convergence. © 1992 John Wiley & Sons, Inc.     

带p-Laplacian算子的四阶四点边值问题迭代解的存在性

考虑带p-Laplacian算子的四阶四点边值问题(φp(x″(t)))″=f(t,x(t),x″(t)),t∈[0,1],x(0)-αx′(0)=0,x(1)+βx′(1)=0,φp(x″(ξ))-γ(φp(x″(ξ)))′=0,φp(x″(η))+δ(φp(x″(η)))′=0,其中φp(s)=s p-2s,p>1;0<ξ,η<1;f∈C([0,1]×R2,R).通过建立上下解方法得到迭代解的存在性.