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1.
A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.  相似文献   

2.
In certain polymer-penetrant systems, nonlinear viscoelasticeffects dominate those of Ficlcian diffusion. By introducinga dependence of the chemical potential on concentration history,this behaviour can be modelled by a memory integral. The mathematicalframework presented is a moving boundary-value problem wherethe boundary separates the polymer into two distinct states:glassy and rubbery. In each region, a different operator holdsat leading order. The problem which results is not solvableby similarity solutions, but can be solved by perturbation andintegral equation techniques. By introducing a new model wherethe diffusion coefficient changes with phase, asymptotic solutionsare obtained where sharp fronts initially move like t3/2. This‘super-Case II’ behaviour is found in various non-Fickianpolymer-penetrant systems.  相似文献   

3.
The aim of the present paper is to analyse the behaviour ofthe stress and displacement fields in the vicinity of the tipof a crack moving along a bi-material interface. For simplicity,we consider a straight interface of infinite extent. We assumethat the two phases are separated by a thin layer which is either‘soft’ or ‘stiff’ compared to the othertwo phases. We derive the transmission conditions which takeinto account the material properties of the layer and modelthe way the load is transferred across the layer from one phaseto the other. We assume that the point of interchange in theboundary/transmission conditions coincides with the crack tipthat moves along the interface boundary with a constant speed.We develop an integral equation formulation and derive asymptoticformulae for the out-of-plane displacement and the Mode-IIIstress intensity factor associated with such a motion of thecrack inside the interphase layer. The theoretical results areillustrated by numerical examples.  相似文献   

4.
The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both  相似文献   

5.
The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.  相似文献   

6.
Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.  相似文献   

7.
This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method.  相似文献   

8.
Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.  相似文献   

9.
In this paper the boundary integral expression for a one-dimensional wave equation with homogeneous boundary conditions is developed. This is done using the time dependent fundamental solution of the corresponding hyperbolic partial differential equation. The integral expression developed is a generalized function with the same form as the well-known D'Alembert formula. The derivatives of the solution and some useful invariants on the characteristics of the partial differential equation are also calculated.The boundary element method is applied to find the numerical solution. The results show excellent agreement with analytical solutions.A multi-step procedure for large time steps which can be used in the boundary element method is also described.In addition, the way in which boundary conditions are introduced during the time dependent process is explained in detail. In the Appendix the main properties of Dirac's delta function and the Heaviside unit step function are described.  相似文献   

10.
The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.  相似文献   

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