Nonlinear spectral problem for a Hamiltonian system of differential equations with redundant conditions

We consider a nonlinear spectral problem for a self-adjoint Hamiltonian system of differential equations. The boundary conditions correspond to a self-adjoint problem. It is assumed that the input data (the matrix of the system and the matrices of the boundary conditions) satisfy certain conditions of monotonicity with respect to the spectral parameter. In addition to the main boundary conditions, a redundant nonlocal condition given by a Stieltjes integral is imposed on the solution. For the nontrivial solvability of the over-determined problem thus obtained, the original problem is replaced by an auxiliary problem that is consistent with the entire set of conditions. This auxiliary problem is obtained from the original one by allowing a discrepancy of a specific form. We study the resulting problem and give a numerical method for its solution.     

On numerical study of periodic solutions of a delay equation in biological models

Some results are presented of the numerical study of periodic solutions of a nonlinear equation with a delayed argument in connection with themathematical models having real biological prototypes. The problem is formulated as a boundary value problem for a delay equation with the conditions of periodicity and transversality. A spline-collocation finite-difference scheme of the boundary value problem using a Hermitian interpolation cubic spline of the class C ¹ with fourth order error is proposed. For the numerical study of the system of nonlinear equations of the finitedifference scheme, the parameter continuation method is used, which allows us to identify possible nonuniqueness of the solution of the boundary value problem and, hence, the nonuniqueness of periodic solutions regardless of their stability. By examples it is shown that the periodic oscillations occur for the parameter values specific to the real molecular-genetic systems of higher species, for which the principle of delay is quite easy to implement.     

Computing eigenvalues of Sturm-Liouville problems via optimal control theory

In this paper, we shall address three problems arising in the computation of eigenvalues of Sturm-Liouville boundary value problems. We first consider a well-posed Sturm-Liouville problem with discrete and distinct spectrum. For this problem, we shall show that the eigenvalues can be computed by solving for the zeros of the boundary condition at the terminal point as a function of the eigenvalue. In the second problem, we shall consider the case where some coefficients and parameters in the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coefficients and parameters by reformulating the problem as a combined optimal control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal control software, MISER. The last problem extends the first to nonstandard boundary conditions such as periodic or interrelated boundary conditions. To illustrate the efficiency and the versatility of the proposed methods, several non-trivial numerical examples are included.     

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.     

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.     

A self-adaptive projection method for contact problems with the BEM

A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.     

Analysis of Numerical Differentiation Formulas in a Boundary Layer on a Shishkin Grid

A problem of numerical differentiation of functions with large gradients in a boundary layer is investigated. The problem is that for functions with large gradients and a uniform grid the relative error of the classical difference formulas for derivatives may be considerable. It is proposed to use a Shishkin grid to obtain a relative error of the formulas that is independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform with respect to the small parameter. In the case of a uniform grid, a boundary layer region is indicated outside of which the numerical differentiation formulas have an error that is uniform with respect to the small parameter. The results of numerical experiments are presented.     

Irregular Boundary Value Problems with Discontinuous Coefficients and the Eigenvalue Parameter

In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation is also studied.     

Identifiability analysis of an epidemiological model in a structured population

We investigate the parameter identifiability problem for a system of nonlinear integro-partial differential equations of transport type, representing the spread of a disease with a long infectious but undetectable period in an individual population. After obtaining the expression of the model input-output relationships, we give sufficient conditions on the initial and boundary conditions of the system that guarantee the parameter identifiability on a finite time horizon. We finally illustrate our findings with numerical simulations.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.     

Numerical Leak Detection in a Pipeline Network of Complex Structure with Unsteady Flow

An inverse problem for a pipeline network of complex loopback structure is solved numerically. The problem is to determine the locations and amounts of leaks from unsteady flow characteristics measured at some pipeline points. The features of the problem include impulse functions involved in a system of hyperbolic differential equations, the absence of classical initial conditions, and boundary conditions specified as nonseparated relations between the states at the endpoints of adjacent pipeline segments. The problem is reduced to a parametric optimal control problem without initial conditions, but with nonseparated boundary conditions. The latter problem is solved by applying first-order optimization methods. Results of numerical experiments are presented.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

The artificial boundary conditions for incompressible materials on an unbounded domain

Summary. In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective. ReceivedJune 7, 1995 / Revised version received August 19, 1996     

On a family of initial-boundary value problems for the heat equation

We consider the heat problem with nonlocal boundary conditions containing a real parameter. For the zero value of the parameter, this problem is well known as the Samarskii-Ionkin problem and has been comprehensively studied. We analyze the spectral problem for the operator of second derivative subjected to the boundary conditions of the original problem. By separation of variables, we prove the existence and uniqueness of a classical solution for any nonzero value of the parameter. The obtained a priori estimates for a solution imply the stability of the problem with respect to the initial data.     

Local Artificial Boundary Conditions for the Incompressible Viscous Flow in a Slip Channel

1.IntroductionManyboundaxyvaJueproblemsofpartialdiffereotialequationsinvo1vingunboundeddomainoccurinmanyareasofapplications,e-g.lfluidflowaroundobstacles,couplingofstructureswithfoundationandsoon.Forgettingthenumericalsolutionsoftheproblemsonunboundeddomian,anaturalapproachistocutoffanunboundedpartofthedomainbyintroducinganartificialboundaryandsetupanaPpropriatear-tificialboundaryconditiononthearti%ialboundaryThentheoriginalproblemisapproximatedbyaproblemonbou.d.dfdomain.Inthelastteny6aJrs,b…     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

Numerical solution to a linearized KdV equation on unbounded domain

Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial‐boundary value problem defined only on a finite interval. A dual‐Petrov‐Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

DTM-BF方法和可渗透收缩壁面上带滑移速度的非稳态磁流体力学流动

研究了运动的粘性导电流体中可渗透收缩壁面上非稳态磁流体边界层流动,利用解析和数值方法对问题进行了研究,并考虑了壁面速度滑移的影响．提出了一个新的解析方法（DTM-BF）,并将其应用于求解带有无穷远边界条件的非线性控制方程的近似解析解．对所有的解析结果和数值结果进行了对比,结果显示两者非常吻合,从而证明了DTM-BF方法的有效性．另外,对不同的参数,得到了控制方程双解和单解的存在范围．最后,分别讨论了滑移参数、非稳态参数、磁场参数、抽吸/喷注参数和速度比例参数对壁面摩擦、唯一解速度分布和双解速度分布的影响．