More on the Problem and the Solution of the Optimal Shape of a Javelin

The analogy between the optimal javelin problem and the problem of determining the optimal shape of the free rotating rod has been established and employed to determine the optimal shape of the javelin via Pontryagin's maximum principle. Five distinct variational principles are constructed for boundary value problem describing optimal shape of the javelin. The first integral for this nonlinear system is found. An a priori estimate of the cross-sectional area is obtained. The optimal shape of the javelin or free rotating rod is determined by numerical integration.     

Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.     

Riemann–Hilbert-type Boundary Value Problems on a Half Hexagon

In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann–Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed.Finally, we investigate the Schwarz problem and the Riemann–Hilbert problem for inhomogeneous Cauchy–Riemann equations.     

Optimum shape of columns with general conservative end loading

In this paper, the optimal shape of a column with the most general conservative state of loading is determined. The boundary conditions of a column contain, in general, 16 parameters; it is shown that the boundary values of a column with general conservative loading may, at most, depend on nine constants. The influence of these parameters on the optimal shape and corresponding buckling load is also obtained.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.     

Shape derivatives in differential forms I: an intrinsic perspective

We treat Zolésio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher-order shape derivatives of domain and boundary integrals and avoids the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective of differential forms perfectly fits second-order boundary value problems (BVPs). We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic BVPs with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions.     

Electroencephalography in ellipsoidal geometry

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy’s type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.

     

边界元近奇异积分计算的迭代Sinh-Sigmoidal组合式变换法

精确有效地消除积分的近奇异性是三维边界元法在工程应用中的首要问题.当源点与三角形积分单元间的距离无限趋近于零时,会出现近奇异积分问题,积分单元的形状和投影点的位置都是影响近奇异积分计算精度的重要因素.现有的非线性变换法大多只关注径向上积分的近奇异性,而忽略了角度方向和积分单元形状的影响,在投影点接近三角形积分单元边界的情况下,无法获得令人满意的计算精度,并且对子三角形积分单元的形状非常敏感.因此提出了一种改进的基于自适应分块技术和不同坐标变换的迭代sinh sigmoidal组合式变换法,分别消除径向和角度方向积分的近奇异性,在确保计算精度的同时,大大减小了计算规模.数值算例验证了该方法的有效性.     

Error estimates for the finite element solution of elliptic problems with nonlinear newton boundary conditions

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous pioblem is a consequence of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numeriral integration for the evaluation of boundary integrals. The error estimates for the solution of the discrete finite element problem are derived     

Optimal shape analysis of a column structure under various loading conditions by using differential transform method (DTM)

The effects of loading on the optimal shape of an Euler-Bernoulli column is investigated by considering four loading conditions which are mainly classified as eccentric compressive and follower type. The governing equations obtained from the structural stability condition of the column are used as a constraint to determine the minimum value of the volume by applying Hamilton principle. In the preceding task, the analysis is presented in step-by-step manner. The calculations are carried out by using differential transform method (DTM) which is a seminumerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the non-linear constrained governing equations are reduced to recurrence relations and related boundary conditions are transformed into a set of algebraic equations. The optimal distribution of cross-sectional area along column-length is obtained. Then, the volume of such column is calculated and compared to that of the uniform column which is also stable under given loading. The results obtained revealed out that DTM is a quite powerful solution technique for optimal shape analysis of a column structure.     

Integral equations for the mixed boundary value problem in plane elastostatics

Six formulations of the mixed boundary value problem of plane elastostatics integral equations are presented. All equations are of purely second kind and are characterized by a uniform structure of the kernels with respect to geometrical and statical boundary values. The kernels of two formulations are regular, the remaining formulations contain Cauchy principal value integrals.     

Optimal boundary control by displacement at one end of a string under a given elastic force at the other end

The problem of optimal boundary control by displacement at one end of a string under a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from an infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved.     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

Positive solutions to a system of second-order nonlocal boundary value problems

In this paper we study the existence of positive solutions to a system of second-order nonlocal boundary value problems by using fixed point index theory in a cone. Our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems, and our boundary value conditions are expressed in terms of possibly nonlinear functions of Riemann–Stieltjes integrals, thus generalizing and unifying the boundary value conditions in the literature. Therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature.     

A variational method applied to a linear hyperbolic initial boundary value problem

This paper deals with the application of a variational method to a boundary value problem of the wave equation. Starting with an initial boundary value problem (which is given) introduction of a boundary condition at the final time leads to a boundary value problem with one of the initial conditions redundant. This redundant initial condition is used by the trial function of the direct method (of the Ritz type) which is employed to stationarize the variational principle.     

Optimal boundary force control at one end of a string for a given displacement mode at the other end

We solve the problem of optimal boundary force control at one end of a string for the case of a given displacement mode at the other end. The problem is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control from infinitely many feasible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and the uniqueness of the solution is proved.     

Invariant imbedding,iterative linearization,and multistage countercurrent processes. VI. Complex distillation column and multipoint boundary value problem

The complex distillation column problem is formulated as a multipoint boundary value problem in difference equations. It is shown that this nonlinear multipoint boundary value problem can be solved easily by the quasilinearization technique. With very rough initial approximations, only ten iterations are needed to obtain a four digit accuracy in concentrations. The advantage of this approach is that the complex column can be solved in essentially the same way as the simple column with approximately the same computation requirements.     

Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers

A boundary value method for solving a class of nonlinear singularly perturbed two point boundary value problems with a boundary layer at one end is proposed. Using singular perturbation analysis the method consists of solving two problems; namely, a reduced problem and a boundary layer correction problem. We use Pade’ approximation to obtain the solution of the latter problem and to satisfy the condition at infinity. Numerical examples will be given to illustrate the method.