Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.     

Numerical integration of systems of delay differential-algebraic equations

The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

Implicit algorithms for solving the Cauchy problem for the equations of the dynamics of mechanical systems

It is shown that in the numerical solution of the Cauchy problem for systems of second-order ordinary differential equations, when solved for the highest-order derivative, it is possible to construct simple and economical implicit computational algorithms for step-by-step integration without using laborious iterative procedures based on processes of the Newton-Raphson iterative type. The initial problem must first be transformed to a new argument — the length of its integral curve. Such a transformation is carried out using an equation relating the initial parameter of the problem to the length of the integral curve. The linear acceleration method is used as an example to demonstrate the procedure of constructing an implicit algorithm using simple iterations for the numerical solution of the transformed Cauchy problem. Propositions concerning the computational properties of the iterative process are formulated and proved. Explicit estimates are given for an integration stepsize that guarantees the convergence of the simple iterations. The efficacy of the proposed procedure is demonstrated by the numerical solution of three problems. A comparative analysis is carried out of the numerical solutions obtained with and without parametrization of the initial problems in these three settings. As a qualitative test the problem of the celestial mechanics of the “Pleiades” is considered. The second example is devoted to modelling the non-linear dynamics of an elastic flexible rod fixed at one end as a cantilever and coiled in its initial (static) state into a ring by a bending moment. The third example demonstrates the numerical solution of the problem of the “unfolding” of a mechanical system consisting of three flexible rods with given control input.     

Riemann boundary-value problem with index minus infinity on a rectifiable curve

The author investigates the piecewise-continuous Riemann boundary-value problem with index minus infinity on a closed rectifiable Jordan curve; the index of the problem is a measure of the integral effect exerted on the solvability of the problem by the argument and modulus of the coefficient, and also by the properties of the junction curve. Discontinuities of the second kind are admitted in the logarithm of the coefficient and in the free term. The solution of the problem is constructed explicitly in the class of functions having a weak power singularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1350–1356, October, 1990.     

Infinite Systems of Hyperbolic Functional Differential Equations

We consider initial-value problems for infinite systems of first-order partial functional differential equations. The unknown function is the functional argument in equations and the partial derivations appear in the classical sense. A theorem on the existence of a solution and its continuous dependence upon initial data is proved. The Cauchy problem is transformed into a system of functional integral equations. The existence of a solution of this system is proved by using integral inequalities and the iterative method. Infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators.     

Reducing factorization of a semiprime number to the integration of highly oscillatory functions

We reduce the problem of factoring a semiprime integer to the problem of (numerically) integrating a certain highly oscillatory function. We provide two algorithms for addressing this problem, one based on the residue theorem and the other on the (extended) Cauchy argument principle. We show that in the former algorithm, computing the residue of the function at a certain pole leads to us obtaining the factors of the semiprime integer. In the latter, we consider a contour integral for which we are able to obtain an analytical solution with several branches. The computational difficulty reduces to that of discovering the branch of the solution which gives the precise integral. We address this problem by numerically computing an upper and a lower bound of the integral and then considering the branch that fits these bounds. The time complexity of the algorithms is left as an open problem.     

Boundary value problem for a mixed type equation with an advanced-retarded argument

We consider the Tricomi problem for the Lavrent??ev-Bitsadze equation with a mixed deviation of the argument. The uniqueness theorem for the problem is proved under constraints on the deviation of the argument. The existence of a solution is related to the solvability of a difference equation. We obtain integral representations of solutions in closed form.     

Riemann boundary problem with index plus-infinity on a rectifiable curve

A piecewise-continuous Riemann boundary problem with index plus-infinity on a closed rectifiable Jordan curve is studied; here the index of the problem takes into account the integral influence on its solvability of the argument and modulus of the coefficient of the problem and also the properties of the line of conjugation. One permits discontinuities of the second kind in the logarithm of the coefficient and the free term of the problem. The solution of the problem is constructed explicitly in the class of functions admitting a weak-power singularity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1204–1213, September, 1990.     

Boundary value problem for a mixed-type equation with a difference-differential operator

The Tricomi problem for a mixed-type equation with retarded argument in an unbounded domain is considered. The unique solvability of the problem is proved without restrictions on the delay magnitude. The existence of a solution follows from the solvability of a difference equation. Closed-form integral representations for the solutions are derived.     

An Investigation of a Pair of Integral Equations for the Biharmonic Problem

The paper contains an elementary investigation of the questionof uniqueness of a pair of integral equations connected withthe plane biharmonic problem. It is shown that for two particularexceptional geometries of the boundary curve the pair of integralequations does not have a unique solution. This defect can beremoved by adding two supplementary integral conditions whichthe solution of the integral equations must satisfy. As an illustrationthe integral equations are solved numerically with and withoutthese extra conditions.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

线性流量阀的内孔设计模型

在详细分析线性流量阀工作原理的基础上,应用平面解析几何、微积分等相关概念,给出了不存在呈严格线性的流量阀的数学论证.在设计近似线性流量阀时,首先构造了"线性误差函数"用以刻画"过流面积"与角度之间的线性误差.之后在分析内孔为对称直线、对称1/2次曲线的基础上,设计出内孔为倒"S"形内孔曲线图,通过最小化线性误差函数,得到内孔曲线的最佳参数.最后针对外孔有磨损时,给出了设计方案.     

The Isis problem as an experimental probe and teaching resource

The Isis problem, which has a link with the Isis cult of ancient Egypt, asks: “Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result.” Since the solution requires minimal technical mathematics, the problem is accessible to a wide range of students. Further, it is notable for the variety of proofs (empirically grounded, algebraic, geometrical) using different forms of argument, and their associated representations, and it provides an instrument for probing students’ ideas about proof, and the interplay between routine and adaptive expertise. A group of 39 Flemish pre-service mathematics teachers was confronted with the Isis problem. More specifically, we first asked them to solve the problem and to look for more than one solution. Second, we invited them to study five given contrasting proofs and to rank these proofs from best to worst. The results highlight a preference of many students for algebraic proofs as well as their rejection of experimentation. The potential of the problem as a teaching tool is outlined.     

Tricomi problem for an advance–delay equation of mixed type with variable deviation of the argument

We study a boundary value problem for an equation of mixed type with the Lavrent’ev–Bitsadze operator in the leading part and with variable deviation of the argument in lower-order terms. The general solution of the equation is constructed. We prove a uniqueness theorem without any conditions on the value of the deviation. The problem is uniquely solvable. We derive integral representations of the solutions in closed form in the elliptic and hyperbolic domains.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

The cauchy problem for mechanical systems with a finite number of degrees of freedom as a problem of continuation on the best parameter

The Cauchy problem for a system of ordinary differential equations is formulated as a problem of continuation on the best parameter. It is proved that the length of an integral curve of the problem is such a parameter. The merits of the proposed transformation are demonstrated by a test example in which a stiff system of equations describing the perturbed motion of an aircraft is solved numerically.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

Infinite Systems of Quasilinear Differential Difference Inequalities and Applications

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type.