A complete uniform asymptotic expansion is found for the integral ?S ?2 udxdy, where u is the solution of the Neumann problem with a delta-function-like derivative on the boundary. A physical application of the result is discussed.

On measurement of electrical conductivity

A uniform asymptotic expansion of the solution of a two-dimensional elliptic problem with mixed boundary conditions is found. A physical application of the result is discussed.

Asymptotics of the solution of Laplace’s equation with mixed boundary conditions

The electric connection of crystals in polycrystalline to locate the contact points on the surface has been simulated using the solution of the boundary problem for an elliptic equation A chain model and the tensor components of the quasi-monocrystalline graphite conductance have been used to calculate the size of the plates artificial graphite We consider the parameters influencing its value and the kind of temperature dependence of the specific electric resistance and evaluate the integral dissipation of electric energy along and across the layers

Simulation of the electric current flow in artificial graphite

В статье приведены результаты математического моделирования протекания электрического тока в пластинчатом поликристалле графита в магнитном поле. В отличие от простой модели с учетом только контактного сопротивления между чешуйками, в настоящей модели также учтено омическое сопротивление вдоль и поперек
слоeв графита на основе цепочечной модели протекания электрического тока в поликристалле графита. В результате получена формула с уточненными поправочными коэффициентами, которая позволяет подобрать параметры поликристалла
для расчета температурной зависимости УЭС и магнетосопротивления. Проведено
сравнение с простой моделью, учитывающей только контактное сопротивление, различие составляет до 50 % от прежней оценки. Отличие отнесено к особенностям
подключения отдельных кристаллов в поликристалле. Это позволяет анализировать размеры кристаллов и их подключения в искусственном графите исходя из
электрофизических свойств.

Определение размеров блоков мозаики и анизометрии чешуек искусственного графита по магнетосопротивлению

We consider a conductive body in the form of a parallelepiped with small square contacts attached to its ends. The potential of the electric current is modeled by a boundary value problem for the Laplace equation in a parallelepiped. The zero normal derivative is assigned on the boundary except for the areas under the contacts, where the derivative is a nonzero constant. Physically, this condition corresponds to the presence of a low-conductivity film on the surface of the contacts. The problem is solved by separation of variables, and then the electrical resistance is found as a functional of the solution in the form of the sum of a double series. Our main aim is to study the dependence of the resistance on a small parameter characterizing the size of the contacts. The leading term of the asymptotics that expresses this dependence is the contact resistance. The mathematical problem is to treat the singular dependence of the sum of the series corresponding to the resistance on the small parameter: the series diverges as the small parameter vanishes. We solve this problem by replacing the series with a two-dimensional integral. We find the leading term of the asymptotics and estimate the remainder. It turns out that the main contribution to the remainder is made by the difference between the two-dimensional integral and the double sum.

Contact Resistance of a Square Contact

A uniform asymptotic expansion is found for the integral ∫∫s▽2udxdy, where u is the solution of the Neumann problem with a delta-function-like derivative on the boundary. A physics application of the result is discussed.

Asymptotics of the solution to the Neumann problem with a delta-function-like boundary function

A harmonic function is considered in a three-dimensional bounded domain. Its normal derivative is given on nearly the entire boundary of the domain, while the value of the harmonic function is specified on the remaining small portion. The method of matched asymptotic expansions is used to construct a complete uniform asymptotic expansion of the function in powers of a small parameter characterizing the size of the boundary portion with a specified function value. The asymptotic expansion is rigorously substantiated.

A. A. Ershov

Papers

On measurement of electrical conductivity

Asymptotics of the solution to the Neumann problem with a delta-function-like boundary function

Asymptotics of the solution of Laplace’s equation with mixed boundary conditions

Mixed problem for a harmonic function