System Of Integral-differential Equations Research Articles

Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach

Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

Physical-Topological Modeling of Electronic Devices with Induction Heating of Particle Emitters

The general principles of physical and topological modeling and construction of algorithms for calculations of a wide range of electronic devices with induction heating of functional elements up to temperatures at which thermoelectronic emission and/or evaporation of the working substance in the atomized state are considered. The direction of physical-topological modeling was chosen due to the possibility of detailed analysis, based on the primary principles for related physical processes in devices, taking into account the impact of physical properties of functional elements and their design and topological (i.e. geometric) parameters. Thus, the results are obtained by solving a system of fundamental equations, which usually include Newton's and Maxwell's equations, the conservation laws of particles, charge, energy and momentum, as well as material properties, boundary and initial conditions. The set of equations is determined by the number of processes that significantly affect the operation of devices. The construction of the device model is based on the consistent hierarchy of elementary physical processes with the sequential transfer of the results of calculations for lower-level models to higher-level models in the form of initial conditions. The original mathematical models of elementary processes are represented by systems of integral-differential equations with partial derivatives in continuous space and continuous time and belong to the class of distributed mathematical models. Methods for solving the system of equations are considered. On the example of a vacuum metal evaporator with induction heating for thermovacuum coating deposition, the procedure of decomposition of the general physical process in the evaporator with a concentrator as a step-down transformer is given and the hierarchy of elementary processes is clarified. Typical initial and boundary conditions for calculating related physical processes are determined. The available application computer soft packages for the calculation of physical and topological models of various induction devices are considered. In the final part of the article it is considered the structure and structure of physical-topological models of devices with induction heating of particle emitters: i) the thermoionic metal evaporator with ionization of vapor by electrons emitted by a thermocathode from an alloy with low electron output, the cathode is made in the form of an insert on the upper end of the crucible, and ii) the X-ray tube with an inductively heated thermoelectron cathode. Calculation results of the electromagnetic field and current distribution, heat exchange in vapor and electron emitters and their emission fluxes, as well as the trajectories of emitted electrons are presented. Analysis of electron trajectories allowed to optimize the topology and design of these devices.

Об одном методе численного решения вырожденных интегро-дифференциальных уравнений со слабой особенностью в ядре

Statements of many applied problems often include differential equations and Volterra integral equations of the first and second kind. By joining such equations together, we obtain a system of integral differential equations with a singular matrix multiplying the leading part. Such systems are commonly referred to as singular integral differential equations. If they do not contain an integral part, then they are called differential-algebraic equations. If there is no term with a derivative, then they are usually called integral algebraic equations. Such mathematical problem statements arise in simulation of processes occurring in electrical and hydraulic circuits, various dynamic systems, in particular, multibody systems. Therefore, qualitative study and numerical solution of such problems are quite relevant, and the results of research remain in demand in practice. In this paper, on the basis of the theory of matrix pencils, as well as using research schemes developed for differential algebraic and integral algebraic equations, the conditions for the existence and uniqueness of the solution of singular integral-differential equations with a weakly singular kernels are analyzed and a numerical method for their solution is proposed. The method was coded in MATLAB and tested on model examples.

UNAMBIGUOUS SOLVABILITY OF A PARTICULAR CASE OF SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS WITH A PULSED KAEV DISTANCE CONTAINING A PARAMETER

We consider a special case of systems of integro-differential equations with a momentum boundary condition containing a parameter when the derivative of the desired function is contained in the right side of the equation. By integrating in parts, an integro-differential equation with a pulsed boundary condition is reduced to a loaded integrodifferential equation with a pulsed boundary condition. it is given in the system of integral-differential equations with impulse boundary conditions parametrically loaded. Then, by entering new parameters, as well as passing to new variables based on these parameters, the problem is reduced to an equivalent problem. Switching to new variables makes it possible to get the initial conditions for the equation. Based on this, the solution of the problem is reduced to solving a special Cauchy problem and a system of linear equations. Using the fundamental matrix of the main part of the differential equation, an integral equation of the Volterra type is obtained. The method of sequential approximation determines the unique solution of the integral equation. Based on this, we find a solution to the special Cauchy problem and put it in the boundary conditions. On the basis of the obtained system of linear equations, necessary and sufficient conditions for an unambiguous solution of the initial problem are established.

On mathematical modelling of temporal spatial spread of epidemics

In the present work a generalized epidemic model containing a system of integral-differential equations is described. Using different transformations the system is reduced to a single nonlinear multidimensional integral equation. For the obtained equation the existence and uniqueness results are proved. Based on theoretical convergence results several application examples are presented with corresponding numerical results.

Комплексы эллиптических цилиндров с характеристическим многообразием образующего элемента в виде координатных прямых

The complex (three-parameter family) of elliptic cylinders is investigated in the three-dimensional affine space, in which the characteristic multiplicity of the forming element consists of three coordinate axes. The focal variety of the forming element of the considered variety is geometrically characterized. Geometric properties of the complex under study were obtained. It is shown that the studied manifold exists and is determined by a completely integrable system of differential equations. It is proved that the focal variety of the forming element of the complex consists of four geometrically characterized points. The center of the ray of the straight-line congruence of the axes of the cylinder, the indicatrix of the second coordinate vector, the second coordinate line and one of the coordinate planes are fixed. The indicatrix of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the second coordinate vector. The end of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the third coordinate vector. The indicatrix of the third coordinate vector and its end describe congruences of planes parallel to the first coordinate plane. The points of the first coordinate line and the first coordinate plane describe one-parameter families of planes parallel to the coordinate plane indicated above.

Correlation functions of Coulomb branch operators

We consider the correlation functions of Coulomb branch operators in four-dimensional mathcal{N} = 2 Superconformal Field Theories (SCFTs) involving exactly one antichiral operator. These extremal correlators are the “minimal” non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt∗ equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in mathcal{N} = 2 SQCD.

Linear inverse problem of the reactor dynamics

The aim of this work is the study transient processes in nuclear reactors. The mathematical model of the reactor dynamics excluding reverse thermal coupling is investigated. This model is described by a system of integral-differential equations, consisting of a non-stationary anisotropic multispeed kinetic transport equation and a delayed neutron balance equation. An inverse problem was formulated to determine the stationary part of the function source along with the solution of the direct problem. The author obtained sufficient conditions for the existence and uniqueness of a generalized solution of this inverse problem.

Волновая модель распространения и изменения лингвистической информации в индоевропейском модельном языковом сообществе

Paper considers the wave mathematical model describing the distribution and change of linguistic information in Indo-European model linguistic community. Brief information about the Anatolian and Kurgan hypothesis of formation of Proto-Indo-Europeans (PIE) is given. The mathematical model of wave propagation and changing of information is described by system of integral-differential equations. Results of the preliminary theoretical analysis and computer modeling are given. In particular, an estimate of the maximum possible time (24000 years) of development/emergence of considered Indo-European “family” is obtained in the framework of this model. Results of computer modeling show, that from two main hypotheses of formation of the PIE - Anatolian and Kurgan, the latter better matches obtained by us temporary estimates. Finally, data on a hypothetical PIE alphabet - namely, the number of possible “characters/symbols” (or “letters”) alphabet of PIE on the basis of our data were obtained. Based on the results, it was found that the range of possible values for alphabetic characters is limited to about values: 3 32. In addition, there is a good correlation of our data with the findings of geneticists on a possible time of occurrence (14000-20000 years ago) of the haplogroup Y-DNA R1a (it is believed that PIE language speakers were primarily men with this haplogroup).

A method of determination of port terminal capacity under irregular cargo delivery and pickup

There was built up and analyzed a stochastic model of a work of a port terminal that takes into consideration irregularity of delivery and pickup of a cargo. It is supposed that a terminal consists of n interchangeable moorages, in which there is carried out loading to ships. The ships arrive to a terminal to take a cargo independently on each other, their total number is equal to N. Time from departure of any loaded ship to the moment of its arrival to a terminal is a random variable that is distributed according to the exponentional law. All cargoes, that come to a terminal with a help of land transport, are immediately unloaded to a storehouse. It is supposed that a stream of incoming cargoes is described with a model of the compound Poisson process with zero drift. From a storehouse cargoes are loaded to any shipl that is in a moorage, with the rate W. With use of non-standard type of the Markov process with drift for finding of limit join distribution of number of ships, that are in moorages, and amount of cargo, that is in a storehouse, there is got a system of integral-differential equations together with relevant boundary conditions. There is given a method of solving of this boundary-value problem, that is based on use of the Laplace-Stieltjes transformation for getting of a solution in a closed form. It gives a possibility to get simple calculation formulae for assessment of indices of capacity of a terminal: the average number of ships in moorages, the average amount of cargo in a storehouse, possibility of demurrage of ships because of absence of cargoes in a storehouse and etc. There are given examples of practical use of the got theoretical results, namely: a method of calculation of necessary capacity of a storehouse, assessment of a term of recoupment of a project of construction of a terminal. They showed that the worked out method of calculation of capacity of a port terminal in conditions of irregularity of a work of transport can be used in project calculations.

Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

Poisson and integrable systems are orbitally equivalent through the Nambu bracket. Namely, we show that every completely integrable system of differential equations may be expressed into the Poisson-Hamiltonian formalism by means of the Nambu-Hamilton equations of motion and a reparametrisation related by the Jacobian multiplier. The equations of motion provide a natural way for finding the Jacobian multiplier. As a consequence, we partially give an alternative proof of a recent theorem in [13]. We complete this work presenting some features associated to Hamiltonian maximally superintegrable systems.

Investigation of the Unsteady-State Hydraulic Networks by Means of Singular Systems of Integral Differential Equations

E. V. Chistyakova, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russian Federation, chistyak@gmail.com, Nguyen Due Bang, National Research Irkutsk State Technical University, Irkutsk, Russian Federation, ducbang@mail.ru Елена Викторовна Чистякова, кандидат физико-математических наук, научный сотрудник, Институт динамики систем и теории управления имени В.М. Матросова СО РАН (г. Иркутск, Российская Федерация), chistyak@gmail.com. Нгуен Дык Ванг, аспирант, Национальный исследовательский Иркутский го-сударственный технический университет (г. Иркутск, Российская Федерация), ducbang@mail.ru.

Credit line takedown and endogenous bank capital

This paper investigates the implications of the uncertain timing and usage of loan commitments for the optimal level of bank capital. We use trended Brownian motion to proxy the stochastic takedown of credit lines. Relying on “time to first passage” mathematics, we derive a probability density function for the time to depletion of the bank credit line as well as the likelihood for the time to exhausting the sources of liquidity that fund the loan takedown. Armed with these analytical results, we solve for the optimal level of bank capital within a simultaneous equation framework in order to capture the interrelationships of the endogenous variables. The optimality conditions produce a system of integral differential equations which refuse to yield reduced form solutions and provide no immediate intuition. Therefore, the maximizing values of the bank’s decision variables were simulated over a host of realistic scenarios. We document the comparative static behavior of the bank’s decision variables when equity is unencumbered by capital requirements and, also, examine the impact of the same parametric changes on bank behavior when equity is a fixed proportion of lending. Further simulations produce the expected time to liquidity depletion under different capital requirement schemes.

Regularizing properties of difference schemes for singular integral-differential equations

Systems of integral-differential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integral-differential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.

The solution of system of integral differential equations and its application in the theory of elasticity

AbstractA contact problem of the theory of elasticity of infinite compound and infinite orthotropic plates with an elastic semi‐infinite or finite inclusion of variable rigidity is considered. The problem is reduced to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, using the methods of analytical functions to get exact solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion. In some case using the method of orthogonal polynomials we obtain the dual infinite system of linear algebraic equations. We can manage to investigate the obtained system on the quasi‐regularity and the method of reduction for approximate solution is developed.

A computational approach to pricing a bank credit line

Using trended Brownian motion to characterize borrower cash needs over time, we are able to derive a probability density function for the time to depletion of a bank credit line as well as the likelihoods for the time to exhausting the sources of liquidity that fund the loan. Armed with these analytic results, we solve for the credit line mark-up rate and the configuration of stored liquidity that maximizes the bank’s intertemporal expected profits from the loan. The optimality conditions produce a system of integral differential equations whose solutions we then simulate over a host of scenarios.

Asymptotic solution of the mathematical model of nonlinear oscillations of absolutely elastic inextensible weightless string

The mathematical model of nonlinear oscillations of weightless string is analyzed. To find an asymptotic solution of the problem, uniformly valid in a long interval of time, an averaged system of integral differential equations has been constructed. A method for constructing special approximations of its solutions is proposed.

Specific features of families of invariant manifolds of conservative systems

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.

Neural field model of rat's cortex based on realistic connectivity from diffusion weighted MRI and neural morphology

Generative models of neural circuits may help to create a link between neural mechanisms and observable data. We propose a model of rat's cortex using a neural field model containing biologically plausible anatomical connections from tractography based on dwMRI data and from the neural morphological database NeuroMorpho [1]. There are three principal types of anatomical connections in the cortex: Local, long-range and distal connections [2]. For specifying local connections we use neural morphologies from [1]. We consider each voxel in the model as a neural mass and distribute randomly drawn neurons from the database therein. After that we use bootstrap methods to determine the total number and variability of synaptic contacts. For the distal connectivity we estimated the degree of anatomical connectedness using white matter tractography on the basis of diffusion weighted MRI [3]. Our neural field consists of 5 layers. For each layer we assume three different neural masses: pyramidal cells, excitatory and inhibitory interneurons [4]. The mutual interactions between neural masses will be described by a system of integral differential equations: where V is the state vector (mean membrane potentials), T the time delays in the dendritic arbors, S the sigmoidal output function, W the connection coefficients between the neural elements, I the input, and t(d) is the distance dependent time delays. Figure ​Figure1A1A shows the estimated spatial dependency of local connectivity, which is in accordance with anatomical observations [2]. The distal connectivity map of 28 regions is displayed in Fig. ​Fig.1B.1B. An example map of simulated activity on the cortex in response to a stimulus to somatosensory cortex is illustrated in Fig. ​Fig.1C.1C. Figure 1 (A) Local connectivity profile between a pyramidal cell in layer V and an interneuron in layer III (grid size 250 µm). (B) Distal connectivity map estimated from dwMRI between 28 regions. (C) Spatio-temporal activity pattern 100 ms after the onset ... To summarize, we developed a method for estimating the local connectivity and constructed a neural field model of the entire cortex enriched by estimated local and distal connectivities. With this model we are able to simulate spatio-temporal activity patterns. This is a first step for a comprehensive dynamic brain model and thereby for understanding complex brain processes.

UMA DISCUSSÃO SOBRE MODELOS MECÂNICOS DE LARINGE PARA SÍNTESE DE VOGAIS

UMA DISCUSSÃO SOBRE MODELOS MECÂNICOS DE LARINGE PARA SÍNTESE DE VOGAIS