In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized ...solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media.
Abstract
The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations of convolution type in Banach spaces. The principal idea of the method ...proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination of lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and, in the case of existence of the latter, the method coincides with this technique. A generalized solution is constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and the function, which includes both the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data will provide for the resolvability of the initial problem in the class of functions characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved. The abstract results have been illustrated via examples of initial-boundary value problems of applied character (from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates).
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This paper presents the numerical methods for calculating the hydrodynamic characteristics of a mechanical seal, taking into account the possibility of a phase transition in its gap. In the ...numerical model, the solutions are used for the flow of both liquid and gas in the microgap of compaction obtained using the finite volume method. Regularities of the influence of the size and taper of the gap, the depth of the microgrooves, the pressure drop and the value of the saturated vapor pressure on the hydrodynamic characteristics of the seal are revealed. A method has been developed for calculating leaks, the stiffness of the lubricant layer, and the friction power in the seal gap, taking into account vaporization and thermal deformations of the sealing rings. Recommendations for the selection of the geometric parameters of the seal are offered. The cause of pulsating seal leakage has been identified. The developed technique is applicable both for cryogenic liquids and for water.
A sample of more than one million K +/- ? p + p - e +/- ? (K e 4) decay candidates with less than one percent background contamination has been collected by the NA48/2 experiment at the CERN SPS in ...2003-2004, allowing a detailed study of the decay properties. The branching ratio, inclusive of K e 4 gamma decays, is measured to be BR (K e 4) = (4.257 +/- 0.016 exp +/- 0.031 ext) X 10 - 5 with a total relative error of 0.8 % . This measurement complements the study of S- and P-wave hadronic form factors by assigning absolute values to the relative hadronic form factors obtained earlier in a simultaneous analysis of the pp scattering lengths conducted on the same data sample. The overall form factor normalization f s = 5.705 +/- 0.017 exp +/- 0.031 ext is obtained with a total relative precision of 0.6 % .
The paper considers the initial value problem for a differential equation with the derivatives of the functionals in Banach spaces. The operator of the elder derivative has the structure of ...projector, i.e. its core is infinite-dimensional. The solution is constructed in the space of generalized functions with the support bounded on the left in the form of convolution of the fundamental solution of the differential operator with the right-hand side of the equation, which includes a free function and some initial conditions of the initial problem. The process of construction of the fundamental solution is realized with the aid of a fundamental operator function for a specially constructed matrix differential operator with an irreversible (generally speaking) matrix in the derivative, i.e. with the operator of finite index. Analysis of the generalized solution constructed by this technique allows one to obtain the sufficient conditions of solvability for our initial-value problem in the classes of finite smoothness functions, and also propose constructive formulas needed to restore such a solution. An illustrative example is given.
In this paper, using methods of the theory of generalized functions in Banach spaces, we examine the Cauchy problem for an abstract integrodifferential equation of a specific type. Under the ...assumption that there exist the complete Jordan structure of the differential part of the equation and the order of the zero of the kernel of the integral part, the fundamental operator-valued function (the fundamental solution) is constructed for the corresponding integrodifferential operator, which is used for further investigation of the problem.
Pion–Muon separation with a RICH prototype for the NA62 experiment Angelucci, B.; Anzivino, G.; Avanzini, C. ...
Nuclear instruments & methods in physics research. Section A, Accelerators, spectrometers, detectors and associated equipment,
09/2010, Letnik:
621, Številka:
1
Journal Article
Recenzirano
The NA62 experiment at CERN, aimed to measure
K
+
→
π
+
ν
ν
¯
branching fraction (O(10
−10)), relies on a Neon based RICH detector for
π
/
μ
separation, time measurement and level 0 trigger. The ...experimental requirements for this detector are: a muon contamination in pion samples lower than 5×10
−3 in the momentum range 15–35
GeV/
c and a time resolution on the charged track better than 100
ps. A prototype of such a detector was built and tested in 2009; it consists of a full length (
≈
18
m
) Ne filled vessel equipped with a spherical mirror and 414 PMs on its focal plane, located about 17
m upstream of the mirror. This prototype was tested at CERN SPS on a positive hadron beam, in the required momentum range, to measure the
π
/
μ
separation and to confirm the time resolution obtained with a previous prototype; the
μ
misidentification probability is about 0.7% and the time resolution is better than 100
ps in the whole momentum range.
The possibility of reducing friction torque in face seals is shown based on theoretical and experimental studies. The operational, technological, and constructive means and techniques of reducing the ...friction torque in face seals are determined. The most economically promising methods for reducing the friction torque are revealed.
In this paper, we present a simulation model of a contact face seal for a multimode turbomachine in which the processes of hydrodynamics, deformations, and wear of sealing surfaces are considered in ...common. The effect of the gap shape on the characteristics of the contact face seal is studied. It was revealed that in the multimode operation of a turbomachine, the combined action of deformations and wear of the contacting sealing surfaces can qualitatively change the gap shape, as well as the contact pressure, heat release, and the amount of leakage. When calculating seals, it is necessary to take into account the history of loading to predict the lifetime and the tightness level of the seals. Complex gap shapes that arise when changing the operating mode of the turbomachine lead to significant leaks of the operating fluid and intense wear. The created software package, which allows simulating the operation of a multimode contact face seal and determining the specified parameters of the operating modes of the turbomachine to the seal life up to a certain limit, as well as the change in temperature, friction power, and leakage through the seal during the operating time, is described. The simulation model can be used to develop methods for calculating the characteristics of a seal, estimate the degree of influence on the characteristics of operational and design factors, develop accelerated test methods with a joint consideration of deformations and wear of sealing surfaces. The use of the model at the stage of fine-tuning the design of the sealing unit and the assignment of operating modes will increase the information content of tests and reduce design costs by reducing the volume of experimental research. The structure of the simulation model allows one to change individual modules by choosing the most suitable design model and allows both simulation and full-scale modeling when individual physical processes during the calculation process will be simulated on bench installations and the results are transmitted to the calculation cycle.
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