Loaded equations, arising during synthesis of control of rod heating by moving sources

2021 ◽  
Vol 21 (2) ◽  
pp. 15-27
Author(s):  
V.A. Hashimov ◽  
Keyword(s):  
Differential Equation ◽  
Parabolic Type ◽  
Optimization Methods ◽  
Point Sources ◽  
Linear Feedback ◽  
Heat Sources ◽  
Right Hand ◽  
Gradient Of The Functional ◽  
The Right ◽  
The Given

The article proposes an approach to solving the problem of synthesis of motion and power control of lumped sources. For concreteness, the problem of linear feedback control of moving heat sources during heating of the rod is considered. The powers and motion of point sources, participating in the right-hand side of the differential equation of parabolic type, are determined depending on the measured values of the process state at the points of measurement. As a result, the right-hand side of the differential equation linearly depends on the values of the process state at the given points of the bar. Formulas for the components of the gradient of the functional with respect to the parameters of linear feedback are obtained, which make it possible to use first-order optimization methods for the numerical solution of synthesis problems.

A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

2019 ◽  
Vol 16 (07) ◽  
pp. 1850115 ◽  
Author(s):  
Nizami A. Gasilov ◽  
Müjdat Kaya
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Boundary Value ◽  
Real Life ◽  
Boundary Values ◽  
Right Hand ◽  
Linear Differential ◽  
The Right

In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from [Formula: see text] to [Formula: see text] through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.

scholarly journals Double Controlled Metric Type Spaces and Some Fixed Point Results

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 320 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Nabil Mlaiki ◽  
Hassen Aydi ◽  
Nizar Souayah
Keyword(s):  
Fixed Point ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Type Space ◽  
Control Functions ◽  
Right Hand ◽  
Type Spaces ◽  
The Right ◽  
The Given

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle inequality, that is, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z ∈ X . Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and ϕ -nonlinear type contractions in the setting of double controlled metric type spaces.

scholarly journals On nonlinear boundary value problems with deviating arguments and discontinuous right hand side

1993 ◽  
Vol 6 (1) ◽  
pp. 83-91
Author(s):  
B. C. Dhage ◽  
S. Heikkilä
Keyword(s):  
Differential Equation ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Deviating Arguments ◽  
Deviating Argument ◽  
Right Hand ◽  
Nonlinear Boundary Value ◽  
The Right ◽  
Generalized Iteration

In this paper we shall study the existence of the extremal solutions of a nonlinear boundary value problem of a second order differential equation with general Dirichlet/Neumann form boundary conditions. The right hand side of the differential equation is assumed to contain a deviating argument, and it is allowed to possess discontinuities in all the variables. The proof is based on a generalized iteration method.

Further Notes on the Sculpture of the Later Temple of Artemis at Ephesus

1914 ◽  
Vol 34 ◽  
pp. 76-88
Author(s):  
W. R. Lethaby
Keyword(s):  
British Museum ◽  
Female Figure ◽  
Left Hand ◽  
Right Hand ◽  
The Subject ◽  
Symmetrical Group ◽  
The Right ◽  
Vertical Joint ◽  
The Given

The Square Pedestals.—In some notes on the sculpture from the Artemision at the British Museum, printed in the last volume of this Journal (p. 87), I suggested that the fragment No. 1201 most probably belonged to a relief representing either Herakles in the Garden of the Hesperides or Herakles and the Hydra. Subsequent examination and the attempt to make a restoration from the given data have made me sure that the former was the subject of the sculpture. Only this would account for the quiet action of the left hand of Herakles and for the closely associated female figure. If this were indeed the subject, how could its normal elements be arranged so as to suit the conditions of the square pedestal having a vertical joint in the centre, and making proper use of the existing fragment of which Fig. 1 is a rough sketch? This question I have tried to answer. The fragment is now fixed in the side of a built-up pedestal close to its left-hand angle, but there is nothing which settles this position and it is a practically impossible one, for there is not room left in which to complete the figure of Herakles. If, however, we shift the piece to the right hand half of the pedestal, and sketch in the completion of the two figures, we at once see how perfectly the tree and serpent would occupy the centre of the composition and leave the left-hand space for the two other watching maidens—the whole making a symmetrical group.

NEW METHOD OF MASSIVE FEYNMAN DIAGRAMS CALCULATION

1991 ◽  
Vol 06 (08) ◽  
pp. 677-692 ◽  
Author(s):  
A.V. KOTIKOV
Keyword(s):  
Differential Equation ◽  
Feynman Diagrams ◽  
New Method ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
Feynman Parameter ◽  
Right Hand ◽  
The Right

A new method of massive Feynman integrals calculation which is based on the rule of integration by parts is presented. This rule is expanded to the massive case. By applying this rule, we obtain a differential equation with respect to the mass for the initial diagram. The right-hand side of the equation contains simpler diagrams (i.e., containing only loops, not chains). This can be done by applying the procedure consecutively. These loops can be calculated either by the standard Feynman-parameter procedure or by a procedure which decreases the number of loops step-by-step. We demonstrate the capacities of this method for various complicated diagrams and make an attempt to analyze other possible massive Feynman diagrams calculations.

CONTINUOUS DEPENDENCE ON A PARAMETER OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS IN A BANACH SPACE

1993 ◽  
Vol 03 (04) ◽  
pp. 477-483
Author(s):  
D.D. BAINOV ◽  
S.I. KOSTADINOV ◽  
NGUYEN VAN MINH ◽  
P.P. ZABREIKO
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Small Parameter ◽  
Continuous Dependence ◽  
Impulsive Differential Equation ◽  
Impulsive Differential Equations ◽  
Right Hand ◽  
Dependence On A Parameter ◽  
The Right

Continuous dependence of the solutions of an impulsive differential equation on a small parameter is proved under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.

scholarly journals Bilt-in turbulence modelling solution method

2020 ◽  
Vol 22 (5) ◽  
pp. 28-40
Author(s):  
L. E. Melamed
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Analytical Form ◽  
Characteristic Functions ◽  
Modeling And Analysis ◽  
Right Hand ◽  
Internal Structures ◽  
Geometric Objects ◽  
Physical Fields ◽  
The Right

THE PURPOSE. The aim of the work is to find a method for mathematical modeling and analysis of inhomogeneous physical fields and the influence of internal structures on these fields. Solutions are sought in areas in which there are subdomains with already known behavior (“embedded” areas and embedded solutions). The goal is to find a modeling method that does not require a change in existing software and is associated only with the modification of the right-hand sides of the equations under consideration. METHODS. The proposed method of mathematical modeling is characterized by the use of characteristic functions for specifying the geometric location and shape of embedded areas, for specifying systems of embedded areas (for example, spherical fillings or turbulent vortices) without specifying them as geometric objects, for modifying the calculated differential equation within the embedded areas. RESULTS. A theorem is formulated and proved (in the form of a statement) that formalizes the essence of the proposed method and gives an algorithm for its application. This algorithm consists in a) representation of the differential equation of the problem in another analytical form; in this form, a term is added to the original differential equation (to its right-hand side), in the presence of which this equation gives a predetermined ("built-in") solution in the necessary regions and b) a representation of the desired solution (using the characteristic function) in the form in which this solution takes the form of either the desired function (in the main area) or the specified functions (in the embedded areas). Examples of calculations from two physical and technical areas - thermal conductivity and hydrodynamics are presented. The result of the work is also the calculation of a turbulent flow in a pipe, in which a system of ball vortices, the speed and direction of rotation of these vortices are specified. CONCLUSION. The proposed method makes it possible to simulate complex physical processes, including turbulence, has been tested, is quite simple and indispensable in cases where embedded structures can be specified only by software.

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