ASYMPTOTICS OF THE FUNDAMENTAL SYSTEM SOLUTIONS OF DIFFERENTIAL EQUATION WITH BLOCK-TRIANGULAR OPERATOR POTENTIAL

2017 ◽  
Vol 102 (10) ◽  
pp. 2359-2366
Author(s):  
Aleksandr M. Kholkin
Keyword(s):  
Differential Equation ◽  
Fundamental System ◽  
Operator Potential ◽  
Triangular Operator

scholarly journals Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Aleksandr Mikhailovich Kholkin
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Operator Potential ◽  
Triangular Operator

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

scholarly journals ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

2020 ◽  
Vol 70 (2) ◽  
pp. 53-58
Author(s):  
P.B. Beisebay ◽  
◽  
G.H. Mukhamediev ◽  
Keyword(s):  
Differential Equation ◽  
Complex Analysis ◽  
Arbitrary Order ◽  
Fundamental System ◽  
Homogeneous Differential Equation ◽  
Complex Roots ◽  
Linear Homogeneous Differential Equation ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Homogeneous Linear

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

Dynamic analysis of rotationally flexible cam mechanisms

1999 ◽  
Vol 213 (6) ◽  
pp. 537-550 ◽  
Author(s):  
J D G Balkwill ◽  
D Morrey
Keyword(s):  
Differential Equation ◽  
Dynamic Analysis ◽  
Rotational Speed ◽  
Fundamental System ◽  
Equation Of Motion ◽  
Dynamic Parameters ◽  
Ic Engine ◽  
First Time ◽  
Level Of Agreement ◽  
Theory And Experiment

A novel mechanism for the actuation of valves in an internal combustion (IC) engine is described. Previous publications are reviewed and it is shown that the mechanism has received very little attention, in particular no dynamic analysis of it has been published. The mechanism features a shaft driving a cam through a coupling of sufficient rotational flexibility to permit significant angular offsets to develop during the cycle. A differential equation of motion of a simplified system is derived and solved using two methods. The theory requires a knowledge of twelve fundamental system dynamic parameters from which a prediction of angular offset of the camlobe throughout the cycle is produced. A dataset was obtained from a purpose built apparatus which recorded the angular offset of the camlobe with respect to the camshaft for a range of rotational speed values. Results graphs are presented and the level of agreement between theory and experiment is discussed. The work demonstrates for the first time that useful predictions of simplified mechanism performance can be obtained from the model developed. Recommendations for further work in this area are made.

scholarly journals On linear homogeneous differential equation of Chebyshev type

2008 ◽  
Vol 48 ◽  
Author(s):  
Eduard Kiriyatzkii
Keyword(s):  
Differential Equation ◽  
Fundamental System ◽  
Chebyshev System ◽  
Homogeneous Differential Equation ◽  
Linear Homogeneous Differential Equation

Let L[y] = y(n)(z)+gn-1(z)y(n-1)(z)+. . .+g1(z)y(1)(z)+g0(z)y(z) = 0  be a differential equation of nth order with analytic in circle |z| < R coefficients. We will call above equation by equation of Chebyshev type in |z| < R, if fundamental system of its solution is a Chebyshev system in circle |z| < R . In present paper the conditions with the fulfillment of which the equation L[y] = 0 is of Chebyshev type.  

The asymptotic expansion of solutions of the differential equation u iv +λ 2 [(z 2 + c ) u" + azu '+ bu ]=0 for large IzI

1980 ◽  
Vol 293 (1404) ◽  
pp. 511-533 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Order Differential Equation ◽  
Fundamental System ◽  
Whittaker Function ◽  
Fundamental Solutions ◽  
Gauss Hypergeometric Function ◽  
Linear Combinations ◽  
Asymptotic Expansion Of Solutions ◽  
Asymptotic Character

The asymptotic expansion of solutions of the fourth order differential equation u iv + λ 2 [( z 2 + c ) u " + azu ' + bu ] = 0 are investigated for | z | -> oo where the parameters a, b, c and λ are supposed to be arbitrary complex constants with λ, b ^ 0. Exact solutions in the form of Laplace and Mellin-Barnes integrals, involving a Whittaker function and a Gauss hypergeometric function respectively, are used to define a fundamental system of solutions. The asymptotic expansion of these solutions is obtained in a full neighbourhood of the point at infinity and their asymptotic character is found to be either exponentially large or algebraic in certain sectors of the z -plane. The expansions corresponding to certain special values of the parameters a and b which yield logarithmic expansions are also treated. Linear combinations of these fundamental solutions which possess an exponentially small expansion for | z |->oo in a certain sector are discussed.

scholarly journals Fractional linear maps in general relativity and quantum mechanics

2021 ◽  
pp. 2150157
Author(s):  
Vito Flavio Bellino ◽  
Giampiero Esposito
Keyword(s):  
Differential Equation ◽  
General Relativity ◽  
Quantum Mechanics ◽  
Limit Point ◽  
Fundamental System ◽  
Second Order ◽  
Positive Real ◽  
Linear Maps ◽  
Point Condition ◽  
Positive Real Line

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

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