Completely discrete schemes for the homogeneous equation

2006 ◽  
pp. 92-105
Keyword(s):  
Homogeneous Equation

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

scholarly journals On a boundary value problem for scalar linear functional differential equations

2004 ◽  
Vol 2004 (1) ◽  
pp. 45-67 ◽  
Author(s):  
R. Hakl ◽  
A. Lomtatidze ◽  
I. P. Stavroulakis
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Homogeneous Equation ◽  
Functional Differential Equations ◽  
Solution Space ◽  
Fredholm Alternative ◽  
Linear Boundary ◽  
Bounded Operators ◽  
Functional Differential ◽  
Linear Boundary Value Problem

Theorems on the Fredholm alternative and well-posedness of the linear boundary value problemu′(t)=ℓ(u)(t)+q(t),h(u)=c, whereℓ:C([a,b];ℝ)→L([a,b];ℝ)andh:C([a,b];ℝ)→ℝare linear bounded operators,q∈L([a,b];ℝ), andc∈ℝ, are established even in the case whenℓis not astrongly boundedoperator. The question on the dimension of the solution space of the homogeneous equationu′(t)=ℓ(u)(t)is discussed as well.

scholarly journals TOTAL ENERGY OF HARMONIC OSCILLATOR WITH IMPULSE ACTION

2020 ◽  
pp. 56-59
Author(s):  
K. Elgondiyev ◽  
S. Matmuratova ◽  
V. Borodin ◽  
L. Vovk
Keyword(s):  
Periodic Function ◽  
Harmonic Oscillator ◽  
Total Energy ◽  
Homogeneous Equation ◽  
External Perturbation ◽  
Oscillatory System ◽  
Time Variable ◽  
Harmonic Oscillations ◽  
Impulse Action ◽  
Periodic Impulse

The problem of finding the total energy of a harmonic oscillator with pulsed action at fixed moments of time is considered. Both for the case of the homogeneous equation of harmonic oscillations and for the case of the equation of harmonic oscillations in the presence of external perturbation, formulas for the total energy of the oscillatory system are obtained. The case of periodic impulse effects is analyzed. The conditions under which in this oscillatory system there are periodic modes are specified. It is shown that under the fulfillment of these conditions on the values of impulse action and external perturbation, the total energy of the vibrational system is also a periodic function of the time variable.

scholarly journals Thermal condition of cohesion in a two-layer roll of a rolling mill

2019 ◽  
pp. 45-48
Author(s):  
Olga Panteleivna Demyanchenko ◽  
Viktor Lyashenko
Keyword(s):  
Heat Exchange ◽  
Temperature Distribution ◽  
Heat Conductivity ◽  
Rolling Mill ◽  
Thermal Condition ◽  
Thermal Contact ◽  
Homogeneous Equation ◽  
Target Setting ◽  
Radial Section ◽  
Ideal Thermal Contact

A condition of heat exchange between the layershaving different thermalphysic properties in a two-layercylindrical roll of a rolling mill is analyzed foe an ideal thermalcontact. It can be realized with application of the condition ofheat balance of one of the layers in the cylindrical area for ahomogeneous equation of heat conductivity. Analyzed was asimplified target setting in the radial section with a supposition,regarding an averaged in radius temperature distribution in theouter layer. By applying the condition of the thermal balance andby integrating the homogeneous equation of heat conductivity inthe two-layer area a condition of cohesion of an impedance typein case of an ideal thermal contact between the layers wasconstructed.

scholarly journals Time Impact on Residue in a Non-homogeneous Equation of Statics in the Theory of Elastic Mixture

2021 ◽  
Vol 09 (05) ◽  
Author(s):  
Ebikiton Ndiwari ◽  
Keyword(s):  
Meromorphic Function ◽  
Homogeneous Equation ◽  
Principal Component ◽  
Variable Coefficients ◽  
Component Mixture ◽  
Second Order Differential Equations ◽  
Elastic Mixture ◽  
The Earth ◽  
Time Relationship ◽  
Two Component

Residual stress in continuum has not been quantified because time relationship with residues has not been proven analytically. This is achieved in this paper by analyzing a two component mixture with the non-homogeneous equation of statics in the theory of elastic mixture, and second order differential equations with variable coefficients. A dry mixture of sand and cement is transformed into a continuum, which is been determined as an entire or a meromorphic function, as a result of the existence of residues that are contained in the principal component of the mixture obtained directly from the earth. The time relationship with residue, in these two functions are determined. Our result shows that time places a limit on residues, making the meromorphic function prone to implosion..

scholarly journals Comparative Study of Wellhole Surrounding Rock under Nonuniform Ground Stress

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Zhongyu Jiang ◽  
Guoqing Zhou
Keyword(s):  
Homogeneous Equation ◽  
Pressure Coefficient ◽  
Circumferential Stress ◽  
Surrounding Rock ◽  
Special Solution ◽  
Operator Matrix ◽  
Specific Expression ◽  
Symplectic Algorithm ◽  
Ground Stress ◽  
The Relationship

The stress analysis of the wellhole surrounding rock and the regular failure of the wellhole has always been a concern for the well builders. Firstly, the Hamilton canonical equations are obtained by using the Hamiltonian variational principle in the sector domain, and the zero eigensolution and nonzero eigensolutions of the homogeneous equation are solved. According to the Hamiltonian operator matrix with the orthogonal eigenfunction system, the special solution form of the nonhomogeneous boundary condition equation is obtained. Then, according to the principle of the same coefficient being equal, the relationship equation between the direction eigenvalue and the angle coefficient is obtained, from which the specific expression of the special solution of the equation can be determined. Furthermore, the analytical solution of the wellhole surrounding rock problem under nonuniform ground stress is obtained by using the linear elastic accumulative principle. Finally, a concrete example is given to compare the finite element method and the symplectic algorithm. The results are consistent, which ensures the accuracy and the reliability of the symplectic algorithm. The relationship between the circumferential stress distribution around the hole and the lateral pressure coefficient is further analyzed.

scholarly journals Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

2018 ◽  
Vol 23 (4) ◽  
pp. 76
Author(s):  
Julia Gregori ◽  
Juan López ◽  
Marc Sanz
Keyword(s):  
Differential Equations ◽  
Homogeneous Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mean Square ◽  
Recent Contribution ◽  
Transform Method ◽  
Order Linear ◽  
Random Power Series ◽  
Linear Differential

The objective of this paper is to complete certain issues from our recent contribution (Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv. Differ. Equ. 2018, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

Stability and Controllability of Euler-Bernoulli Beams With Intelligent Constrained Layer Treatments

1996 ◽  
Vol 118 (1) ◽  
pp. 70-77 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Laplace Transform ◽  
Closed Form ◽  
Characteristic Equation ◽  
Matrix Equation ◽  
Homogeneous Equation ◽  
Transform Domain ◽  
Threshold Control ◽  
Control Gain ◽  
The Laplace Transform ◽  
The Stability

This paper studies the stability and controllability of Euler-Bernoulli beams whose bending vibration is controlled through intelligent constrained layer (ICL) damping treatments proposed by Baz (1993) and Shen (1993, 1994). First of all, the homogeneous equation of motion is transformed into a first order matrix equation in the Laplace transform domain. According to the transfer function approach by Yang and Tan (1992), existence of nontrivial solutions of the matrix equation leads to a closed-form characteristic equation relating the control gain and closed-loop poles of the system. Evaluating the closed-form characteristic equation along the imaginary axis in the Laplace transform domain predicts a threshold control gain above which the system becomes unstable. In addition, the characteristic equation leads to a controllability criterion for ICL beams. Moreover, the mathematical structure of the characteristic equation facilitates a numerical algorithm to determine root loci of the system. Finally, the stability and controllability of Euler-Bernoulli beams with ICL are illustrated on three cantilever beams with displacement or slope feedback at the free end.

scholarly journals ℬ𝒞-total stability and almost periodicity for some partial functional differential equations with infinite delay

2005 ◽  
Vol 2005 (3) ◽  
pp. 259-274 ◽  
Author(s):  
Abdelhai Elazzouzi ◽  
Khalil Ezzinbi
Keyword(s):  
Periodic Solution ◽  
Infinite Delay ◽  
Homogeneous Equation ◽  
Linear Part ◽  
Functional Differential Equations ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Functional Differential ◽  
Total Stability ◽  
Nonhomogeneous Equation

In this paper, we study the existence of an almost periodic solution for some partial functional differential equation with infinite delay. We assume that the linear part is nondensely defined and satisfies the known Hille-Yosida condition. We prove if the null solution of the homogeneous equation is ℬ𝒞-total stable, then the nonhomogeneous equation has an almost periodic solution.

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