• Title/Summary/Keyword: Fundamental Solutions

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ON GENERALIZED FLOQUET SYSTEMS II

  • EI-Owaidy, H.;Zagrout, A.A.
    • Kyungpook Mathematical Journal
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    • v.27 no.1
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    • pp.35-41
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    • 1987
  • Consider the system (i) x'=Ax. Let ${\Phi}$ be its fundamental matrix solution. If there is w>0 such that A(t+w)-A(t) commutes with ${\Phi}$ for all t, then we call this system a "generalized" Floquet system or a "G. F. system". We show that $A(t+w)-A(t)=B_1$=constant if and only if $A(t)=C+B_1t/w+Q(t)$, Q is periodic of period w>0. For this A(t) We prove that if all eigenvalues of $B_1$ have negative real parts, then the origin is asumptotically stable. We find a growth condition for a continuous D(t) which guarantees that all solutions of z'=[A(t)+D(t)]z are bounded if all solutions of the G. F. system (i) are bounded. Combining the foregoing results yield a class of perturbed G. F. operators all of whose solutions are bounded.

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A Boundary Integral Equation Formulation for an Unsteady Anisotropic-Diffusion Convection Equation of Exponentially Variable Coefficients and Compressible Flow

  • Azis, Mohammad Ivan
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.557-581
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    • 2022
  • The anisotropic-diffusion convection equation with exponentially variable coefficients is discussed in this paper. Numerical solutions are found using a combined Laplace transform and boundary element method. The variable coefficients equation is usually used to model problems of functionally graded media. First the variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation is then Laplace-transformed so that the time variable vanishes. The Laplace-transformed equation is consequently written as a boundary integral equation which involves a time-free fundamental solution. The boundary integral equation is therefore employed to find numerical solutions using a standard boundary element method. Finally the results obtained are inversely transformed numerically using the Stehfest formula to get solutions in the time variable. The combined Laplace transform and boundary element method are easy to implement and accurate for solving unsteady problems of anisotropic exponentially graded media governed by the diffusion convection equation.

ON THE STABILITY OF DIFFERENTIAL SYSTEMS INVOLVING 𝜓-HILFER FRACTIONAL DERIVATIVE

  • Limpanukorn, Norravich;Ngiamsunthorn, Parinya Sa;Songsanga, Danuruj;Suechoei, Apassara
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.3
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    • pp.513-532
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    • 2022
  • This paper deals with the stability of solutions to 𝜓-Hilfer fractional differential systems. We derive the fundamental solution for the system by using the generalized Laplace transform and the Mittag-Leffler function with two parameters. In addition, we obtained some necessary conditions on the stability of the solutions to linear fractional differential systems for homogeneous, non-homogeneous and non-autonomous cases. Numerical examples are also given to illustrate the behavior of solutions.

Out-of-Plane Vibration Analysis of Curved Beams Considering Shear Deformation Using DQM (전단변형이론 및 미분구적법을 이용한 곡선보의 면외 진동해석)

  • Kang, Ki-Jun;Kim, Jang-Woo
    • Journal of the Computational Structural Engineering Institute of Korea
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    • v.20 no.4
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    • pp.417-425
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    • 2007
  • The differential quadrature method(DQM) is applied to computation of eigenvalues of the equations of motion governing the free out-of-plane vibration for circular curved beams including the effects of rotatory inertia and transverse shearing deformation. Fundamental frequencies are calculated for the members with clamped-clamped end conditions and various opening angles. The results are compared with exact solutions or numerical solutions by other methods for cases in which they are available. The DQM provides good accuracy even when only a limited number of grid points is used.

Fundamental Studies for the Removal and Recovery of Silver from Waste Photo-Developing Solution by Solvent Extraction (사진폐액으로부터 용매추출에 의한 은의 제거 및 회수에 대한 기초연구)

  • Lee, Sun-Hwa;Kim, Dong-Su;Lee, Hwa-Young
    • Journal of Korean Society on Water Environment
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    • v.22 no.1
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    • pp.122-127
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    • 2006
  • Fundamental studies were carried out for an effective removal and recovery of silver from waste photo-developing solution by solvent extraction. The organic solvents examined for silver-extraction were ALIQUAT 336, D2EHPA, KELEX 100, and TBP. ALIQUAT 336, which is an anionic exchanger, was found to be efficient for the extraction of silver and the reason for this was considered to be due to the chloride ion contained in its structure. The extent of silver extraction was examined to increase with the concentration of ALIQUAT 336 until it reached 0.6 M and no more extraction was observed above this concentration. The extraction of silver by ALIQUAT 336 was found to reach its pseudo-equilibrium within a few minutes after the reaction started and additional slight increase in silver extraction was observed until 30 minutes of reaction time. The observed differences in silver extraction for artificial and actual waste solutions were considered to be based upon the different ionic form of silver-containing species in these solutions.

Numerical analysis for free vibration of hybrid laminated composite plates for different boundary conditions

  • Benhenni, Mohammed Amine;Daouadji, Tahar Hassaine;Abbes, Boussad;Abbes, Fazilay;Li, Yuming;Adim, Belkacem
    • Structural Engineering and Mechanics
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    • v.70 no.5
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    • pp.535-549
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    • 2019
  • This study aimed to develop a high-order shear deformation theory to predict the free vibration of hybrid cross-ply laminated plates under different boundary conditions. The equations of motion for laminated hybrid rectangular plates are derived and obtained by using Hamilton's principle. The closed-form solutions of anti-symmetric cross-ply and angle-ply laminates are obtained by using Navier's solution. To assess the validity of our method, we used the finite element method. Firstly, the analytical and the numerical implementations were validated for an antisymmetric cross-ply square laminated with available results in the literature. Then, the effects of side-to-thickness ratio, aspect ratio, lamination schemes, and material properties on the fundamental frequencies for different combinations of boundary conditions of hybrid composite plates are investigated. The comparison of the analytical solutions with the corresponding finite element simulations shows the good accuracy of the proposed analytical closed form solution in predicting the fundamental frequencies of hybrid cross-ply laminated plates under different boundary conditions.

Representation of fundamental solution and vibration of waves in photothermoelastic under MGTE model

  • Rajneesh Kumar;Nidhi Sharma;Supriya Chopra;Anil K. Vashishth
    • Ocean Systems Engineering
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    • v.13 no.2
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    • pp.123-146
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    • 2023
  • In this paper, Moore-Gibson-Thompson theory of thermoelasticity is considered to investigate the fundamental solution and vibration of plane wave in an isotropic photothermoelastic solid. The governing equations are made dimensionless for further investigation. The dimensionless equations are expressed in terms of elementary functions by assuming time harmonic variation of the field variables (displacement, temperature distribution and carrier density distribution). Fundamental solutions are constructed for the system of equations for steady oscillation. Also some preliminary properties of the solution are explored. In the second part, the vibration of plane waves are examined by expressing the governing equation for two dimensional case. It is found that for the non-trivial solution of the equation yield that there exist three longitudinal waves which advance with the distinct speed, and one transverse wave which is free from thermal and carrier density response. The impact of various models (i)Moore-Gibson-Thomson thermoelastic (MGTE)(2019), (ii) Lord and Shulman's (LS)(1967) , (iii) Green and Naghdi type-II(GN-II)(1993) and (iv) Green and Naghdi type-III(GN-III)(1992) on the attributes of waves i.e., phase velocity, attenuation coefficient, specific loss and penetration depth are elaborated by plotting various figures of physical quantities. Various particular cases of interest are also deduced from the present investigations. The results obtained can be used to delineate various semiconductor elements during the coupled thermal, plasma and elastic wave and also find the application in the material and engineering sciences.

Dynamic Boundary Element Analysis of Underground Structures Using Multi-Layered Half-Plane Fundamental Solutions (2차원 다층 반무한해를 이용한 지하구조계의 동적 경계요소 해석)

  • 김문겸;이종우;조성용
    • Journal of the Earthquake Engineering Society of Korea
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    • v.1 no.4
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    • pp.59-68
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    • 1997
  • In analysis of underground structures, the effects of artificial boundary conditions are considered as one of the major reasons for differences from experimental results. These phenomena can be overcome by using the boundary elements which satisfy the multi-layered half space conditions. The fundamental solutions of multi-layered half-space for boundary element method is formulated satisfying the transmission and reflection of waves at each layer interface and radiation conditions at bottom layer. The governing equations can be obtained from the displacements at each layer which are expressed in terms of harmonic functions. All types of waves can be included using the complete response from semi-infinite integrals with respect to horizontal wavenumbers using expansion of Fourier series and Hankel transformation. Two dimensional Green's functions are derived from cylindrical Navier equations and potentials performing infinite integration in y-direction. In this case, it is effective to transform into two dimensional problem using semi-analytical integration and sinusoidal Bessel function. Some verifications are given to show the accuracy and efficiency of the developed method, and numerical examples to demonstrate the dynamic behavior of underground with various properties.

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Analysis of Linear Consolidation Problems by the Boundary Element Method (경계요소법에 의한 선형 압밀문제의 해석)

  • 서일교
    • Computational Structural Engineering
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    • v.8 no.4
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    • pp.129-136
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    • 1995
  • This paper presents a boundary element method for obtaining approximate solutions of 2-dimensional consolidation problems based on the Biot's linear theory. Laplace transform is applied to differential equation system in order to eliminate the time dependency. The boundary integral equations in transformed space are formulated and the fundamental solutions are shown in a closed form. In order to convert the transformed solutions to the ones in real space, the Hosono's numerical Laplace transform inversion method is applied. As a numerical example, a half-space consolidation problem subjected to a strip local load is selected and the applicability of the method is demonstrated through the comparison with the exact solutions.

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Improving Hydraulic System Design by Analysis Model of a Self-propelled Spinach Harvester (자주식 시금치 수확장치 해석모델을 활용한 유압시스템 개선 설계 제안)

  • Noh, Dae Kyung;Lee, Dong Won;Lee, Jong Su;Jang, Joo Sup
    • Journal of Drive and Control
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    • v.19 no.1
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    • pp.69-75
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    • 2022
  • This study aimed to develop solutions for the intermittent performance deterioration of self-propelled spinach harvesters through analysis model. The study was conducted in the following manner. First, changes in performance deterioration and surplus flow, which result from oil temperature changes, were analyzed by simulating actual sequential harvesting movements, which involve driving with actuators operated simultaneously, by analysis model developed in a previous study. Second, fundamental solutions for surplus flow problems were presented. Third, the solutions were applied to a virtual environment to present their practicality and quantitative effects. The two solutions based on the study results were as follows. First, a closed center-type directional control valve was applied to the hydraulic circuit. Second, an unloading system was set up through an on-off solenoid valve.