• Title/Summary/Keyword: the variation-of-constants formula

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SOLUTIONS OF HIGHER ORDER INHOMOGENEOUS PERIODIC EVOLUTIONARY PROCESS

  • Kim, Dohan;Miyazaki, Rinko;Naito, Toshiki;Shin, Jong Son
    • Journal of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1853-1878
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    • 2017
  • Let $\{U(t,s)\}_{t{\geq}s}$ be a periodic evolutionary process with period ${\tau}$ > 0 on a Banach space X. Also, let L be the generator of the evolution semigroup associated with $\{U(t,s)\}_{t{\geq}s}$ on the phase space $P_{\tau}(X)$ of all ${\tau}$-periodic continuous X-valued functions. Some kind of variation-of-constants formula for the solution u of the equation $({\alpha}I-L)^nu=f$ will be given together with the conditions on $f{\in}P_{\tau}(X)$ for the existence of coefficients in the formula involving the monodromy operator $U(0,-{\tau})$. Also, examples of ODEs and PDEs are presented as its application.

BOUNDEDNESS IN THE PERTURBED DIFFERENTIAL SYSTEMS

  • Goo, Yoon Hoe
    • The Pure and Applied Mathematics
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    • v.20 no.3
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    • pp.223-232
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    • 2013
  • Alexseev's formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In recent years M. Pinto introduced the notion of $h$-stability. S.K. Choi et al. investigated $h$-stability for the nonlinear differential systems using the notion of $t_{\infty}$-similarity. Applying these two notions, we study bounds for solutions of the perturbed differential systems.

BOUNDEDNESS IN THE FUNCTIONAL NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

  • GOO, YOON HOE
    • The Pure and Applied Mathematics
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    • v.22 no.2
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    • pp.101-112
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    • 2015
  • Alexseev's formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In this paper, we investigate bounds for solutions of the functional nonlinear perturbed differential systems using the two notion of h-stability and $t\infty$-similarity.

A study on failure probability characteristic based on the reliability analysis according to the variation of boundary conditions (신뢰성 기반 쉴드터널의 경계조건 변화에 따른 파괴확률 특성에 관한 연구)

  • Gyu-Phil Lee;Young-Bin Park
    • Journal of Korean Tunnelling and Underground Space Association
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    • v.25 no.6
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    • pp.447-458
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    • 2023
  • In this study, a comparison model considering the stochastic characteristics of the load and member resistance of the shield tunnel segment lining as well as the variability of the boundary condition was selected and reliability analysis was performed, and the adequacy of the limit state design was analyzed by calculating the probability of failure and reviewing the structural safety. For the analysis considering the probability characteristics of these ground constants, the ground spring coefficient was considered as the mean value by calculating the quantitative value by applying the Muirwood formula, and the coefficient of variation was selected based on the existing research data to review the models according to the change of ground boundary conditions. Through the structural analysis of these models and the reliability analysis using MCS technique, the failure probability and reliability index were calculated to examine the changes in the failure probability due to changes in ground boundary conditions.