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Figure 3 | Boundary Value Problems

Figure 3

From: Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues

Figure 3

Graphs of \(\pmb{\varphi_{+}(\kappa)}\) and \(\pmb{\varphi _{-}(\kappa)}\) . Solid red lines (——) represent \(\varphi_{+}(\kappa)\), and dashed blue lines (-  -  -) represent \(\varphi_{-}(\kappa)\). \(\varphi_{+}\) increases on \(( h^{-1} ( 2\pi n + \pi/2 ), h^{-1} ( 2\pi(n+1) + \pi/2 ) )\) from −∞ to ∞, and \(\varphi_{-}\) decreases on \(( h^{-1} ( 2\pi n - \pi/2 ), h^{-1} ( 2\pi(n+1) - \pi/2 ) )\) from ∞ to −∞. \(\varphi_{\pm}( h^{-1} ( 2\pi n ) ) = \exp \{ L \cdot h^{-1}(2\pi n) \}\), \(\varphi_{\pm}( h^{-1}(2\pi n + \pi) ) = -\exp \{ L \cdot h^{-1}(2\pi n + \pi) \}\), \(\varphi_{\pm}( h^{-1} ( 2\pi n \mp\pi/2 ) ) = 0\).

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