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Boundary Value Problems
Figure 2 | Boundary Value Problems

Figure 2

From: Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

Figure 2

Graphs of \(\varphi(s)\) and \(\varphi_{\hat{s}}(s)\) for \(\hat{s} = \hat{s}_{\mathrm{max}}, \hat{s}_{0}\) in Example 5.1. \(\overline{s} = \sigma k\) and \(\lim_{s \to\overline{s}-}{\varphi(s)} = 0\) since \({\sigma k < s_{\rho}= \infty}\). The interval between \(s_{\mathrm{min}}\) and \(s_{\mathrm{max}}\), depicted as a thick line segment in the s-axis, represents the set \(\{ s \in [ 0, \overline{s} ) \mid\varphi(s) \geq \varphi_{\hat{s}_{0}}(s) \}\). The values of \(\hat{s}_{\mathrm{max}}\), \(s_{*}\) are determined by \({\varphi ( s_{*} ) = \varphi_{\hat{s}_{\mathrm{max}}} ( s_{*} )}\) and \({\varphi^{\prime}( s_{*} ) = \varphi_{\hat{s}_{\mathrm {max}}}^{\prime}( s_{*} )}\)

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