15. Singular boundary integral equations of supersonic transport BVP

Тheorem15.1. If the classical solution of BVP satisfies the Holder's conditions at D+, i.e., ∃ C > 0, β > 0 that

$$\left|\mathfrak{u}\_{\flat}(\mathfrak{x},\mathfrak{z}) - \mathfrak{u}\_{\flat}(\mathfrak{y},\mathfrak{z})\right| < \mathsf{C} \|\mathfrak{x} - \mathfrak{y}\|^{\beta}, \quad \mathfrak{x}, \mathfrak{y} \in \mathsf{S}.$$

then it satisfies the singular boundary integral equation at D+

$$\begin{aligned} 0, \mathsf{5u}\_i(\mathbf{x}, z) &= \sum\_{k=1}^2 \int\_{S\_i^\delta(\mathbf{x}')} \theta(\mathbf{z} - m\_k r) dS(\mathbf{y}) \int\_0^{z - m\_k r} \left\{ \mathcal{U}\_i^j(\mathbf{x} - \mathbf{y}, z - \tau) \, p\_j(\mathbf{y}, \tau) - \mathcal{U}\_j^j(\mathbf{x}) \right\} d\tau \\ &- H\_i^{jd}(\mathbf{x} - \mathbf{y}, z - \tau, n(\mathbf{y})) \mu\_{j \cdot z}(\mathbf{y}, \tau) \right\} d\tau - \\ &- V.P. \int\_{S\_i^\delta(\mathbf{x})} H\_i^{j\delta}(\mathbf{x} - \mathbf{y}, z, n(\mathbf{y})) \mu\_j(\mathbf{y}, z - m\_k r) dS(\mathbf{y}), r = \|\mathbf{x} - \mathbf{y}\| \\\\ &\le \delta^k C(\delta \mathbf{x} - \mathbf{y}, \delta \mathbf{y}, \tau) \|\mathbf{x} - \mathbf{y}\| \delta \mathbf{x} \delta \delta \mathbf{y} \delta \delta \mathbf{x} \end{aligned}$$

$$S^k\_\tau(x') = \{(y,\tau) : m\_k r < z - \tau\}, \quad S^k\_\tau(x) = \{(y) : m\_k r < z\}$$

Proof. The desired assertion follows from Theorem 14.1 and Theorem 11.1 for tensor H by analogy of the proof of Theorem 12.1 about singular boundary integral equations in the subsonic case. Full proofs of these theorems can be found in [7].

This theorem gives us a resolving system of integral equations for definition of unknown values of boundary displacements in the supersonic case.

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Moreover, the Somigliana formula for displacements was obtained for generalized functions. But since they are regular, from the Dubois-Reymond lemma ([3]: 97), this solution is classical. However, if the acting loads are described by singular generalized functions, which often takes place in physical problems, then one should use a representation of a generalized solution in the convolution form (Eq. (65)) with the evaluation of convolutions by the definition (see [3]: 133).
