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Spline Galerkin approximation methods for the Sherman-Lauricella integral equation on simple closed piecewise smooth contours are studied, and necessary and sufficient conditions for their stability are obtained. It is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the contour are invertible. Numerical experiments demonstrate a good convergence of the spline Galerkin methods and validate theoretical results. Moreover, it is shown that if all corners of the contour have opening angles located in interval (0.1π, 1.9π), then the corresponding Galerkin method based on splines of order 0, 1 and 2 is always stable. These results are in strong contrast with the behaviour of the Nyström method, which has a number of instability angles in the interval mentioned.

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Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators R τ associated with the corner points τ . The operators R τ do not depend on the shape of the contour but only on the opening angles of the corner points τ . The invertibility of these operators is studied numerically via the stability of the method on model curves, all corner points of which have the same opening angle. The case of the splines of order 0, 1 and 2 is considered. It is shown that no opening angle located in the interval [0.1π, 1.9π] can cause the instability of the method. This result is in strong contrast with the Nyström method, which has four instability angles in the interval mentioned. Numerical experiments show a good convergence of the methods even if the right-hand side of the equation has discontinuities located at the corner points of the contour.

In this work, we introduce a nonlinear Lanchester model of Network centric warfare (NCW)-type and study a problem of finding the optimal fire allocation for this model. A Blue party B will fight against a Red party consisting of A and R, where A is an independent force and R fights with supports from a supply unit N. Optimal fire allocation will then be sought in the form of piece-wise constant functions so that the remaining force of B is as large as possible. For this model, we also introduce a notion of threatening rates which are computed for A,R,N at each stage of the battle. These rates will then be used to derive the optimal fire allocation for B. Several numerical experiments are presented to justify the theoretical findings.