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We consider several types of boundary value problems, including Goursat problem, mixed initial–boundary value problem, and centered wave problem for the two-dimensional steady relativistic Euler system with a general equation of state p = p ( ρ ) which is assumed to satisfy 0 < p ′ ( ρ ) < 1 and p ′ ′ ( ρ ) [ p ( ρ ) + ρ ] + 2 p ′ ( ρ ) [ 1 − p ′ ( ρ ) ] > 0 . Global classical solutions to these boundary value problems are constructed by the method of characteristic decomposition. Using these results, we also construct a global supersonic relativistic jet flow out of a semi-infinite convex duct intovacuum. [ABSTRACT FROM AUTHOR]

We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck–Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising 3-permutable varieties of universal algebras. [ABSTRACT FROM AUTHOR]

In this paper a necessary and sufficient condition, such as the Pontryagin’s maximum principle for an optimal control problem with distributed parameters, is given by a hyperbolic equation of the second order with L p ( x ) -coefficients. The results can be used in the theory of optimal processes for distribution Pontryagin maximum principle for various controlled processes described by hyperbolic equations of second order with discontinuous coefficients in variable exponent Sobolev spaces with dominant mixed derivatives. [ABSTRACT FROM AUTHOR]

This paper is concerned with a two-dimensional Riemann problem for the nonlinear wave system which gives rise to an interaction of two planar rarefaction waves. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum, which is often interpreted as the dam collapse problem in hydraulics. We establish the global existence of smooth solutions in the interaction region of rarefaction waves up to a non-trivial vacuum boundary. [ABSTRACT FROM AUTHOR]

Abstract: We find the exact analytical solution to a Goursat PDE system governing the kernels of a backstepping-based boundary control law that stabilizes a constant-coefficient 2×2 system of first-order hyperbolic linear PDEs. The solution to the Goursat system is related to the solution of a simpler, explicitly solvable Goursat system through a suitable infinite series of powers of partial derivatives which is summed explicitly in terms of special functions, including Bessel functions and the generalized Marcum -functions of the first order. The Marcum functions are common in certain applications in communications but have not appeared previously in control design problems. The dependence of the explicit solutions with respect to system parameters is analyzed through several examples, including the stabilization of a constant equilibrium for a quasi-linear plant. [Copyright &y& Elsevier]

Abstract: In this paper we study normal Hopf subalgebras of a semisimple Drinfeld double. This is realized by considering an analogue of Goursat’s lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension. [Copyright &y& Elsevier]

Abstract: In this paper we propose a method to construct numerically the solution-pair of a second order hyperbolic partial differential equation when two types of boundary conditions on a triangular domain are imposed. Specifically, the PDE is , where the unknown functions are and . One set of boundary conditions refer to one set of the triangle’s boundaries (we call them Goursat type), and the other set of boundary conditions both refer to the third boundary of the triangle (we call them Cauchy type). To solve numerically this boundary value problem we make a change of dependent variables and place a mesh on the triangular domain with lines that are the characteristic curves of the hyperbolic PDE. Then we shall integrate along the characteristic curves of the hyperbolic PDE in such a way that numerical values of and will be produced at mesh nodes located on vertical lines. The calculations will proceed from the vertical boundary of the domain going backwards to the origin O of the Oxt coordinate system, from one vertical line to another. The trapezoidal formula will be used for numerical integration. Numerical examples will be provided. [Copyright &y& Elsevier]

Abstract: We consider the problem of classifying the orbits within a tower of fibrations with -fibers that generalize the Monster Tower due to Montgomery and Zhitomirskii. The action on the tower is given by prolongations of diffeomorphism germs of 3-space. As a corollary we give the first steps towards the problem of classifying Goursat 2-flags of small length. In short, we classify the orbits within the first four levels of the Monster Tower and show that there is a total of 34 orbits in the fourth level of the tower. [Copyright &y& Elsevier]

Abstract: We define relative Goursat categories and prove relative versions of the equivalent conditions defining regular Goursat categories. These include 3-permutability of equivalence relations, preservation of equivalence relations under direct images, a condition on so-called Goursat pushouts, and the denormalised 3×3 Lemma. This extends recent work by Gran and Rodelo on a new characterisation of Goursat categories to a relative context. [Copyright &y& Elsevier]

Abstract: For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge–Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: We give a local integral formula, valid on general curved spacetimes, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin. The derivation of the formula is based on the work of Friedlander (1975) for the wave equation. A parametrix for the square of the Dirac operator, which is a spinor wave equation, is built using Friedlander''s construction. Deriving the representation formula obtained in function of the characteristic data for this particular wave equation gives an integral formula for the Goursat problem. The results obtained by Penrose (1963) in the flat case in are recovered directly. [Copyright &y& Elsevier]

This paper deals with the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C1 functions. Some physical and mechanical applications are given. [Copyright &y& Elsevier]