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In this paper, we study the expansion of Noble-Abel gas into a vacuum around the convex corner of the two-dimensional compressible magnetohydrodynamic system. We reduce this problem to the interaction of a centered simple wave emanating from the convex corner with a backward planar simple wave. Mathematically, this is a Goursat problem. By using the method of characteristic decomposition and construction of invariant regions, combining C0 and C¹ estimation as well as hyperbolicity estimation, we obtain the existence of a global classical solution by extending the local classical solution [ABSTRACT FROM AUTHOR]

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo's g H -differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo's g H -differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo's g H -differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform. [ABSTRACT FROM AUTHOR]

The hyperbolic partial differential equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with variable coefficients. The Goursat PDE is transformed into a second kind of linear Volterra integral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maximum norm is estimated. The paper includes numerical examples to prove the method's effectiveness and precision. [ABSTRACT FROM AUTHOR]

In this paper, we use the fractional-order Erdélyi–Kober operator as a transmutation operator. Applying this operator, the formulated Goursat problem for the Boussinesq–Love equation with singular coefficients has been reduced to a similar problem for the Boussinesq–Love equation without singular coefficients. The solution to this auxiliary problem has been constructed explicitly by the Riemann method. Then, on its basis, an exact solution to the original problem is constructed. [ABSTRACT FROM AUTHOR]

We consider two types of wave interactions arising from the expansion of a wedge of gas to vacuum: interaction of rarefaction waves and interaction of shock-rarefaction composite waves. In order to construct the solutions to these wave interactions, we consider a standard Goursat problem and a discontinuous Goursat problem for the two-dimensional self-similar Euler system. Global solutions to these Goursat problems are constructed by the method of characteristics. The main difficulty to construct the global solutions is that the type of the 2D self-similar Euler system is a priori unknown. By constructing invariant regions of the characteristic angles, we obtain the hyperbolicity of the system in the interaction regions. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems of the Euler equations with general equations of state. [ABSTRACT FROM AUTHOR]

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. [ABSTRACT FROM AUTHOR]

This paper explains a connection between the famous Poncelet problem, the Abel equation, a new moment problem, problems of solution existence and uniqueness of boundary value problems for the string equation and some others in short presentation. We prove a connection of above problem with one problem of Goursat-type for a hyperbolic-like equation of fourth order. We consider a fourth-order differential equation with two multiple characteristics and parameters in the lower terms and a Goursat-type homogeneous boundary value problem with two real boundary conditions in some space of real entire functions. It is proven that this problem has a nontrivial solution in some space if and only if a parameter belongs denumerable dense set of real number. [ABSTRACT FROM AUTHOR]

This article is concerned in establishing the existence and regularity of solution of semihyperbolic patch problem for two‐dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is very common in the numerical solution of Riemann problems. We use the idea of characteristic decomposition and bootstrap method to prove the existence of a global smooth solution, which is uniformly C1,12$C^{1, \tfrac{1}{2}}$ continuous up to the sonic curve. We also prove that the sonic curve is C1,12$C^{1, \tfrac{1}{2}}$ continuous. Further, we show the formation of shock as an envelope for positive characteristics before reaching their sonic points. [ABSTRACT FROM AUTHOR]

The paper devoted to the boundary value problem with parameter for second order system of hyperbolic equations. We study of a questions for existence and uniqueness of solution to the problem and a construction of algorithms for finding its solution. Conditions for the unique solvability to problem with parameter are established in the terms of fundamental matrix and initial data. [ABSTRACT FROM AUTHOR]

In this paper, we construct a conformal scattering theory for Dirac fields in the interior of a Reissner–Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution. [ABSTRACT FROM AUTHOR]

We show asymptotic completeness for the charged Klein–Gordon equation in the exterior De Sitter–Reissner–Nordström spacetime when the product of the charge of the black with the charge of the Klein-Gordon field is small enough. We then interpret scattering as asymptotic transports along principal null geodesics in a Kaluza–Klein extension of the original spacetime. [ABSTRACT FROM AUTHOR]

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge–Wheeler and Zerilli equations obtained recently by Anderson et al. (Ann. Henri Poincaré 21:61–813, 2020) are sufficient to obtain the conformal scattering. [ABSTRACT FROM AUTHOR]

The main topic of this paper is the Goursat problem at the horizon for the Klein–Gordon equation on the De Sitter–Kerr metric when the angular momentum (per unit of mass) of the black hole is small. Indeed, we solve the Goursat problem for fixed angular momentum n of the field (with the restriction that n is not zero in the case of a massless field). [ABSTRACT FROM AUTHOR]

We state a new problem for a fourth-order factorized differential equation, whose operator is the product of first- and third-order differential operators, and prove its unique solvability. We construct the solution in terms of the Riemann function of the corresponding third-order pseudoparabolic differential operator, determining one of functions that enter in the formula for the problem solution from resolvents of two integral equations. [ABSTRACT FROM AUTHOR]

We apply new approach for the Goursat and Cauchy problems with additional recovering of coefficients of equation for selection of the cases of solvability of these problems in quadratures. Instead of introduction of additional boundary value conditions we offer restrictions on the structure of the equation connected with possibility of its factorization. [ABSTRACT FROM AUTHOR]

In this paper, we examine the solvability of a mixed problem with an integral condition for a third-order equation whose principal part contains the wave operator. The existence and uniqueness of a classical solution to this problem are proved by the Riemann method. [ABSTRACT FROM AUTHOR]

The main subjects of the present paper are the Goursat and Darboux boundary-value problems for hyperbolic systems with two independent variables. We show that obtained by T.V. Chekmarev in terms of successive approximations formulas for solution of the Goursat problem can be built also by the Riemann method, work out an analog of the Riemann–Hadamard method for the system, and introduce its Riemann–Hadamard matrix. We solve the Darboux problem in terms of the introduced matrix. [ABSTRACT FROM AUTHOR]

We describe a method for constructing general integrals for some nonstrictly hyperbolic quasilinear equations and prove a nonlinear analog of Asgeirsson's mean value theorem. Using a general integral, we study the nonlinear version of the Goursat characteristic problem. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness of solution of a class of semi-linear wave equations with initial data prescribed on the light-cone with vertex the origin of a Minkowski space-time. The nonlinear term is assumed to satisfy a nullity condition which guarantees that the neighborhood of the initial cone on which we obtain our solution does not shrink to zero as one approaches infinity. This result is applied to wave maps on Minkowski space-time with [ABSTRACT FROM AUTHOR]

In this note, we prove that D8 × C2n−3 is the non-elementary abelian 2-group of order 2n, n ≥ 3, whose number of subgroups of possible orders is maximal. This solves a conjecture by Haipeng Qu. A formula for counting the subgroups of an (almost) extraspecial 2-group is also presented. [ABSTRACT FROM AUTHOR]

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of ow patches appear frequently in transonic ow problems. In order to use the method of characteristic decomposition to construct such a ow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations. [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 the third-order Bianchi equation with coefficients from variable exponent Lebesgue spaces. The statement of an optimal control problem is studied by using a new version of the increment method that essentially uses the concept of the adjoint equation of the integral form. [ABSTRACT FROM AUTHOR]

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux-Laplace transforms for second-order equations with two independent variables to the case of three-dimensional linear hyperbolic equations of the third order. In this paper we construct examples of such transformations. We consider applications to the problem of orthogonal curvilinear coordinate systems in ℝ3. [ABSTRACT FROM AUTHOR]

This paper studies two types of wave interactions arising from two-dimensional (2D) Riemann problems for the full Euler equations: interaction of jump-fan composite waves and interaction of fan-jump-fan composite waves, which are absent in polytropic gases, but appear in van der Waals gases. Because of the occurrence of shock waves, the flow considered here is unsteady, nonisentropic, and rotational. In order to construct the solutions to these wave interaction problems, we consider two types of discontinuous Goursat problems for a 2D self-similar Euler system. Global piecewise smooth solutions to these discontinuous Goursat problems are obtained by the method of characteristics. The 2D self-similar Euler system is a mixed-type system, and the type in each point is determined by the local pseudo-Mach number. So, one of the main difficulties for the global existence is that hyperbolicity is hard to control in the composite wave interaction zones. By constructing some invariant regions of wave characteristic angles, we prove that the system is uniformly hyperbolic in these zones. In order to obtain the a priori gradient estimates of solutions to these discontinuous Goursat problems, we derive some characteristic decompositions for the 2D self-similar full Euler system. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems. [ABSTRACT FROM AUTHOR]

In terms of coefficients of an equation under consideration, we derive 16 variants of collections of conditions of resolvability of this equation in quadratures. Every collection consists of three identities that interconnect five coefficients appearing in left-hand side of an equation. [ABSTRACT FROM AUTHOR]

For functional-differential quasilinear hyperbolic equation not investigated earlier, we consider problems with conditions on characteristics and obtain conditions of solvability. [ABSTRACT FROM AUTHOR]

Numerical solutions to inverse Sturm-Liouville problems by three spectra, on a finite interval are considered. Dirichlet boundary conditions are assumed to hold at the end-points of the interval , and either a Dirichlet or a Robin-type interior point condition is imposed at an arbitrary interior point of . We demonstrate that the numerical solution of the inverse three spectra problem corresponding to an interior point depends continuously on . This statement is particularly useful when measurements of the input data cannot be obtained at the interior point , but they can be obtained at a nearby interior point. Consequently, we could solve approximately the original inverse problem by departing slightly from . We validate this statement through numerical experiments for sequences of interior points converging to . [ABSTRACT FROM AUTHOR]

We consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [11]. Here we prove the uniqueness of weak solutions. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional (2D) steady jet generated by gas streaming in parallel supersonic flow out of a duct and into the atmosphere. When the gas is a van der Waals gas, at the corners at the exit of the duct, the parallel supersonic flow expands symmetrically into centered simple waves, jump-fan (JF) composite waves, fan-jump composite waves, or fan-jump-fan (FJF) composite waves. This paper studies the interaction of the symmetric centered FJF composite waves in the jet. The interaction of the FJF composite waves is more involved in comparison to our earlier work on the interaction of the symmetric centered JF composite waves, since the flow in the FJF composite wave interaction zone is rotational. To construct the flow in the interaction zone, we consider a Goursat-type boundary value problem with discontinuous boundary data associated with the 2D and isentropic steady Euler equations. The existence of a global piecewise smooth solution to this Goursat problem is obtained by a new method based on characteristics. [ABSTRACT FROM AUTHOR]

We construct an expansion of a solution to the Goursat problem for the sine-Gordon equation with respect to a small parameter that is involved in the data on characteristics. We prove that the expansion obtained is an asymptotic series. [ABSTRACT FROM AUTHOR]

The evolution of the concentrations of particles of two types that annihilate at collision is considered. The kinetic model describing the dynamics of the mixture is represented by a system of two first-order nonlinear partial differential equations. It is shown that the solutions of this model are related to the solutions of the inhomogeneous transport equations by the Bäcklund transform. Analytic solutions of the problem about penetration of particles of the first type from the left half-plane into the right half-plane occupied by the particles of the second type (the two-dimensional penetration problem or molecular beam problem) and of the problem of outflow of the particles of the first type from a circular source into a domain occupied by the particles of the second type are obtained. Possible generalizations of the model are discussed. [ABSTRACT FROM AUTHOR]

A nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients is considered. For solving this problem, a priori estimates in the differential and difference forms are obtained. The a priori estimates imply the uniqueness and stability of the solution on a layer with respect to the initial data and the right-hand side and the convergence of the solution of the difference problem to the solution of the differential problem. [ABSTRACT FROM AUTHOR]

The local fractional differential transform method (LFDTM) and local fractional decomposition method (LFDM) are applied to implement the homogeneous and nonhomogeneous Goursat problem involving local fractional derivative operators. The approximate analytical solution of this problem is calculated in form of a series with easily computable components. Examples are studied in order to show the accuracy and reliability of presented methods. We demonstrate that the two approaches are very e ective and convenient for finding the analytical solutions of partial differential equations with local fractional derivative operators. [ABSTRACT FROM AUTHOR]

We study the global solutions to the two-dimensional Riemann problem for a system of conservation laws. The initial data are three constant states separated by three rays emanating from the origin. Under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity, this problem is classified into five cases. By the self-similar transformation, the reduced system changes type from being elliptic near the origin to being hyperbolic far away in selfsimilar plane. Then in hyperbolic region, applying the generalized characteristic analysis method, a Goursat problem is solved to describe the interactions of planar contact discontinuities. While, in elliptic region, a boundary value problem arises. It is proved that this boundary value problem admits a unique solution. Based on these preparations, five explicit solutions and their corresponding criteria can be obtained in selfsimilar plane. [ABSTRACT FROM AUTHOR]

The Goursat problem for the fractional telegraph equation with Caputo derivatives is studied. An existence and uniqueness theorem for the solution of the problem is proved. [ABSTRACT FROM AUTHOR]

Under consideration is the sub-Riemannian geodesic flow for the Goursat distribution. We find the level surfaces of the first integrals that are in involution and study the trajectories in the phase space whose projections to the horizontal plane are closed curves. [ABSTRACT FROM AUTHOR]

We consider a version of the Goursat problem for a previously unstudied third-order equation and prove the existence and uniqueness of the solution of this problem. [ABSTRACT FROM AUTHOR]

In this paper, we address the general problem of approximating, in a certain optimal way, non-admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider systems modeled by a subriemannian Goursat structure, a particular case being the well-known system of a car with trailers, along with the associated parallel parking problem. Several authors approximate the successive Lie brackets using trigonometric functions. By contrast, we show that more natural optimal motions are related with closed hyperelliptic plane curves with a certain number of loops. [ABSTRACT FROM AUTHOR]

A nonlocal problem of Goursat type for a second-order quasilinear equation with rectilinear characteristics is investigated. [ABSTRACT FROM AUTHOR]

The Cauchy and Goursat problems for a quasi-linear second-order hyperbolic equation with admissible order and type degeneracy are considered. The conditions for nonexistence of a solution of the Cauchy problem are established. However, in conditions of the existence, the solution of the problem is constructed explicitly. In addition, nonlinear versions of the Goursat problem are examined and solutions of these problems are constructed in the explicit form. For the Cauchy problem, as well as for both of the characteristic problems, the structure of the domain of definition of the solution is described. [ABSTRACT FROM AUTHOR]

By reduction to Goursat problems, we obtain various versions of conditions ensuring that the solution of the system in question can be constructed in closed form. [ABSTRACT FROM AUTHOR]

The classical Goursat lemma is extended to classes of differentiable mappings of domains of real Banach spaces to algebras of linear operators acting on these spaces. We clarify the modification of the form and content of this lemma in dependence on the choice of the original object. The case in which the original space is a Banach algebra and Fréchet derivatives are operators of multiplication by elements of this algebra is considered separately. [ABSTRACT FROM AUTHOR]

We consider a kind of Goursat problem for a fourth-order equation in a form not studied yet and prove the existence and uniqueness of the solution of this problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for the construction of the solution of the Goursat problem for the Toda system associated with the Lie algebra C2 based on Hamiltonian reduction of the Wess-Zumino model. [ABSTRACT FROM AUTHOR]

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel $$K_{1}$$ of the transmutation operator relating $$A=-\frac{d^{2} }{dx^{2}}+q_{1}(x)$$ and $$B=-\frac{d^{2}}{dx^{2}}$$ turns out to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating $$C=-\frac{d^{2}}{dx^{2}}+q_{2}(x)$$ and $$B$$ where $$q_{2}$$ is obtained from $$q_{1}$$ by a Darboux transformation. We prove an expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators we obtain a new representation for their kernels. Several examples are given. Moreover, approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed. [ABSTRACT FROM AUTHOR]