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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]

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 in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method’s effectiveness and precision.

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

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]

For the 3D quasilinear wave equation − (1 + (∂ t ϕ) p) ∂ t 2 ϕ + Δ ϕ = 0 with the short pulse initial data (ϕ , ∂ t ϕ) (1 , x) = (δ 2 − ε 0 ϕ 0 (r − 1 δ , ω) , δ 1 − ε 0 ϕ 1 (r − 1 δ , ω)) , where p ∈ N , p ≥ 2 , 0 < ε 0 < 1 , r = | x | , ω = x r ∈ S 2 , and δ > 0 is sufficiently small, under the outgoing constraint condition (∂ t + ∂ r) k ϕ (1 , x) = O (δ 2 − ε 0 − k max ⁡ { 0 , 1 − (1 − ε 0) p }) for k = 1 , 2 , we will establish the global existence of smooth large data solution ϕ when p > p c with p c = 1 1 − ε 0 being the critical exponent. In the forthcoming paper, when 1 ≤ p ≤ p c , we show the formation of the outgoing shock before the time t = 2 under the same outgoing constraint condition and the other suitable assumption of (ϕ 0 , ϕ 1). [ABSTRACT FROM AUTHOR]

The problem of existence and uniqueness of nonlinear Goursat is studied by Claude Wagschal in different spaces: holomorphic spaces, partially holomorphic, continuous-Gevrey and Gevrey-holomorphic. In this paper, we replace the sequence (n!)d−1 in the case Gevery with suitable sequence (Mn) in the case Carleman. After that, we study the problem of existence and uniqueness of nonlinear Goursat problem in Denjoy–Carleman space.

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s gH-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s gH-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 gH-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.

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]

The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.

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]

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]

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, 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.

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]