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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 $ C^{0} $ and $ C^{1} $ estimation as well as hyperbolicity estimation, we obtain the existence of a global classical solution by extending the local classical solution.

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.

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]

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]

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.

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 review article is devoted to a class of linear equations with a dominant (leading) partial derivative of the form \((D+M)u=f\), where \(Du\) is a mixed partial derivative, and \(M\) is a linear differential operator containing the derivatives of the function \(u\) obtained from \(D\) by discarding at least one differentiation. We can point out the structural similarity of such linear equations with linear ordinary differential equations. We present the Riemann method for linear equations with a dominant partial derivative, which is a natural generalization of the well-known Riemann method for a second-order hyperbolic equation with two independent variables. The article deals with the main provisions of the theory developed for the equation with the dominant partial derivative of the general form, allowing the interested reader to apply the obtained results to the task that interests him. The definition of the Riemann function as a solution of the Volterra integral equation is given. The main differential identity is discussed, and the process of obtaining a formula for solving the Cauchy problem in terms of the Riemann function by integrating the specified identity over the corresponding domain in \(n\)-dimensional space is demonstrated. An example of constructing a solution to the Cauchy problem for the third-order equation is given. The Riemann method is described below for a fairly wide class of linear systems of hyperbolic equations (including those with multiple characteristics). This method is ideologically very close to the Riemann method for linear equations with a dominant partial derivative. Applications of the Riemann method to the study of new problems for partial differential equations are discussed. In particular, using the Riemann method, the correctness of new boundary value problems for factorized hyperbolic equations is proved, the solvability of integral equations with partial integrals is investigated, and a certain modification of the Riemann method allows us to develop the Riemann--Hadamard method for Darboux problems. The explicit representation of solutions of hyperbolic systems in terms of the Riemann matrix allows us to study new boundary value problems, in particular, problems with the assignment of normal derivatives of the desired functions on the characteristics, problems with conditions on the entire boundary of the domain, and Darboux problems. The Riemann method described here for linear equations with a dominant partial derivative is obviously transferred to matrix equations. In this regard, some cases are indicated when the Riemann matrix is constructed explicitly (in terms of hypergeometric functions) for such matrix equations. The paper provides a review of the literature, briefly describes the history of the development of this direction in Russia and in foreign countries.

For a degenerate first-order hyperbolic equation of the second order containing a term with a lower derivative, we study two boundary value problems with an offset that generalize the well-known first and second Darboux problems. Theorems on an existence of the unique regular solution of problems are proved under certain conditions on given functions and parameters included in the formulation of the problems under study. The properties of all regular solutions of the equation under consideration are revealed, which are analogues of the mean value theorems for the wave equation.

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.

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]

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]

For a fourth-order equation with two independent variables a variant of the Goursat problem with data on two intersecting characteristics is considered. It includes not only the construction of the desired function, but also the coefficients of the equation. Thus, we are talking about the inverse problem of determining the coefficients of the equation. The method of construction of conditions providing allocation of infinite number of sets of this type equations is offered, for which the problem under consideration is solvable in quadratures. Instead of introducing additional boundary conditions, restrictions on the structure of the equation are proposed, related to the possibilities of its factorization.

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 paper we consider the Goursat problem for the two-dimensional system of high order. The purpose is to find sufficient conditions of solvability of the considered problem in quadratures. The method of finding solutions of these problems in explicit calculation based on factorization of equations is devised. As a result the initial problem is reduced to 5 simpler problems: to four Goursat problems for equation and the Goursat problem for hyperbolic system of the second order. The final result in terms of the coefficients of the original system is formulated in two theorems.

A linear integral equation of the general type with three independent variables, for which the special case of this equation is known, has been studied along with indication of certain variants of solving it in quadratures. In this paper, new variants of such cases have been found by developing the technique used previously by the author for similar equations with two independent variables. In the first step of reasoning, a method has been suggested for reduction of the original integral equation to a differential one, the Goursat boundary values for which are calculated in quadratures under certain conditions. Then, two approaches have been applied to the Goursat problem: direct construction of its solutions based on the already known results; factorization of the left side of the differential equation by using the first- and second-order operators. Each of these approaches allows to single out a particular class of integral equations of the considered type that are solved in quadratures.

In this paper, we construct the solution to the Goursat problem for a generalized telegraph equation of fractional order with variable coefficients.

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]

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]

In the paper we study a loaded degenerate hyperbolic equation of the second order with variable coefficients. The principal part of the equation is the Gellerstedt operator. The loaded term is given in the form of the trace of desired solution on the degenerate line. The latter is located in the inner part of the domain. We investigate a boundary value problem. The boundary conditions are given on a characteristics line of the equation under study. For the model equation (when all subordinated coefficients are zero) we construct an explicit representation for solution of the Goursat problem. In the general case, we reduce the problem to an integral Volterra equation of the second kind. We apply the method realized by Sven Gellerstedt solving the second Darboux problem. In both cases, model and general, we use widely properties of the Green-Hadamard function.

In this paper the characteristic problem for the wave equation loaded with a special shift. A theorem on the uniqueness of the solution of the Goursat problem and find necessary conditions for its solvability.