Euler-Poisson-Darboux Equation Research Articles

On the singular sector of the Hermitian random matrix model in the large N limit

The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler–Poisson–Darboux equations.

Linearized perturbations of the Kerr spacetime and outer boundary conditions in numerical relativity

This paper discusses linearized vacuum gravitational perturbations of the Kerr spacetime in a neighbourhood of future null infinity . Unlike earlier discussion of perturbations of the Kerr spacetime we avoid the use of spheroidal harmonics and harmonic time dependence. Instead we develop the theory in terms of Hertz potentials and spherical harmonics with coupling between modes. The ‘master equation’ is a single complex scalar wave equation which, in the Minkowski limit, reduces to the Euler–Poisson–Darboux equation. We solve this by Picard iteration making extensive use of the flat spacetime Riemann–Green function. As an application we consider the problem of outer boundary conditions for numerical relativity and generalize earlier results of Buchman and Sarbach (2006 Class. Quantum Grav. 23 6709–44, 2007 Class. Quantum Grav. 24 S307–26) for the Schwarzschild case.

Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler–Poisson–Darboux equation

It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler–Poisson–Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock-type singularities are presented.

A singular Cauchy problem for the Euler–Poisson–Darboux equation

We consider a singular Cauchy problem for the Euler–Poisson–Darboux equation of Fuchsian type in the time variable with ramified Cauchy data. In this paper we establish an expansion of the solutions in a series of hypergeometric functions and then investigate the nature of the singularities of the solutions.

The Euler–Poisson–Darboux equation for relativists

The Euler–Poisson–Darboux (EPD) equation is the simplest linear hyperbolic equation in two independent variables whose coefficients exhibit singularities, and as such must be of interest as a paradigm to relativists. Sadly it receives scant treatment in the textbooks. The first half of this review is didactic in nature. It discusses in the simplest terms possible the nature of solutions of the EPD equation for the timelike and spacelike singularity cases. Also covered is the Riemann representation of solutions of the characteristic initial value problem, which is hard to find in the literature. The second half examines a few of the possible applications, ranging from explicit computation of the leading terms in the far-field backscatter from predominantly outgoing radiation in a Schwarzschild space–time, to computing explicitly the leading terms in the matter-induced singularities in plane symmetric space–times. There are of course many other applications and the aim of this article is to encourage relativists to investigate this underrated paradigm.

Quenching for a generalization of semilinear Euler–Poisson–Darboux equations

A generalization of semilinear Euler–Poisson–Darboux equations is studied; criteria for solutions of the generalization of semilinear Euler–Poisson–Darboux equations with initial and boundary conditions to quench in finite time are obtained. Furthermore, criteria for the derivatives of solutions to blow up are given.

Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation

We establish sharp estimates for distributional solutions to the Euler–Poisson–Darboux equation posed in a periodic domain. These equations are highly singular, and setting the Cauchy problem requires a precise understanding of the nature of the singularities that may arise in weak solutions. We consider initial data in a space of functions with fractional derivatives such that weak solutions are solely integrable, and we derive sharp continuous dependence estimates for solutions to the initial-value problem. Our results strongly depend on a key parameter arising in the Euler–Poisson–Darboux equation.

Inversion formulae for the spherical mean in odd dimensions and the Euler–Poisson–Darboux equation

The paper contains a simple proof of the Finch–Patch–Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. Inversion of that transform is important for thermoacoustic tomography and represents a challenging mathematical problem. The argument relies on the idea of analytic continuation and known properties of the Erdélyi–Kober fractional integrals. The result is applied to the solution of the Cauchy problem for the Euler–Poisson–Darboux equation with initial data on the cylindrical surface.

THE SCALAR WAVE EQUATION IN A NON-COMMUTATIVE SPHERICALLY SYMMETRIC SPACE-TIME

Recent work in the literature has studied a version of non-commutative Schwarzschild black holes where the effects of non-commutativity are described by a mass function depending on both the radial variable r and a non-commutativity parameter θ. The present paper studies the asymptotic behavior of solutions of the zero-rest-mass scalar wave equation in such a modified Schwarzschild space-time in a neighborhood of spatial infinity. The analysis is eventually reduced to finding solutions of an inhomogeneous Euler–Poisson–Darboux equation, where the parameter θ affects explicitly the functional form of the source term. Interestingly, for finite values of θ, there is full qualitative agreement with general relativity: the conformal singularity at spacelike infinity reduces in a considerable way the differentiability class of scalar fields at future null infinity. In the physical space-time, this means that the scalar field has an asymptotic behavior with a fall-off going on rather more slowly than in flat space-time.

Exact solutions for a different version of the nonhomogeneous E.P.D equation

In this article we construct explicit solutions involving hypergeometric functions to a different version of the nonhomogeneous Euler–Poisson–Darboux equation. We also study the singularities of the solution and give its analytical continuation. The proof of these results relies on the theory of hypergeometric functions.

Linear Differential Relations Between Solutions of the Class of Euler-Poisson-Darboux Equations

All the linear first-order relations of the form $$u^{(\beta )} = A(r,z)\frac{{\partial u^{(\alpha )} }}{{\partial r}} + B(r,z)\frac{{\partial u^{(\alpha )} }}{{\partial z}} + C(r,z)u^{(\alpha )}$$ between solutions u = u(α) and u = u(β) of the class of Euler-Poisson-Darboux (EPD) equations are obtained. We consider applications of the obtained relations for obtaining identities between the EPD operators, recursive relations for the Bessel function, and general solutions of the EPD equation in special cases in application to the gas dynamics of a polytropic gas.

Singularities of solutions to some second-order partial differential equations which include the E.P.D. equation

In this article we construct singular solutions for some second–order partial differential equations which include the Euler–Poisson–Darboux equation in a complex domain.

An Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

An Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

Some Classification Results for Hyperbolic Equations F(x, y, u, ux, uy, uxx, uxy, uyy)=0

We provide a contact invariant characterization for equations of the formuxy+a(x, y, u)ux+b(x, y, u)uy+c(x, y, u)=0,uxy+a(x, y)ux+b(x, y)uy+c(x, y, u)=0,uxy+c(x, y, u)=0,uxy=0. We classify all equations of the form uxy+f(x, y, u, ux, uy)=0 for which the two Ovsiannikov's invariants are constants. These results include characterization of the Klein–Gordon equation uxy=u, the Liouville equation uxy=eu, and the class of Euler–Poisson–Darboux equations. It is shown that the wave equation uxy=0, Liouville equation, and the linear equation uxy=2u/(x+y)2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge–Ampére equation Darboux integrable at level one is equivalent to an equation of type uxy+f(x, y, u, ux, uy)=0. We prove that the hyperbolic Fermi–Ulam–Pasta (FPU) equation uyy=κ(ux)2uxx is contact equivalent to a linear equation of type uxy=c(x+y)u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type uxy=F(u, ux) that describe pseudo-spherical surfaces.

Explicit and approximate solutions of second-order evolution differential equations in Hilbert space

The explicit closed-form solutions for a second-order differential equation with a constant self-adjoint positive definite operator coefficient A (the hyperbolic case) and for the abstract Euler–Poisson–Darboux equation in a Hilbert space are presented. On the basis of these representations, we propose approximate solutions and give error estimates. The accuracy of the approximation automatically depends on the smoothness of the initial data. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 111–131, 1999

$q$-difference analogue of the Euler-Poisson-Darboux equation and its Laplace sequence

$q$-difference analogue of the Euler-Poisson-Darboux equation and its Laplace sequence

On Finite Element Methods for the Euler–Poisson–Darboux Equation

In this work we deal primarily with the derivation of various convergence estimates for some semidiscrete and fully discrete procedures which might be used in the approximation of exact solutions of initial-boundary value problems with homogeneous Dirichlet boundary conditions for the Euler–Poisson–Darboux equation. These procedures include the ordinary Galerkin method based on conforming finite element subspaces as well as certain methods which do not require such restrictions. Although the equation is of hyperbolic type, the results are somewhat analogous to those known for parabolic equations. This is due to the presence of a limited “smoothing” property. This paper contains $L_2 $ estimates, maximum norm estimates, negative norm estimates, interior estimates of difference quotients and superconvergence estimates of the error. Most of the proofs are based on different modifications of an energy method, some of them intrinsically depend on the Weinstein recursion formulae and on known results for the as...

On a Generalized Euler–Poisson–Darboux Equation

Solutions of the following Cauchy problem are obtained by means of Fourier transform methods: \[\begin{gathered} \sum_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_i^2 }} - \frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{\lambda }{t}\frac{{\partial u}}{{\partial t}}} = c^2 u, \hfill \left. \begin{gathered} u(x,t) = f(x) \hfill u_t (x,t) = 0 \hfill \end{gathered} \right\}\quad {\text{at }} t = 0. \hfill \end{gathered} \] Special cases for the parameters $\lambda $ and c are considered and the regularity of the solutions is studied in some detail.

Lie theory and separation of variables. 11. The EPD equation

We show that the Euler–Poisson–Darboux equation {∂tt−∂rr−[(2m+1)/r]∂r}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C).

Asymptotic Evaluation of Fractional Integral Operators with Applications

A technique is developed which yields an asymptotic expansion in the two limits $\lambda \to 0^ + $ and $\lambda \to \infty $ for the fractional integral operator of order $\mu $ with respect to the function $\lambda ^p $ given by $I_{\lambda ^p }^\mu f(\lambda )\frac{1}{{\Gamma (\mu )}}\int_0^\lambda {(\lambda ^p - \xi ^p )^{\mu - 1} p\xi ^{p - 1} f(\xi )d\xi } ,$ under general conditions that f be algebraically dominated near 0 and $\infty $. Representing $I_{\lambda ^p }^\mu f(\lambda )$ as a convolution of Mellin transforms the domain of the transform is extended by analytic continuation. By moving the contour of integration to the right or the left an asymptotic expansion for $I_{\lambda ^p }^\mu f(\lambda )$ can be systematically generated for $\lambda \to \infty $ or $\lambda \to 0^ + $. The technique is illustrated by the asymptotic expansion of fractional integral operators derived from the Euler–Poisson–Darboux equation and generalized axially symmetric potential theory.