Euler-Poisson-Darboux Equation Research Articles

Numerical study of distributed-order Bessel fractional derivative with application to Euler–Poisson–Darboux equation

The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts.

Non-reflective Propagation of Kink Waves in Magnetic-Flux Tubes in the Solar Atmosphere

We study the non-reflective propagation of kink waves in inhomogeneous magnetic-flux tubes. We use the thin-tube and zero-beta plasma approximations. The wave equation with the variable velocity is reduced to the Euler–Poisson–Darboux equation. This equation contains one dimensionless parameter. There are two infinite sequences of this parameter, one monotonically increasing and the other monotonically decreasing, when exact analytical solutions for the Euler–Poisson–Darboux equation can be obtained. For the monotonically increasing sequences the Euler–Poisson–Darboux equation becomes the equation describing spherically symmetric waves in multi-dimensional spaces. The general results are applied to kink-wave propagation in coronal magnetic loops. We consider a coronal magnetic loop of a half-circular shape. We find that for a fixed loop height there is a one-parametric family of dependences of the loop cross-sectional radius on the coordinate along the loop corresponding to the non-reflective kink-wave propagation.

On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations

On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations

On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations

On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations

Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

We consider an abstract Euler–Poisson–Darboux equation containing powers of an unbounded operator that is the generator of a Bessel operator function. Sufficient conditions for the unique solvability of the Dirichlet problem on the half-line are obtained. The question concerning the convergence of the solution to zero at infinity is investigated. Examples are given.

Time–space fractional Euler–Poisson–Darboux equation with Bessel fractional derivative in infinite and finite domains

In this paper, we study the time-fractional Euler–Poisson–Darboux equation with the Bessel fractional derivative. The Laplacian operator of this equation is considered in the ordinary and fractional derivatives and also in different coordinates. For the multi-dimensional Euler–Poisson–Darboux equation in the infinite domain (the whole space), we use the joint modified Meijer–Fourier transforms and establish a complex inversion formula for deriving the fundamental solution. The fractional moment of this solution is also presented in different dimensions. For studying the time-fractional Euler–Poisson–Darboux equation by the numerical methods in finite domain, we sketch the semi- and fully-discrete methods along with the matrix transfer technique to analyze the equation with fractional Laplacian operators in the cartesian, polar and spherical coordinates. The associated error and convergence theorems are also discussed. The illustrative examples are finally presented to verify our results in different coordinates.

Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

Dirichlet Problem on the Half-Line for an Abstract Euler–Poisson–Darboux Equation Containing Powers of an Unbounded Operator

Euler–Poisson–Darboux equations and iterated fractional Brownian motions

Euler–Poisson–Darboux equations and iterated fractional Brownian motions

On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms

On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms

Solving the Euler–Poisson–Darboux Equation of Fractional Order

Solving the Euler–Poisson–Darboux Equation of Fractional Order

On the Unique Solvability of Boundary Value Problems for an Abstract Euler–Poisson–Darboux Equations on a Finite Interval

On the Unique Solvability of Boundary Value Problems for an Abstract Euler–Poisson–Darboux Equations on a Finite Interval

Uniqueness of the Solution of the Cauchy Problem for the General Euler–Poisson–Darboux Equation

Uniqueness of the Solution of the Cauchy Problem for the General Euler–Poisson–Darboux Equation

Acceleration of gossip algorithms through the Euler–Poisson–Darboux Equation

Abstract Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of Boyd et al. (2006) converges to the heat equation, and the accelerated Jacobi polynomial iteration of Berthier et al. (2020) converges to the Euler–Poisson–Darboux (EPD) equation—a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to $t$—the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $\sqrt{t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.

FRACTIONAL WEIGHTED SPHERICAL MEAN AND MAXIMAL INEQUALITY FOR THE WEIGHTED SPHERICAL MEAN AND ITS APPLICATION TO SINGULAR PDE

In this paper we establish a mean value property for the functions which is satisfied to Laplace–Bessel equation. Our results involve the generalized divergence theorem and the second Green’s identities relating the bulk with the boundary of a region on which differential Bessel operators act. Also we design a fractional weighted mean operator, study its boundedness, obtain maximal inequality for the weighted spherical mean and get its boundedness. The connection between the boundedness of the spherical maximal operator and the properties of solutions of the Euler–Poisson–Darboux equation with Bessel operators is given as an application.

On the Solvability of Initial Problems for an Abstract Generalized Euler–Poisson–Darboux Equation of Higher Order

On the Solvability of Initial Problems for an Abstract Generalized Euler–Poisson–Darboux Equation of Higher Order

On the Relationship between the Solutions of an Abstract Euler–Poisson–Darboux Equation and Fractional Powers of the Operator Coefficient in the Equation

On the Relationship between the Solutions of an Abstract Euler–Poisson–Darboux Equation and Fractional Powers of the Operator Coefficient in the Equation

ЗАДАЧА РIК’Є ДЛЯ ОДНОГО МОДЕЛЬНОГО РIВНЯННЯГIПЕРБОЛIЧНОГО ТИПУ 4-ГО ПОРЯДКУ

Integral operators that transform arbitrary functions into regular solutions of hyperbolic equations of the second and higher orders are applied to solving boundary value problems. In particular, the Riquet problem for the Euler–Poisson–Darboux equation of the 4th order is posed and solved.

On the exact distributions of the maximum of the asymmetric telegraph process

In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval [0,t] under the conditions that the initial velocity V(0) is either c1 or −c2 and the number of changes of direction is odd or even. For the case V(0)=−c2 the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on c1 and c2, but not on the current time t. We obtain also the unconditional distribution of the maximum for either V(0)=c1 or V(0)=−c2 and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler–Poisson–Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper.We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications.

Mean Value Properties of Solutions to the Helmholtz and Modified Helmholtz Equations

Mean value properties of solutions to the m-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given. It involves the Euler–Poisson–Darboux equation. Despite the similar form of these properties for both equations, their consequences distinguish essentially. The restricted mean value property of harmonic functions is amended so that a function satisfying this property in a bounded domain of a special class solves the modified Helmholtz equation in this domain.

Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator

In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper.