The Wright–Fisher equation, which was introduced as a model to study demography in the presence of diffusion, has had a renaissance as a model for the migration of alleles in the genome. Our goal in this paper is to give a careful analysis of the fundamental solution to the Wright–Fisher equation, with particular emphasis on its behavior for a short time near the boundary.

The signed fundamental solution of a scalar conservation law is constructed explicitly or implicitly when its flux is nonconvex. The flux is assumed to have finite number of inflection points. The fundamental solution constructed consists of a series of rarefaction waves, contact discontinuities and a shock. These analytically constructed fundamental solutions are also compared with numerical a...

In this work, we introduce and analyze two new techniques for obtaining the Q factor in the QR factorization of some (or all) columns of a fundamental solution matrix Y of a linear diierential system. These techniques are based on elementary Householder and Givens transformations. We implement and compare these new techniques with existing approaches on some examples.

Abstract. The procedure of solving the system of the Navier-Stokes equations is proposed. The initial conditions: the velocity divergence (dilatation) being zero and the temperature being the known function, the initial density being constant. The problem is set in the infinite space. The solution of the Navier-Stokes equations was reduced to the solution of integral equations of the Volterra t...

The first complete orbital solution for the double-lined spectroscopic binary system γ 2 Velorum, obtained from measurements with the Sydney University Stellar Interferometer (SUSI), is presented. This system contains the closest example of a Wolf-Rayet star and the promise of full characterisation of the basic properties of this exotic highmass system has subjected it to intense study as an ar...

This paper provides a solution to the fundamental linear fractional order differential equation, namely, cdtqx(t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the induct...

We present a new preconditioner for the iterative solution of linear systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electormagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel ...

Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hypersphere SR with R > 0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in ter...

There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. In this paper we study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. Three extra shock types are introduced other than the usual genuine shock, which are left, right and double sided contacts. There are three kinds of phenomena of t...

We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K (x, y) = (xy)λ/2, around the steady state f (x) = x−(3+λ)/2 with λ ∈ (1, 2). Detailed estimates on its asymptotics are obtained. Some consequences are deduced for the flux properties of the particles distributions described by such models.