The uniqueness of hierarchically extended backward solutions of the Wright–Fisher model

The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the Kolmogorov forward and backward equations, with a leading term that degenerates at the boundary. This degeneracy has the consequence that standard PDE tools do not apply, and solutions lack regularity properties. In this paper, we develop a regularizing blow-up scheme for the iteratively extended global solutions of the backward Kolmogorov equation presented in a previous paper, which are constructed from a known class of solutions, and establish their uniqueness for the stationary case. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the occurring singularities result from the loss of an allele. While in an analytical approach, this provides substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularizes the solution via a carefully constructed iterative transformation of the domain.     

The extended Cantor–Bendixson analysis of trees

The Hausdorff analysis of chains is an instance of the Cantor–Bendixson analysis of topological spaces. Using the methods of point-free topology I obtain a considerable extension of Hausdorffs methods applicable to all trees.Received January 27, 2004; accepted in final form August 3, 2004.     

General solution of the Bagley–Torvik equation with fractional-order derivative

This paper investigates the general solution of the Bagley–Torvik equation with 1/2-order derivative or 3/2-order derivative. This fractional-order differential equation is changed into a sequential fractional-order differential equation (SFDE) with constant coefficients. Then the general solution of the SFDE is expressed as the linear combination of fundamental solutions that are in terms of α-exponential functions, a kind of functions that play the same role of the classical exponential function. Because the number of fundamental solutions of the SFDE is greater than 2, the general solution of the SFDE depends on more than two free (independent) constants. This paper shows that the general solution of the Bagley–Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.     

The solution gap of the Brezis–Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).     

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.     

Positive periodic solution of the discrete Lasota–Wazewska model with impulse

By using coincidence degree theory, some conditions are obtained for the existence of positive periodic solution of the discrete Lasota–Wazewska model with impulse.     

The existence of an extremal solution to a nonlinear system with the right-handed Riemann–Liouville fractional derivative

A monotone iterative method is applied to show the existence of an extremal solution for a nonlinear system involving the right-handed Riemann–Liouville fractional derivative with nonlocal coupled integral boundary conditions. Two comparison results are established. As an application, an example is presented to demonstrate the efficacy of the main result.     

The influence of perceived stock value price histories in the mean–variance-instability model

The model introduced by H. Talpaz, A. Harpaz and J.B. Penson (1984. European Journal of Operational Research 14, 262–269) extends the mean–variance model introducing the concept of instability. In this way it is possible to see an investor's attitude towards predicted instability. In this paper we show how optimisation procedures based on penalty (or preferred) weighted instability matrices can be interpreted in terms of real time utility functions which depend on an `actual' and a `remembered' time series due to fading memory. This approach justifies some bounded normalised functions used to represent the investors preference between the irregular frequency fluctuations.     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

1-soliton solution of the Zakharov–Kuznetsov equation with dual-power law nonlinearity

In this paper, the Zakharov–Kuznetsov equation, with dual-power law nonlinearity is solved by using the solitary wave ansatze and 1-soliton solution is obtained. Using this soliton solution, a couple of conserved quantities, of this equation, are calculated.     

Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

On the fundamental solution of the Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators.     

A particle approximation of the solution of the Kushner–Stratonovitch equation

We construct a sequence of branching particle systems α _n convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve numerically the filtering problem. We prove that the rate of convergence of the algorithm is of order n ^?. This paper is the third in a sequence, and represents the most efficient algorithm we have identified so far. Received: 4 February 1997 / Revised version: 26 October 1998     

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

In this study, both the dual reciprocity boundary element method and the differential quadrature method are used to discretize spatially, initial and boundary value problems defined by single and system of nonlinear reaction–diffusion equations. The aim is to compare boundary only and a domain discretization method in terms of accuracy of solutions and computational cost. As the time integration scheme, the finite element method is used achieving solution in terms of time block with considerably large time steps. The comparison between the dual reciprocity boundary element method and the differential quadrature method solutions are made on some test problems. The results show that both methods achieve almost the same accuracy when they are combined with finite element method time discretization. However, as a method providing very good accuracy with considerably small number of grid points differential quadrature method is preferrable.     

Topological 1-soliton solution of the Biswas–Milovic equation with power law nonlinearity

This paper studies the topological or dark solitons due to the Biswas–Milovic equation with power law nonlinearity. The coefficients of the dispersion and nonlinearity terms are time dependent. The damping (gain) term with time-dependent coefficient is also taken into consideration. The solitary wave ansatz is used to carry out the integration. The only requirement is that the coefficient of the damping term is Riemann integrable.     

A closed-form solution of the Black–Litterman model with conditional value at risk

We consider a portfolio optimization problem of the Black–Litterman type, in which we use the conditional value-at-risk (CVaR) as the risk measure and we use the multi-variate elliptical distributions, instead of the multi-variate normal distribution, to model the financial asset returns. We propose an approximation algorithm and establish the convergence results. Based on the approximation algorithm, we derive a closed-form solution of the portfolio optimization problems of the Black–Litterman type with CVaR.     

Special two-soliton solution of the generalized Sine–Gordon equation with a variable coefficient

A new special two-soliton solution to the generalized Sine–Gordon equation with a variable coefficient is constructed analytically, by using the self-similar method and Hirota bilinear method. To construct this special solution, we do not utilize the pairs of one-soliton solutions, as is customarily done when solving the Sine–Gordon equation, but introduce two auxiliary self-similar variables in Hirota’s procedure. We also study features of this solution by choosing different self-similar variables. The results obtained confirm that the behavior of such Sine–Gordon solitons can be easily controlled by the selection of the self-similar variables.