On Riccati Matrix Differential Equations

In this paper square Riccati matrix differential equations are considered. The coefficients can be arbitrary time—dependent matrices and need not satisfy any symmetry conditions. Contributions to the basic problems — existence and asymptotic behaviour of solutions — are presented based on two new methods. The first one is the usage of maximum principles for second order linear differential equations, the second one is a variety of possibilities for the parametric representation of solutions of Riccati differential equations.     

Solutions of Semilinear Differential Equations Related to Harmonic Functions

This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of ^d and let (D), (D) and (D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1].     

Solving Differential Matrix Equations using Parareal

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDEs

Regular Functions Satisfying Irregular Ordinary Differential Equations

This paper deals with entire solutions to linear ordinary differential equations in the complex domain. We show that certain entire solutions to singular equations, cannot satisfy any normalized equation without singularities. We provide two proofs of this result, one based on the indicial equation and the other using the Frobenius notion of irreducibility. Our examples include the entire Bessel function.     

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {P_n}_n, with respect to a weight W, that are eigenfunctions of a second-order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second-order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.     

Almost Automorphic Functions in Frechet Spaces and Applications to Differential Equations

In this paper we first develop a theory of almost automorphic functions with values in Frechet spaces. Then, we consider the semilinear differential equation x'(t) = A x(t) + f(t, x(t)), t ∈ ℝ in a Frechet space X, where A is the infinitesimal generator of a C₀-semigroup satisfying some conditions of exponential stability. Under suitable conditions on f, we prove the existence and uniqueness of an almost automorphic mild solution to the equation.     

时滞矩阵微分方程的等度稳定性

在矩阵值范数定义的广义赋范空间上利用矩阵Liapunov泛函研究了时滞矩阵微分方程的等度稳定性,得出了若干新结果.     

Third-Order Differential Subordinations for Analytic Functions Associated with Generalized Mittag-Leffler Functions

In this paper, third-order differential subordination results are obtained for analytic functions associated with an operator defined by the normalized form of the generalized Mittag-Leffler functions. Some particular cases involving Mittag-Leffler and hyperbolic functions are also considered.     

Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

This paper focuses on the study of differential properties of the symmetric matrix-valued Fischer–Burmeister (FB) function. As the main results, the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian of the symmetric matrix-valued Fischer–Burmeister function are established, which can be utilized in designing implementable Newton-type algorithms for nonsmooth equations involving the symmetric matrix-valued FB function.     

A New Matrix Approach For Solving Second-Order Linear Matrix Partial Differential Equations

The basic aim of this article is to present a novel efficient matrix approach for solving the second-order linear matrix partial differential equations (MPDEs) under given initial conditions. For imposing the given initial conditions to the main MPDEs, the associated matrix integro-differential equations (MIDEs) with partial derivatives are obtained from direct integration with regard to the spatial variable x and time variable t. Hence, operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials are used to reduce the obtained MIDEs to the corresponding algebraic Sylvester equations. Using two well-known subspace Krylov iterative methods (i.e., GMRES(10) and Bi-CGSTAB) we provide two algorithms for solving the mentioned Sylvester equations. A numerical example is provided to show the efficiency and accuracy of the presented approach.

     

Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated Legendre Functions

Second-order linear ordinary differential equations with a large parameter u are examined. Asymptotic expansions involving modified Bessel functions are applicable for the case where the coefficient function of the large parameter has a simple pole. In this paper, we examine such equations in the complex plane, and convert the asymptotic expansions into uniformly convergent series, where u appears in an inverse factorial, rather than an inverse power. Under certain mild conditions, the region of convergence containing the simple pole is unbounded. The theory is applied to obtain exact connection formulas for general solutions of the equation, and also, in a special case, to obtain convergent expansions for associated Legendre functions of complex argument and large degree.     

Efficient Laguerre and Hermite Spectral Methods for Odd-Order Differential Equations in Unbounded Domains

Laguerre dual-Petrov-Galerkin spectral methods and Hermite Galerkin spectral methods for solving odd-order differential equations in unbounded domains are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Numerical results demonstrate the effectiveness of the suggested approaches.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

Mixed Fractional Differential Equations and Generalized Operator-Valued Mittag-Leffler Functions

Mathematical Notes - We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of...