On the Drinfeld-Sokolov hierarchy of type E_6~((1)) and its topological solution

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type E_6~((1)), and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E_6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type F_4~((1)).     

On the optimum choice of weight functions in a class of variational calculations

Summary We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.     

Reciprocal Diffusions and Symmetries of Parabolic PDE: The Nonflat Case

We introduce a new set of Reciprocal Characteristics for the class of reciprocal diffusions naturally associated to a general parabolic second-order linear differential operator. All the coefficients of this operator, including the diffusion matrix, depend on time. This set of reciprocal characteristics is provided by the study of the symmetries of the differential operator. The Riemannian metric defined by the diffusion matrix is of central importance. Our reciprocal characteristics are the natural extension of Ovsiannikov's differential invariants to the time dependent parabolic case. We also show that the symmetries of the PDE coincide with the one parameter families of transformations which leave the usual stochastic Lagrangian as well as a modified Onsager–Machlup Lagrangian invariant.     

Balanced POD for model reduction of linear PDE systems: convergence theory

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3):997?C1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems.     

On a linear nth-order differential equation with special right-hand side in a Banach space

We show that the classical algorithm for finding a particular solution of a scalar linear differential equation can be applied to linear differential equations with bounded operator coefficients in a Banach space.     

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.     

Monotone schemes for a class of nonlinear elliptic and parabolic problems

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients satisfying the strict ellipticity condition, and that the right-hand side is a positive Radon measure. The numerical schemes are not derived from finite difference operators approximating differential operators, but rather from a general principle which ensures the convergence of approximate solutions. The main feature of these schemes is that they possess stencils stretching far from basic grid-rectangles, thus leading to system matrices which are related to M-matrices.     

Rank reduction,normal forms and stokes matrices

We introduce the reduction of the rank of a linear differential system as the unique solution of a natural problem and we relate it to the formal meromorphic classification (normal forms) and to the convergent meromorphic classification (Stokes matrices). The reduction of the rank makes sense for connections as well.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

Picard Iterations for Diffusions on Symmetric Matrices

Matrix-valued stochastic processes have been of significant importance in areas such as physics, engineering and mathematical finance. One of the first models studied has been the so-called Wishart process, which is described as the solution of a stochastic differential equation in the space of matrices. In this paper, we analyze natural extensions of this model and prove the existence and uniqueness of the solution. We do this by carrying out a Picard iteration technique in the space of symmetric matrices. This approach takes into account the operator character of the matrices, which helps to corroborate how the Lipchitz conditions also arise naturally in this context.     

On the eigenvalues of a class of saddle point matrices

We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners. The work of this author was supported in part by the National Science Foundation grant DMS-0207599 Revision dated 5 December 2005.     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

Parameter estimation using minimum norm differential approximation

We present mathematical results which can be used to compute the parameters of a system described by differential equations, using the method of minimum norm differential approximation. The algorithm is described and several examples are given in both the ordinary and partial differential equations cases. The approximating subspaces used in this algorithm are those spanned by certain B-splines of degree 3 in the O.D.E. case and by tensor products of B-splines in the P.D.E. case. Singular value decomposition is used in two distinct ways in the algorithm. The method described can be used on any type of differential operator with constant coefficients, i.e., elliptic, hyperbolic, parabolic, although only in the case of elliptic operators can error bounds between the data function and a generalized solution of the D.E. with the approximated parameters be estimated.     

Fractional-order Legendre functions for solving fractional-order differential equations

In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Rational differential operators and their kernels

A linear differential/integral operator is associated to a rational matrix in a natural way, and the behavior is defined to be the kernel of this operator. Conditions under which two rational matrices have the same behavior are derived.     

Tau-functions,Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.