The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

     

Time-dependent deformations of powers of the wave operator

Conditions under which the linear differential operators of the second order are equivalent to operators not containing “friction” (first partial derivatives) are investigated. One can construct iso-Huygens deformations for powers of the wave operator with time-dependent coefficients. The fundamental solutions of these deformations and conditions under which the Huygens principle holds are found. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 213–228.     

Solvable systems of linear differential equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.     

Propagation of singularities for operators with constant coefficient hyperbolic-elliptic principal part

Summary In this paper we consider partial differential operators of the type P(x, D)= P_m(D)+Q(x, D), where the constant coefficient principal part P_m is supposed to be hyperbolic-elliptic. We study the propagation of Gevrey singularities for solutions u of the equation P(x, D) u=f, for ultradistributions f, finding exactly to which spaces of ultradistribuiions u microlocally belongs. The results are obtained by constructing a fundamental solution for P when the lower order part Q is with constant coefficients, and a parametrix otherwise.     

A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.     

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.     

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.     

One-point commuting difference operators of rank 1

One-point commuting difference operators of rank 1 are considered. The coefficients in such operators depend on one functional parameter, and the degrees of shift operators in difference operators are positive. These operators are studied in the case of hyperelliptic spectral curves, where the base point coincides with a point of branching. Examples of operators with polynomial and trigonometric coefficients are constructed. Operators with polynomial coefficients are embedded in differential operators with polynomial coefficients. This construction provides a new method for constructing commutative subalgebras in the first Weyl algebra.     

Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

On localization of the spectrum of non-self-adjoint differential operators

In this paper, we consider the problem on localization of the spectrum of non-self-adjoint differential operators on unbounded domains with power coefficients. To find the location of spectrum points in the complex plane, we use isospectral deformations of differential operators and the properties of families of closed operators analytic in the Kato sense. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 11–19, 2006.     

Helmholtz operators and symmetric space duality

We consider the property of vanishing logarithmic term (VLT) for the fundamental solution of the shifted Laplace-d’Alembert operators □ + b (b a constant), on pseudo-Riemannian reductive symmetric spaces M. Our main result is that such an operator on the c-dual or Flensted–Jensen dual of M has the VLT property if and only if a corresponding operator on M does. For Lorentzian spaces, where the □ + b are hyperbolic, VLT is known to be equivalent to the strong Huygens principle. We use our results to construct a large supply of new (space, operator) pairs satisfying Huygens’ principle. Oblatum 23-XII-1995 & 22-VIII-1996     

The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients

We constructed a fundamental matrix of solutions of the Cauchy problem and studied its basic properties for a new class of linear parabolic systems with smooth bounded variable coefficients that includes a class of the Shilov-type parabolic systems of partial differential equations with nonnegative genus.     

Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite-differenceapproximations on an arbitrary fixed grid which preserve anynumber of integrals of the partial differential equation andpreserve some of its symmetries. A basis for the space of suchfinite-difference operators is constructed; most cases of interestinvolve a single such basis element. (The ‘Arakawa’Jacobian is such an element, as are discretizations satisfying‘summation by parts’ identities.) We show how thegrid, its symmetries, and the differential operator interactto affect the complexity of the finite difference.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Algebraic properties of Gardner’s deformations for integrable systems

We formulate an algebraic definition of Gardner’s deformations for completely integrable bi-Hamiltonian evolutionary systems. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. We find an exactly solvable two-component extension of the Liouville equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 101–117, July, 2007.     

Systems of equations of Kolmogorov type

We consider one class of degenerate parabolic systems of equations of the type of diffusion equation with Kolmogorov inertia. For systems whose coefficients may depend only on the time variable, we construct a fundamental matrix of solutions of the Cauchy problem and obtain estimates for this matrix and all its derivatives. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1650–1663, December, 2008.     

Construction of isospectral deformations of differential operators with periodic coefficients

Isospectral deformations of differential operators with periodic coefficients are constructed by modifying a method due to Burchnall and Chaundy. If the commutant of a differential operator L of order at least two consists of polynomials in L, then L admits holomorphic families of isospectral deformations of every positive dimension. The methods are independent of the order of the operator L.     

一类二阶二次变系数微分算子的不变子空间及其应用

研究了一类二阶二次变系数微分算子的不变子空间,讨论了这类微分算子不变子空间的应用,并给出了具体应用的一些例子.在这些例子中,构造了大量变系数非线性演化方程的精确解.     

Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method

The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.