Biorthogonal System Of Functions Research Articles

Construction of Solutions of the Helmholtz Equation in a Cylindrical Coordinate System in the Form of Homogeneous Polynomials in Two Biorthogonal Systems of Functions

Construction of Solutions of the Helmholtz Equation in a Cylindrical Coordinate System in the Form of Homogeneous Polynomials in Two Biorthogonal Systems of Functions

AN INVERSE SOURCE PROBLEM FOR A GENERALIZED TIME FRACTIONAL DIFFUSION EQUATION

Abstract This paper is devoted to the study of the inverse problem of finding the time- dependent coefficient of a generalized time fractional diffusion equation, in the case of non- local boundary and integral overdetermination conditions. The existence and uniqueness of the solution of the considered inverse problem are obtained by a method based on the expan- sion of the solution by using a bi-orthogonal system of functions and the fractional calculus. Moreover, we show its continuous dependence on the data. At the end, two examples are presented to illustrate the obtained results.

Construction of the solutions of the Helmholtz equation in cylindrical system of coordinates in the form of homogeneous polynomials by two biorthogonal systems of functions

Construction of the solutions of the Helmholtz equation in cylindrical system of coordinates in the form of homogeneous polynomials by two biorthogonal systems of functions

Second Boundary-Value Problem for the Lavrent’ev–Bitsadze Equation in a Rectangular Domain with Two Degeneration Lines

For a mixed-type equation, we examine the second boundary-value problem and by using the spectral method prove the uniqueness and existence of solutions. The uniqueness criterion is proved based on the completeness property of the biorthogonal system of functions corresponding to the onedimensional spectral problem. A solution of the problem is constructed as the sum of a biorthogonal series.

Combinations of cross-products of Bessel functions, associated addition formulas and biorthogonal systems

This study deals with cross-products of Bessel functions of the first and second kinds χ ν ( p , w ) = Y ν ( w ) J ν ( p ) − Y ν ( p ) J ν ( w ) , ν , p , w ∈ C . Combinations χ ν ( p , w ) are shown to be generating functions for associated Legendre functions. This fact permits to establish new integral and series representations of χ ν ( p , w ) . Under special choices of parameters, the representations become integral transforms or coefficients of orthogonal series whose inverting leads to new addition formulas for Bessel functions and their continuous analogues. On other hand, integral representations in some cases are integral operators with known inversion formulas and can be treated as operator relations between two systems of functions. Hence theorems on expanding functions into χ ν ( p , w ) series are proved and biorthogonal systems of functions are found in special cases. As examples, it is applied to evaluate new integrals and series.

An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions

We consider the inverse problem of determination of the solution and a source term for a time fractional diffusion equation in two dimensional space. The time fractional derivative is the Hilfer derivative. A bi-orthogonal system of functions in L2(Ω), obtained from the associated non-self-adjoint spectral problem and its adjoint problem, is used to prove the existence and uniqueness of the solution of the inverse problem. The stability of the solution of the inverse problem on the given data is proved.

The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type

We study the problem with boundary conditions of the first and second kind on the boundary of a rectangular domain for an equation with two internal perpendicular lines of change of a type. With the use of the spectral method we prove the unique solvability of the mentioned problem. The eigenvalue problem for an ordinary differential equation obtained by separation of variables is not self-adjoint, and the system of root functions is not orthogonal. We construct the corresponding biorthogonal system of functions and prove its completeness. This allows us to establish a criterion for the uniqueness of the solution to the problem under consideration. We construct the solution as the sum of the biorthogonal series.

Neumann problem for the Lavrent’ev–Bitsadze equation with two type-change lines in a rectangular domain

The Neumann problem for an equation with two perpendicular internal type-change lines in a rectangular domain is investigated. Uniqueness and existence theorems are proved by applying the spectral method. The separation of variables yields an eigenvalue problem for an ordinary differential equation. This problem is not self-adjoint, and the system of its eigenfunctions is not orthogonal. A corresponding biorthogonal system of functions is constructed and proved to be complete. The completeness result is used to prove a necessary and sufficient uniqueness condition for the problem under study. Its solution is constructed in the form of the sum of a biorthogonal series.

О полноте одной пары биортогонально сопряженных систем функций

В работе изучена спектральная задача для обыкновенного дифференциального уравнения второго порядка на конечном отрезке с разрывным коэффициентом при старшей производной. На концах отрезка заданы краевые условия первого рода, во внутренней точке заданы условия сопряжения по функции и первой производной. Найдены собственные значения с соответствующей асимптотикой как корни трансцендентного уравнения. Система собственных функций представляет собой тригонометрические синусы на одной половине отрезка и гиперболические синусы - на другой. Система собственных функций неортогональна в пространстве квадратично суммируемых функций. Построена соответствующая ей биортогональная система функций как решение сопряженной задачи. При доказательстве полноты биортогональной системы использована известная теорема Келдыша о полноте системы собственных функций несамосопряженного оператора.

On boundary properties and biorthogonal systems in the spaces A ω 2 ⊂ H 2

The paper is devoted to investigation of some Hilbert spaces A ω 2 which are contained in Hardy’s H 2 of functions holomorphic in |z| < 1. In particular, it is proved that the boundary properties of functions from A ω 2 are characterized by some sharp ω-capacities, and some biorthogonal systems of functions containing small degrees of the Cauchy kernel are found in A ω 2 by means of an explicit isometry between A ω 2 and H 2.

One class of biorthogonal systems of functions that arise in the solution of the Helmholtz equation in the cylindrical coordinate system

We consider one class of systems of functions biorthogonal on a closed contour in a complex domain. We study properties of these systems of functions and conditions for expandability of analytic functions in series in the basic system. Examples of expansion of elementary functions in such series are given.

On biorthogonal systems associated to orthogonal polynomials

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.

The quantum algebraUq(su2) andq-Krawtchouk families of polynomials

The aim of this paper is to study quantum and affine q-Krawtchouk polynomials by means of operators of irreducible representations of the quantum algebra Uq(su2). We diagonalize a certain operator I in such a representation and show that elements of the transition matrix from the initial (canonical) basis to the basis consisting of eigenfunctions of the operator I are expressed in terms of quantum q-Krawtchouk polynomials. Then we find an explicit form of the operator in the basis of the eigenfunctions of I, in which it has the form of a Jacobi matrix. Normalizing this basis and using the operator , we thus derive the orthogonality relations for quantum and affine q-Krawtchouk polynomials. We show that affine q-Krawtchouk polynomials are dual to quantum q-Krawtchouk polynomials. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

Bigq-Laguerre andq-Meixner polynomials and representations of the quantum algebraUq(su1,1)

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are explicitly found. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial Uq(su1,1) basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials Mn(x; b, c; q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the q-Meixner polynomials Mn(x; b, c; q) with b < 0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

The method of homogeneous solutions and biorthogonal expansions in the plane problem of the theory of elasticity for an orthotropic body

In connection with the solution of boundary-value problems of the theory of elasticity for an orthotropic strip the problem of expanding two different limiting functions in series in terms of the characteristic elements of the generalized eigenvalue problem is considered. A system of functions biorthogonal to the system of characteristic elements is constructed. The double completeness of characteristic elements is proved. It is shown that the biorthogonality condition is equivalent to a generalized orthogonality relation of Papkovich type. The form of systems of biorthogonal functions is established. For expansions of a special form the biorthogonal systems are identical with the systems of characteristic elements. Biorthogonal systems of functions are constructed corresponding to expansions of a general form. Using the biorthogonal systems obtained, explicit expressions for the expansion coefficients are found. An example demonstrating the existence of a non-trivial double null expansion is given.

Biorthogonal expansions in problems of mechanics: PMM vol. 43, no. 4, 1979, pp. 698–708

The author has shown in [1] how biorthogonal systems of functions can be used to obtain explicit solutions of problems of mechanics (in the form of series in terms of the specified systems) which could not be obtained by other methods. Below it is shown that such solutions can be constructed for problems which can be described by the following equation: Lg = f ( f ϵ H, g ϵ H') (0.1) where L is a linear operator acting from the Hilbert space H' to the Hilbert space H. The plane contact problem of the theory of elasticity is used to illustrate the method.

A biorthogonal system of functions

In [i] Leont'ev investigated questions of representability of functions in regions of the complex plane by means of series with respect to the systems {e z vZ}, ~f(~ vz)}, {A(z, ~ v)}He constructed systems of functions which are biorthogonal to the indicated systems, and he studied their properties. Later on, the question of represeniability of an arbitrary analytic function by means of a Dirichlet series, or a series of a more general type, was investigated with the aid of these biorthogonal systems. With an analogous attitude we investigate in this paper a system of functions which is constructed in the following manner. Let Dy ~ y(~)(z) + pt(z)y(~-')(z) + ... + p , ( z )g ( z ) , s >t 2, be a linear differential operator whose coefficients Pi(Z) ..... Ps (z) will be assumed to be entire functions, although one could assume that they are analytic in some domain. Let us denote by y(z, #) the solution of the equation Dy = #Sy which satisfies the initial conditions y(z 0, # ) = 1, y ' ( z 0, p ) = # . . . . . y(S-1)(z0, # ) = # s i . Note tha t i f p l ( z ) . . . = p s ( Z ) . ~ 0 , t h e n y ( z , # ) = e#(Z-Z0).

Extended orthogonality relationships for some problems of elasticity theory: PMM vol. 34, n≗5, 1970, pp. 945–951

In connection with the solution of boundary-value problems of the theory of elasticity for an orthotropic strip the problem of expanding two different limiting functions in series in terms of the characteristic elements of the generalized eigenvalue problem is considered. A system of functions biorthogonal to the system of characteristic elements is constructed. The double completeness of characteristic elements is proved. It is shown that the biorthogonality condition is equivalent to a generalized orthogonality relation of Papkovich type. The form of systems of biorthogonal functions is established. For expansions of a special form the biorthogonal systems are identical with the systems of characteristic elements. Biorthogonal systems of functions are constructed corresponding to expansions of a general form. Using the biorthogonal systems obtained, explicit expressions for the expansion coefficients are found. An example demonstrating the existence of a non-trivial double null expansion is given.

Biorthogonal Systems of Functions

Biorthogonal Systems of Functions

Biorthogonal systems of functions

Biorthogonal systems of functions