V.L. Makarov

National Academy of Sciences of Ukraine | ISP · Department of Numerical Mathematics

УДК 519.62, 519.63Запропоновано та проаналiзовано експоненцiально збiжний наближений метод розв’язування диференцiального рiвняння з правосторонньою дробовою похiдною Рiмана – Лiувiлля i необмеженим операторним коефiцiєнтом у банаховому просторi. Застосовано зображення розв’язку за допомогою iнтеграла Данфорда – Кошi по гiперболi, що охоплює спектр...

Graded A 3 B 5-semiconductors are prospective materials for devices with a wide range of applications because of their peculiarities regarding many physical effects. One of these peculiarities , connected with the effect of photon recycling in semiconductors with high internal quantum efficiency, may essentially influence the parameters of devices...

We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the S. Chernikov theorem) connected with the conditions of existence of linear interpolation polynomials in Euclidean spaces.

A homogeneous fractional-differential equation with a fractional Hardy—Titchmarsh integral and an unboun-ded operator coefficient in a Banach space is considered. The conditions for the representation of the solution in the form of a Danford—Cauchy integral are established, and an exponentially convergent approximation method is developed.

We consider the boundary value problems (BVPs) for linear second-order ODEs with a strongly positive operator coefficient in a Banach space. The solutions are given in the form of the infinite series by means of the Cayley transform of the operator, the Meixner type polynomials of the independent variable, the operator Green function, and the Fouri...

УДК 517.988 Запропоновано нові критерiї сумiсностi лiнiйної системи рiвнянь (еквівалентні теоремі Кронекера - Капеллi) та нерівностей (еквівалентні теоремі С. М. Чернікова), пов'язані з умовами існування лінійного інтерполяційного полінома в евклiдових просторах.

The ideas of the method of fictitious domains and homotopy are combined with an aim to reduce the solution of boundary-value problems for multidimensional partial differential equations (PDE) in domains of any shape to an exponentially convergent sequence of PDE in a parallelepiped (or, in the 2D case, in a rectangle). This enables us to decrease t...

We consider the boundary value problems (BVPs) for linear secondorder ODEs with a strongly positive operator coefficient in a Banach space. The solutions are given in the form of the infinite series by means of the Cayley transform of the operator, the Meixner type polynomials of the independent variable, the operator Green function and the Fourier...

A recurrence algorithm for finding particular solutions of the resonant equation of the fourth order connected with the generalization of Laguerre and Legendre polynomials is constructed and substantiated. For this purpose, we use the general theorem on the representation of particular solutions of resonant equations in Banach spaces proved by Maka...

We establish the conditions of existence and determine the general structure of solutions of resonant and iterative equations in Banach spaces and propose their algorithmic realization.

We represent the solution {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an oper...

We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm for finding their particular and general solutions in the closed form. This algorithm is especially suitable for the computer-algebra tools, such as Maple. The re...

We study some resonant equations related to the classical orthogonal polynomials and propose an algorithm for finding their particular and general solutions in the explicit form. The algorithm is especially suitable for the computer algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the eff...

A new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the solution of fourth order Sturm--Liouville problem on a finite interval in the Hilbert space. The eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated. The sufficient cond...

We consider spectral problems for the Schrödinger operator with polynomial potentials in ℝK, K ≥ 2. By using a functional-discrete (FD-)method and the Maple computer algebra system, we determine a series of exact least eigenvalues for the potentials of special form. In the case where the traditional FD-method is divergent (the degree of the polynom...

A number of properties of a special case of Meixner polynomials given by their generating function are investigated. These polynomials arise when applying the Cayley transformation method to solving the first bounda ryvalue problem for an abstract differential equation of the second order with an unbounded operator coefficient.

We consider the Dirichlet boundary value problem for linear fractional differential equations with the Riemann–Liouville fractional derivatives. By transforming the boundary value problem to the integral equation, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of the boundary value pr...

We construct and analyze grid methods for solving the first boundary-value problem for an ordinary differential equation with the Riemann–Liouville fractional derivative. Using Green’s function, we reduce the boundary-value problem to the Fredholm integral equation, which is then discretized by means of the Lagrange interpolation polynomials. We pr...

A new method for solving stiff two-point boundary value problems is described and compared to other known approaches using the Troesch's problem as a test example. The method is based on the general idea of alternate approximation of either the unknown function or its inverse and has a genuine “immunity” towards numerical difficulties invoked by th...

In the present paper, we thoroughly investigate the theoretical properties of the SI-method, which was firstly introduced in arXiv:1601.04272v8 and proved to be remarkably stable when applied to a certain class of stiff boundary value problems. In particular, we provide sufficient conditions for the method to be applicable to the given two-point bo...

A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete (FD-) method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schr\"odinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme...

New exact representations for the solutions of numerous one-dimensional spectral problems for the Schrödinger operator with polynomial potential are obtained by using a technique based on the functional-discrete (FD) method. In the cases where the ordinary FD-method is divergent, we propose to use its modification, which proves to be quite efficien...

A new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the solution of fourth order Sturm--Liouville problem on a finite interval in the Hilbert space. The eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated. The sufficient cond...

We construct an abstract continued fraction of the Thiele type as an interpolating fraction for a nonlinear operator acting from a linear topological space X to an algebra Y with identity. In some special cases, this fraction transforms either into a classical Thiele fraction or into a matrix-valued fraction of the Thiele type depending on many var...

A grid method for solving the first boundary value problem for ordinary and partial differential equations with the Riemann-Liouville fractional derivative is justified. The algorithm is based on using Green's function, the Fredholm integral equation, and the Lagrange interpolation polynomial. The impact of the Dirichlet boundary condition on the a...

The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued fraction contains a standard interpolation continued C-fraction which is used to approximate the functions of...

A new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a se...

We consider a nonlocal problem for the first-order differential equation with unbounded operator coefficient in a Banach space and a nonlinear integral nonlocal condition. We propose an exponentially convergent method for the numerical solution of this problem and justified this method under the assumptions that the indicated operator coefficient A...

A new algorithm for selfadjoint eigenvalue problems for the fractional Jacobi-type ODE with the polynomial potential is proposed. It is shown that the corrections to the eigenfunctions of the FD-method are linear combinations of the generalized Jacobi functions ${}^{+} J_{n}^{(-\alpha ,\beta )} (x)$ with a number of summands depending on the degree...

It is shown that the rate of convergence of difference schemes used for the solution of the Sturm–Liouville problem for linear ordinary differential equations is higher in the immediate vicinity of the boundary. Moreover, we obtain the a priori estimates of accuracy that can be regarded as a quantitative confirmation of this effect. The results of...

A new algorithm for eigenvalue problems for the linear operators of the type A = A + B with a special application to high order ordinary differential equations is proposed and justified. The algorithm is based on the approximation of A by an operator ¯ A = A + ¯ B where the eigenvalue problem for ¯ A is supposed to be simpler then that for A. The a...

We generalize the well-known results on the interpolation of a function of single variable by the Thiele–Hermite fraction with arbitrary multiplicity of each interpolation node to the case of a functional acting from the space of piecewise continuous functions with finitely many discontinuities of the first kind. We obtain an interpolating integral...

A new algorithm for the solution of eigenvalue problems for linear operators of the form A = A + B (with a special application to high-order ordinary differential equations) is proposed and justified. The algorithm is based on the approximation of A by an operator (Formula presented.) such that the eigenvalue problem for Ā is supposed to be simpler...

A new algorithm for eigenvalue problems for linear differential operators with fractional derivatives is proposed and justified. The algorithm is based on the approximation (perturbation) of the coefficients of a part of the differential operator by piecewise constant functions where the eigenvalue problem for the last one is supposed to be simpler...

In this work we obtain necessary conditions for the functional-discrete (FD-) method's super-exponential convergence with respect to eigenvectors. The mentioned method is targeted to solve spectral problems for linear operators in Banach space which can contain eigenvalues of arbitrary multiplicity. (Ukrainian. English summary)

We have constructed and substantiated a generalization of continued Thiele-type fractions to the case of interpolation of nonlinear operators acting from a linear topological space X into an algebra Y with unit element I. It is shown that important particular cases of this generalization are interpolation continued Thiele-type fractions for vector-...

A new algorithm for eigenvalue problems for the linear operators of the type à = A + B with a special application to high order ordinary differential equations is proposed and justified. The algorithm is based on the approximation of à by an operator ¯Ã = A + ¯ B where the eigenvalue problem for ¯Ã is supposed to be simpler then that for Ã. The alg...

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient A is strongly positive and certain exist...

A new algorithm for the eigenvalue problems for linear self-adjont operators in the form of sum $A+B$ with a discrete spectrum in a Hilbert space is proposed and justified. The algorithm is based on the approximation of B by an operator $\overline{B}$ such that the eigenvalue problem for $A+\overline{B}$ is computationally simpler than that for $A+...

A new algorithm for eigenvalue problems for the linear self-adjoint operators in the form of sum $A + В$ with a discrete spectrum in Hilbert space is proposed and justified. The algorithm is based on the approximation of $В$ by an operator $\overline{В}$ such that the eigenvalue problem for $A+\overline{В}$ is computationally simpler than that for...

We consider the problem of interpolating a multivariable function defined on a bounded domain using its traces on parametric hypersurfaces. Our approach is based on the theory of operator polynomial interpolation. We construct the corresponding operator interpolation polynomial for a given function and analyse in detail particular two- and three-di...

This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal problem to its classical initial value analogue, uses more advanced analysis. That is a validation of the corre...

We consider a scalar Sturm-Liouville problem with the Dirichlet boundary conditions where the potential q (x) is assumed to be a derivative of the function with bounded variation. The application of the abstract FDmethod scheme to such eigenvalue problem is studied in the scope of this work. In addition to the general case when the function $\bar{q...

A new algorithm of the functional-discrete method (FD-method) is proposed for solving Sturm–Liouville problems on an interval with Dirichlet–Neumann boundary conditions where the potential is a polynomial. A software implementation of the algorithm using a computer algebra software package shows its high efficiency. (Russian. English summary)

We propose explicit and implicit schemes of the functional-discrete method (FD-method) for the solution of the nonlinear Klein–Gordon equation. The algorithm of the proposed FD-method is described and investigated from the viewpoint of its complexity. The explicit and implicit schemes of the FD-method are compared by using a numerical example.

To solve a regular scalar Sturm-Liouville problem for a second-order ordinary differential equation on the interval [0,1] with the Dirichlet boundary conditions we study the behaviour of the finite difference (FD)-method components of the eigenvalue number n versus the smoothness of the potential q(x) whereas the approximate q ¯(x) is identical to...

The article develops and proves an exponentially convergent numerical-analytical method (the FD-method) for solving Sturm-Liouville problems with a singular Legendre operator and a singular potential. Obtained within are sufficient conditions for convergence of the method and a priori estimates of its accuracy. A detailed algorithm for programmatic...

We prove that the FD-method, when applied to the Sturm-Liouville problem for a second-order ordinary differential equation with Dirichlet boundary conditions, converges faster than as compared with the result of the previous articles by V. L. Makarov and his students. A substantially new algorithm for the FD-method is presented and shown to be high...

A new functional-discrete method is proposed for Sturm-Liouville problems with matrix coefficients in the case when the base problem has multiple eigenvalues. Elements of the matrix coefficients are odd functions about the point 1/2 on the interval [0,1], which creates substantial difficulties for both theoretical investigation and algorithmic impl...

The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown...

We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is an “isosceles right triangle.” As a special case, Bernstein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined.

We survey the main results on the construction and study of exact and truncated compact difference schemes of high-order accuracy for a numerical solution of boundary-value problems for ordinary differential equations and the development of new efficient algorithms for the numerical solution of boundary-value problems with given accuracy and automa...

In the paper we have developed a theory of stability preserving structural transformations of systems of second-order ordinary differential equations (ODEs), i.e., the transformations which preserve the property of Lyapunov stability. The main Theorem proved in the paper can be viewed as an analogous of the Erugin's theorem for the systems of secon...

The scalar BVP d 2 u dx 2 − m 2 u = −f (x, u) , x ∈ (0, ∞) , u (0) = µ 1 , lim x→∞ u (x) = 0, on the infinite interval [0, ∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme of which the solution coincides with the p...

In the paper we present a functional-discrete method for solving the Goursat problem for nonlinear Klein-Gordon equation. The sufficient conditions providing that the proposed method converges superexponentially are obtained. The results of numerical example presented in the paper are in good agreement with the theoretical conclusions.

In the paper we present a functional-discrete method for solving Sturm-Liouville problems with potential including function from L_{1}(0,1) and \delta-function. For both, linear and nonlinear cases the sufficient conditions providing superexponential convergence rate of the method are obtained. The question of possible software implementation of th...

Positive polynomial operator that approximates Urison operator, when integration domain is a "regular triangle" is investigated. We obtain Bernstein Polynomials as a particular case.

In the paper we describe a superexponentially convergent numerical-analytical method for solving the eigenvalue problem for the some class of singular differential operators with boundary conditions. The method (FD-method) was firstly proposed by V. L. Makarov and successfully combines the benefits of using the {\it coefficient approximation method...

This chapter deals with problems associated with differential equations of the first order with an unbounded operator coefficient A in Banach space. The operatorvalued function e-tA (generated by A) plays an important role for these equations. This function is called also an operator exponential.

This chapter is devoted to studying the problems associated with second-order differential equations with an unbounded operator coefficient A in a Banach space. In Section 4.1, we consider these equations with an unbounded operator in either Banach or Hilbert spaces depending on the parameter t. We propose a discretization method with a high parall...

In this chapter we briefly describe some relevant basic results on interpolation, quadratures, estimation of operators and representation of operator-valued functions using the Dunford-Cauchy integral.

One of the important fields of application for modern computers is the numerical solution of diverse problems arising in science, engineering, industry, etc. Here, mathematical models have to be solved which describe e.g. natural phenomena, industrial processes, nonlinear vibrations, nonlinear mechanical structures or phenomena in hydrodynamics and...

In this last chapter we present a variety of mathematical exercises and the corresponding sample solutions by which the reader can test and deepen the knowledge acquired in the previous chapters of this book.

In this chapter we generalize the idea of the exact difference schemes to BVPs which are defined on the half axis.

In this chapter we consider nonlinear monotone ODEs with Dirichlet boundary conditions. Using a non-equidistant grid we construct an EDS on a three-point stencil and prove the existence and uniqueness of its solution. Moreover, on the basis of the EDS we develop an algorithm for the construction of a three-point TDS of rank \(\bar{m}\,=\,2[(m\,+\,1...

Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. Both techniques are based on the successive solution of IVPs on small subintervals and are theoretically supported by a posteriori error estimates. However, the advantage of our difference methods is that a unified theory of a priori estimates can be est...

This chapter deals with a generalization of the results from the previous chapter to the case of systems of second-order ODEs with a monotone operator.

Based on the functional-discrete technique (FD-method), an algorithm for eigenvalue transmission problems with discontinuous flux and integrable potential is developed. The case of the potential as a function belonging to the functional space $L_1$ is studied for both linear and nonlinear eigenvalue problems. The sufficient conditions providing sup...

In the paper a new numerical-analytical method for solving the Cauchy problem for systems of ordinary differential equations of special form is presented. The method is based on the idea of the FD-method for solving the operator equations of general form, which was proposed by V.L. Makarov. The sufficient conditions for the method converges with a...