Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

In this chapter, the connection between general linear interpolation and initial, boundary and multipoint value problems is explained. First, a result of a theoretical nature is given, which highlights the relationship between the interpolation problem and the Fredholm integral equation for high-order differential problems. After observing that the given problem is equivalent to a Fredholm integral equation, this relation is used in order to determine a general procedure for the numerical solution of high-order differential problems by means of appropriate collocation methods based on the integration of the Fredholm integral equation. The classical analysis of the class of the obtained methods is carried out. Some particular cases are illustrated. Numerical examples are given in order to illustrate the efficiency of the method

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms

Nonlocal and multipoint boundary value problems for linear evolution equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e.\ that it admits a unique solution. For the second order case, we also give criteria that guarantee well-posedness.Comment: 28 pages, 4 figure

On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed

On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W23(0,1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed