The Series Expansion of the Greens Function of a Differential Operator with Block-triangular Operator Potential

2016 ◽  
Vol 19 (5) ◽  
pp. 1-12
Author(s):  
A Kholkin
Keyword(s):  
Differential Operator ◽  
Series Expansion ◽  
Operator Potential ◽  
Triangular Operator ◽  
Greens Function

scholarly journals Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Aleksandr Mikhailovich Kholkin
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Operator Potential ◽  
Triangular Operator

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

scholarly journals On solving initial value problems for partial differential equations in maple

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Srinivasarao Thota
Keyword(s):  
Applied Sciences ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Partial Differential Operator ◽  
Initial Value ◽  
Partial Differential ◽  
Greens Function ◽  
The Given

Abstract Objectives In this paper, we present and employ symbolic Maple software algorithm for solving initial value problems (IVPs) of partial differential equations (PDEs). From the literature, the proposed algorithm exhibited a great significant in solving partial differential equation arises in applied sciences and engineering. Results The implementation include computing partial differential operator (), Greens function () and exact solution () of the given IVP. We also present syntax, , to apply the partial differential operator to verify the solution of the given IVP obtained from . Sample computations are presented to illustrate the maple implementation.

scholarly journals Series expansion of the Gamma function and its reciprocal

2021 ◽  
Vol 27 (4) ◽  
pp. 104-115
Author(s):  
Ioana Petkova ◽  
Keyword(s):  
Differential Operator ◽  
Explicit Form ◽  
Series Expansion ◽  
Exponential Function ◽  
Gamma Function ◽  
Linear Functional ◽  
Maclaurin Series

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

scholarly journals Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

2021 ◽  
Author(s):  
Aleksandr Kholkin
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operator ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Spectral Singularities ◽  
Matrix Potential ◽  
Triangular Operator ◽  
Sturm Liouville

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

The influence of wave aberrations: an operator approach

1985 ◽  
Vol 63 (2) ◽  
pp. 250-253 ◽  
Author(s):  
J. Ojeda-Castañeda ◽  
A. Boivin
Keyword(s):  
Differential Operator ◽  
Wave Front ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Complex Amplitude ◽  
Third Order ◽  
Operator Approach

The wave departures from sphericity known as Seidel's primary- or third-order aberrations are represented by a differential operator that modifies the diffraction-limited image. We show that the complex amplitude at any point over the aberrant image can be obtained from certain terms of the Taylor series expansion of the diffraction limited image. This approach allows us to describe the propagation of an aberrant wave front in terms of its aberration coefficients. Furthermore, it can also be applied to implement an optical or an electronic procedure for simulating aberrant images.

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