scholarly journals Groups defined on images in fluid diffusion

1988 ◽  
Vol 30 (1) ◽  
pp. 101-119 ◽  
Author(s):  
A. J. Bracken ◽  
H. S. Green ◽  
L. Bass
Keyword(s):  
Recurrence Relation ◽  
Differential Operators ◽  
Laguerre Polynomials ◽  
Chemical Reactor ◽  
Degree Of Approximation ◽  
Simple Relationship ◽  
Invariance Group ◽  
Third Order ◽  
Point Of Entry ◽  
Method Of Solution

AbstractA method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind

2021 ◽  
Author(s):  
Zulqurnain Sabir ◽  
Hafiz Abdul Wahab
Keyword(s):  
Neural Networks ◽  
Differential Operators ◽  
Layer Structure ◽  
Research Work ◽  
Merit Function ◽  
Third Order ◽  
Intelligent Computing ◽  
Computing Platform ◽  
Hybrid Heuristics ◽  
Hidden Layer

Abstract The presented research work articulates a new design of heuristic computing platform with artificial intelligence algorithm by exploitation of modeling with feed-forward Gudermannian neural networks (FFGNN) trained with global search viability of genetic algorithms (GA) hybrid with speedy local convergence ability of sequential quadratic programing (SQP) approach, i.e., FFGNN-GASQP for solving the singular nonlinear third order Emden-Fowler (SNEF) models. The proposed FFGNN-GASQP intelligent computing solver Gudermannian kernel unified in the hidden layer structure of FFGNN systems of differential operators based on the SNEF that are arbitrary connected to represent the error-based merit function. The optimization objective function is performed with hybrid heuristics of GASQP. Three problems of the third order SNEF are used to evaluate the correctness, robustness and effectiveness of the designed FFGNN-GASQP scheme. Statistical assessments of the performance of FFGNN-GASQP are used to validate the consistent accuracy, convergence and stability.

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

scholarly journals Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
M. L. Mittal ◽  
Mradul Veer Singh
Keyword(s):  
Error Estimation ◽  
Laguerre Polynomials ◽  
Degree Of Approximation ◽  
Continuity Condition ◽  
Frontier Point ◽  
Laguerre Series ◽  
Summability Methods ◽  
The Individual ◽  
Euler Operators ◽  
Cesàro Matrix

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.

scholarly journals On some connection results between Laguerre polynomials via third-order differential operator

2020 ◽  
Author(s):  
Baghdadi Aloui ◽  
wathek chammam ◽  
Jihad Souissi
Keyword(s):  
Differential Operator ◽  
Laguerre Polynomials ◽  
Order Differential Operator ◽  
Third Order ◽  
Orthogonal Sequence

Let $\{L^{(\alpha)}_n\}_{n\geq 0}$, ($\alpha\neq-m, \ m\geq1$), be the monic orthogonal sequence of Laguerre polynomials. We give a new differential operator, denoted here $\mathscr{L}^{+}_{\alpha}$, raises the degree and also the parameter of $L^{(\alpha)}_n(x)$. More precisely, $\mathscr{L}^{+}_{\alpha}L^{(\alpha)}_n(x)=L^{(\alpha+1)}_{n+1}(x), \ n\geq0$. As an illustration, we give some properties related to this operator and some other operators in the literature, then we give some connection results between Laguerre polynomials via this new operator.

scholarly journals Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nemat Dalir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Initial Value ◽  
Third Order ◽  
Partial Differential ◽  
The Third

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

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