scholarly journals On Sheffer polynomial families

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 4
Author(s):  
Sandra Pinelas ◽  
Paolo Emilio Ricci
Keyword(s):  
Differential Equations ◽  
Exponential Functions ◽  
Sheffer Polynomials ◽  
Fractional Type ◽  
Sheffer Polynomial

Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.

scholarly journals New Bell–Sheffer Polynomial Sets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 71 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Paolo Ricci
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Higher Order ◽  
Combinatorial Analysis ◽  
Bell Numbers ◽  
Integer Sequences ◽  
Shift Operators ◽  
Sheffer Polynomials ◽  
Logarithmic Functions ◽  
Sheffer Polynomial

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

Nonlinear Corrections for Edge Bending of Shells

1980 ◽  
Vol 47 (4) ◽  
pp. 861-865 ◽  
Author(s):  
G. V. Ranjan ◽  
C. R. Steele
Keyword(s):  
Differential Equations ◽  
Spherical Shell ◽  
Asymptotic Expansions ◽  
Axial Displacement ◽  
Exponential Functions ◽  
Spherical Cap ◽  
Influence Coefficients ◽  
Edge Loading ◽  
General Geometry ◽  
Edge Influence

Asymptotic expansions for self-equilibrating edge loading are derived in terms of exponential functions, from which formulas for the stiffness and flexibility edge influence coefficients are obtained, which include the quadratic nonlinear terms. The flexibility coefficients agree with those previously obtained by Van Dyke for the pressurized spherical shell and provide the generalization to general geometry and loading. In addition, the axial displacement is obtained. The nonlinear terms in the differential equations can be identified as “prestress” and “quadratic rotation.” To assess the importance of the latter, the problem of a pressurized spherical cap with roller supported edges is considered. Results show that whether the rotation at the edge is constrained or not, the quadratic rotation terms do not have a large effect on the axial displacement. The effect will be large for problems with small membrane stresses.

scholarly journals Computer method to analyze structural-dynamic properties of Rhodobacter sphaeroides reaction centers based on system of differential equations

2019 ◽  
Keyword(s):  
Electron Transfer ◽  
Differential Equations ◽  
Rhodobacter Sphaeroides ◽  
Reaction Centers ◽  
System Of Differential Equations ◽  
Balance Equations ◽  
Exponential Functions ◽  
Structural Dynamic ◽  
Constant Coefficients ◽  
Kinetics Of

Background: Reactions of the natural objects to external influences can be analyzed using balance equations. If such reactions have a multi-exponential character, they can be represented as a sum of exponent components. Such kind of reaction is due both to the influence of hidden parameters, and the influence of the reaction itself on the structure of the object. The problem is that it is often not possible to determine empirically the values of the constants of the velocities of the balance equation, their relation with the parameters of the exponential components of the reaction, the kinetics of the population of the substates of the object. Objectives: The aim of the work is to develop a method of detailed analysis of the reaction of the object to external influence, which allows to determine the kinetics of the population of possible substates of the object by constructing a system of differential equations with constant coefficients. Materials and methods: Isolated reaction centers (RC) of Rhodobacter sphaeroides bacteria, the structure of which is well known, were used as an object. Behavior of the RC under photo-excitation was analyzed by constructing a system of differential equations with constant coefficients. The experimental kinetics of the cyclic electron transfer of the RC was approximated by the sum of three exponential functions. The parameters of these functions were used to determine the balance rate constants solving an optimization problem by a gradient method. The task was to study the RC using the method of constructing the system of differential equations and the method of two expositions. Results: A computer procedure was developed to determine the values of the speed constants of four balance equations, to analyze the kinetics of the population of the bases of the RC using the parameters of three exponential functions of the kinetics of electron transfer. Experimental and calculated kinetics of the donor population after photoexcitation of the RC are in a good agreement. The results of the two methods are correlated. They show that in the process of photo-excitation the maxima of populations of RC states correspond to a range of 3–140 s after the turning on (turning off) the light. Conclusion: RC corresponds to the system of four electron-conformational states. The features of the kinetics of population of the bases of the RC characterize the spatial-temporal characteristics of the RC.

scholarly journals Some new generalizations of $ F- $contraction type mappings that weaken certain conditions on Caputo fractional type differential equations

2021 ◽  
Vol 6 (11) ◽  
pp. 12718-12742
Author(s):  
Naeem Saleem ◽  
◽  
Mi Zhou ◽  
Shahid Bashir ◽  
Syed Muhammad Husnine ◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Integral Boundary ◽  
Integral Boundary Value Problem ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Contraction Type ◽  
Fractional Type

<abstract><p>In this paper, firstly, we introduce some new generalizations of $ F- $contraction, $ F- $Suzuki contraction, and $ F- $expanding mappings. Secondly, we prove the existence and uniqueness of the fixed points for these mappings. Finally, as an application of our main result, we investigate the existence of a unique solution of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order (1, 2).</p></abstract>

scholarly journals Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey †

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 50 ◽  
Author(s):  
Paolo Emilio Ricci
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sheffer Polynomial

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown.

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