scholarly journals Integration method of non-elementary exponential functions using iterated Fubinni integrals

2021 ◽  
Vol 9 (9) ◽  
pp. 169-177
Author(s):  
Odirley Willians Miranda Saraiva ◽  
Gustavo Nogueira Dias ◽  
Fabricio da Silva Lobato ◽  
José Carlos Barros de Souza Júnior ◽  
Washington Luiz Pedrosa da Silva Junior ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Exponential Function ◽  
Taylor Series ◽  
Integration Method ◽  
Exponential Functions ◽  
Integration Methods ◽  
Elliptic Differential Equations ◽  
Double Integrals ◽  
Statistical Functions

The present work presents a new method of integration of non-elementary exponential functions where Fubinni's iterated integrals were used. In this research, some approximations were used in order to generalize the results obtained through mathematical series, in addition to integration methods and double integrals. In addition to the integration methods, the Taylor series was used, where the value found and compatible with the values ​​of the power series that are used to calculate the value of the exponential function demonstrated in the work was verified. In addition to the methods described, a comparison of the values ​​obtained by the series and the values ​​described in the method was improvised, where it was noticed that the higher the value of the variable, the closer the results show a stability for the variable greater than the value 4, described in table 01. The conclusions point to a great improvement, mainly for solving elliptic differential equations and statistical functions.

6. and beyond, to complex things

2020 ◽  
pp. 109-127
Author(s):  
Glen Van Brummelen
Keyword(s):  
Fluid Dynamics ◽  
Exponential Function ◽  
Taylor Series ◽  
Electromagnetic Waves ◽  
Irrational Number ◽  
Exponential Functions ◽  
Wireless Technologies ◽  
Catenary Curve ◽  
De Moivre’S Formula ◽  
Euler’S Identity

‘ … and beyond, to complex things’ first considers the Taylor series for the exponential function. One of the most famous, yet enigmatic, numbers in mathematics, e is an irrational number equal to 2.718281828. … Exponential functions deal with the phenomena of growth and decay. As calculus was starting to become established, curious parallels between the apparently disparate worlds of trigonometry and exponential functions were starting to appear. Imaginary numbers, Euler’s formula, and Euler’s identity are discussed along with the Argand diagram, De Moivre’s formula, hyperbolic trigonometric functions, and the catenary curve. Imaginary numbers are now at the heart of science and technology, and are used in the study of electromagnetic waves, cellular and wireless technologies, and fluid dynamics.

Integrals developable about a singular point of a Hamiltonian system of differential equations. Part II

1925 ◽  
Vol 22 (4) ◽  
pp. 510-533 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Series ◽  
System Of Differential Equations ◽  
Linearly Independent ◽  
Image Position

This paper completes an investigation, of which the first part has already been published, into the integrals of a Hamiltonian system which are formally developable about a singular point of the system. Letbe a system of differential equations of which the origin is a singular point of the first type, i.e. a point at which H is developable in a convergent Taylor series, but at which its first derivatives all vanish. We suppose that H does not involve t, and we consider only integrals not involving t. Let the exponents of this singular point be ± λ1, ± λ2,…±λn. In Part I, I considered the case in which the constants λ1,…λn are connected by no relation of commensurability, i.e. a relation of the formwhere A1…An are integers (positive, negative or zero) not all zero, and showed that the equations (1) possess n, and only n, integrals not involving t which are formally developable as power series in the xk, yk. In this paper I consider the case in which λ1 … λn are connected by one or more relations of commensur-ability. Suppose that there are p, and only p, such relations linearly independent (p > 0): it will be shown that the equations (1) possess (n − p) independent integrals not involving t, formally developable about the origin and independent of H.

scholarly journals Object-Oriented Implementation of Software for Solving Ordinary Differential Equations

1993 ◽  
Vol 2 (4) ◽  
pp. 217-225 ◽  
Author(s):  
Kjell Gustafsson
Keyword(s):  
User Interface ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integration Method ◽  
Object Oriented ◽  
Software Components ◽  
Integration Methods ◽  
Numerical Integration Methods

We describe an object-oriented implementation of numerical integration methods for solving ordinary differential equations. Software components that are common to many different integration methods have been identified and implemented in such a way that they can be reused. This facilitates the design of a uniform user interface and makes the task of implementing a new integration method fairly modest. The sharing of code in this type of implementation also allows for less subjective comparisons of the result from different integration methods.

scholarly journals Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
U. Al Khawaja ◽  
Qasem M. Al-Mdallal
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Series Expansion ◽  
Taylor Series ◽  
Nonlinear Differential Equations ◽  
Series Representation ◽  
Power Series Expansion ◽  
Detailed Comparison ◽  
Radius Of Convergence ◽  
Finite Radius

It is known that power series expansion of certain functions such as sech⁡(x) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech⁡(x) that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

scholarly journals Generalized quantum exponential function and its applications

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4907-4922
Author(s):  
Burcu Silindir ◽  
Ahmet Yantir
Keyword(s):  
Exponential Function ◽  
Difference Equations ◽  
Taylor Series ◽  
Infinite Product ◽  
Existence And Uniqueness Theorem ◽  
Exponential Functions ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Dynamic Wave

This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.

scholarly journals Exponential Functions in Cartesian Differential Categories

2020 ◽  
Author(s):  
Jean-Simon Pacaud Lemay
Keyword(s):  
Differential Equations ◽  
Exponential Function ◽  
Differential Calculus ◽  
Exponential Map ◽  
Dual Numbers ◽  
Exponential Functions ◽  
Smooth Functions ◽  
Exponential Maps ◽  
Differential Categories ◽  
Complex Exponential

Abstract In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

scholarly journals Deformation Exploration in Mass Spring Model using Euler and Verlet Integration Methods

2019 ◽  
Vol 9 (2) ◽  
pp. 119-124
Keyword(s):  
Differential Equations ◽  
Virtual Environments ◽  
Real Time ◽  
Integration Method ◽  
Spring Model ◽  
Integration Methods ◽  
Mass Spring Model ◽  
General Mass ◽  
Mass Spring ◽  
Deformation Error

The paper assigns the firm technique that has been designed for the mesh based simulation by using the concept of mass spring model. The general mass spring model has been utilized in a lot of applications for instance, fashion designing, merging virtual booth and in the basics of cloth simulations, consecutively in order to develop effectual surgical training through virtual environments. Though, virtual simulators necessitate meeting both requirements that are, dynamic to be real time and high realistic. While dissimilar forces have applied on the particles they generate several differential equations. In order to, solve these equations, different kinds of integration methods have been used to get the best results. Here in this paper, it shows the procedure of generating a mesh based simulation using euler and verlet integrations method. Verlet method executes vigorous compared to Euler integration method on the basis of deformation error.

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.

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