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.

A Mixed-Flow Cascade Passage Design Procedure Based on a Power Series Expansion

1983 ◽  
Vol 105 (2) ◽  
pp. 231-242 ◽  
Author(s):  
R. A. Novak ◽  
G. Haymann-Haber
Keyword(s):  
Experimental Data ◽  
Power Series ◽  
Series Expansion ◽  
Taylor Series ◽  
Power Series Expansion ◽  
Design Procedure ◽  
Taylor Series Expansion ◽  
Analytic Solutions ◽  
Mixed Flow ◽  
Flow Passage

Flow passage design and analysis procedures employing a TAYLOR series expansion across the flow field were first suggested as far back as 1908. With very few exceptions the early work addressed only the problem of symmetrical convergent-divergent passages. A generalization and an extension of the approach permits it to be used for the design of arbitrary cascade passages in subsonic, supersonic, or mixed-flow regimes. The method will give fast, realistic solutions in situations where shock-free flow is meaningful, and where the cascade gap-chord ratio is not excessive. It assumes inviscid, irrotational flow and employs a fourth-order TAYLOR series expansion across the cascade passage. Numerous examples are given, with comparisons with experimental data and other analytic solutions.

A Mixed-Flow Cascade Passage Design Procedure Based on a Power Series Expansion

1982 ◽  
Author(s):  
R. A. Novak ◽  
G. Haymann-Haber
Keyword(s):  
Experimental Data ◽  
Power Series ◽  
Series Expansion ◽  
Taylor Series ◽  
Power Series Expansion ◽  
Design Procedure ◽  
Taylor Series Expansion ◽  
Analytic Solutions ◽  
Mixed Flow ◽  
Flow Passage

Flow passage design and analysis procedures employing a Taylor series expansion across the flow field were first suggested as far back as 1908. With very few exceptions the early work addressed only the problem of symmetrical convergent-divergent passages. A generalization and an extension of the approach permits it to be used for the design of arbitrary cascade passages in subsonic, supersonic, or mixed-flow regimes. The method will give fast, realistic solutions in situations where shock-freeflow is meaningful, and where the cascade gap-chord ratio is not excessive. It assumes inviscid, irrotational flow and employs a fourth-order Taylor series expansion across the cascade passage. Numerous examples are given, with comparisons with experimental data and other analytic solutions.

scholarly journals The Lambert function, the quintic equation and the proactive discovery of the Implicit Function Theorem

2021 ◽  
Vol 5 (1) ◽  
pp. 94-114
Author(s):  
Silvia Foschi ◽  
◽  
Daniele Ritelli ◽  
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Undergraduate Students ◽  
Implicit Function Theorem ◽  
Series Representation ◽  
Power Series Expansion ◽  
Implicit Function ◽  
Lambert Function ◽  
Quintic Equation ◽  
Power Series Representation

One of the problems on which a great deal of focus is being placed today, is how to teach Calculus in the presence of the massive diffusion of Computer Algebra tools and online resources among students. The essence of the problem lies in the fact that, during the problem solving activities, almost all undergraduates can be exposed to certain ''new'' functions, not typically treated at their level. This, without being prepared to handle them or, in some cases, even knowing the meaning of the answer provided by the computer system used. One of these functions is Lambert’s \(W\) function, undoubtedly due to the elementary nature of its definition. In this article we introduce \(W\), in a way that is easy to grasp for first year undergraduate students and we provide some general results concerning polynomial-exponential and polynomial-logarithmic equations. Among the many possible examples of its applications, we will see how \(W\) comes into play in epidemiology in the SIR model. In the second part, using more advanced concepts, we motivate the importance of the Implicit Function Theorem, using it to obtain the power series expansion of the Lambert function around the origin. Based on this approach, we therefore also provide a way to obtain the power series expansion of the inverse of a given smooth function \(f(y)\), when it is assumed that \(f(0)=0,\,f'(0)\neq0\), aided by the computational power of Mathematica\(_\circledR\). Basically, in this way, we present an alternative approach to the Lagrange Bürman Inversion Theorem, although in a particular but relevant case, since the general approach is not at an undergraduate level. A number of good references are [<a href="#1">1</a>, pp. 23-28] and [<a href="#2">2</a>], where the Lambert function is applied. Finally, these skills are used to take into consideration the particular quintic equation in the unknown \(y\) presented by F. Beukers [<a href="#3">3</a>]. Namely, we consider \(x(1+y)^5-y=0\) as an example of an equation for which the power series representation of one of its real solutions is known, calculating, with the same method used for the Lambert function, the first terms of its power series representation.

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