scholarly journals Heun Functions and Some of Their Applications in Physics

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
M. Hortaçsu

Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé, and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two-way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable. The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature, will be given.

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes

We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. In order to assess the benefits of this proposal, three nonlinear problems of different kind are solved and compared against the power series solution obtained using an approximative method. The problems are homogeneous Lane-Emden equation ofαindex, governing equation of a burning iron particle, and an explicit differential-algebraic equation related to battery model simulations. The results show that PSEM generates highly accurate handy approximations requiring only a few steps. The main advantage of PSEM is to extend the domain of convergence of the power series solutions of approximative methods as Taylor series method, homotopy perturbation method, homotopy analysis method, variational iteration method, differential transform method, and Adomian decomposition method, among many others. From the application of PSEM, it results in handy easy computable expressions that extend the domain of convergence of high order power series solutions.


2021 ◽  
Vol 25 (8) ◽  
pp. 6075-6082
Author(s):  
Hemanta Mandal ◽  
B. Bira ◽  
D. Zeidan

Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.


2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.


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