power series
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2022 ◽  
Vol 15 (2) ◽  
pp. 481-504
Motahare ZaeamZadeh ◽  
Jafar Ahmadi ◽  
Bahareh Khatib Astaneh ◽  

2022 ◽  
Vol 252 ◽  
pp. 113663
Héctor Martín ◽  
Claudio Maggi ◽  
Marcelo Piovan ◽  
Anna De Rosa ◽  
y Nicolás Martin Gutbrod

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Mine Aylin Bayrak ◽  
Ali Demir ◽  
Ebru Ozbilge

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

Victor Barrera-Figueroa ◽  
Vladimir S. Rabinovich ◽  
Samantha Ana Cristina Loredo-Ramı́rez

Abstract The work is devoted to the asymptotic and numerical analysis of the wave function propagating in two-dimensional quantum waveguides with confining potentials supported on slowly varying tubes. The leading term of the asymptotics of the wave function is determined by an adiabatic approach and the WKB approximation. Unlike other similar studies, in the present work we consider arbitrary bounded potentials and obtain exact solutions for the thresholds, and for the transverse modes in the form of power series of the spectral parameter. Our approach leads to an effective numerical method for the analysis of such quantum waveguides and for the tunnel effect observed in sections of the waveguide that shrink or widen too much. Several examples of interest show the applicability of the method.

2022 ◽  
Vol Volume 44 - Special... ◽  
Jeremy Lovejoy

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

2022 ◽  
pp. 101552
Qipin Chen ◽  
Wenrui Hao ◽  
Juncai He

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 50
Aliaa Burqan ◽  
Rania Saadeh ◽  
Ahmad Qazza

In this article, a new, attractive method is used to solve fractional neutral pantograph equations (FNPEs). The proposed method, the ARA-Residual Power Series Method (ARA-RPSM), is a combination of the ARA transform and the residual power series method and is implemented to construct series solutions for dispersive fractional differential equations. The convergence analysis of the new method is proven and shown theoretically. To validate the simplicity and applicability of this method, we introduce some examples. For measuring the accuracy of the method, we make a comparison with other methods, such as the Runge–Kutta, Chebyshev polynomial, and variational iterative methods. Finally, the numerical results are demonstrated graphically.

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