Differential operators on the class of convergent power series and the Weierstrass auxiliary lemma

1969 ◽  
Vol 2 (3) ◽  
pp. 235-244 ◽  
Author(s):  
V. P. Palamodov
Keyword(s):  
Power Series ◽  
Differential Operators ◽  
Convergent Power Series ◽  
Auxiliary Lemma

scholarly journals Newton polygons and gevrey indices for linear partial differential operators

1992 ◽  
Vol 128 ◽  
pp. 15-47 ◽  
Author(s):  
Masatake Miyake ◽  
Yoshiaki Hashimoto
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Convergent Power Series ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

scholarly journals Note on a special determinant

1937 ◽  
Vol 30 ◽  
pp. xxvii-xviii
Author(s):  
A. C. Aitken
Keyword(s):  
Power Series ◽  
Convergent Power Series ◽  
Image Position

Suppose a polynomial or convergent power seriesis raised to powers j = 0, 1, 2, 3, … The coefficients of xk in [f(x)]j, k = 0, 1, 2, …, may be entered as elements in positions (j, k) in an array or matrix F, thus:By construction all elements in column (k) have weight (sum of suffixes) equal to k.

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

scholarly journals An explicit solution of coupled viscous Burgers' equation by the decomposition method

2001 ◽  
Vol 27 (11) ◽  
pp. 675-680 ◽  
Author(s):  
Doğan Kaya
Keyword(s):  
Perturbation Theory ◽  
Power Series ◽  
Explicit Solution ◽  
Decomposition Method ◽  
Series Solution ◽  
Burgers Equation ◽  
Coupled System ◽  
The Self ◽  
Convergent Power Series ◽  
Weak Nonlinearity

We consider a coupled system of viscous Burgers' equations with appropriate initial values using the decomposition method. In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The decomposition series solution of the problem is quickly obtained by observing the existence of the self-canceling “noise” terms where the sum of components vanishes in the limit.

scholarly journals Rings of convergent power series and Weierstrass Preparation Theorem

1981 ◽  
Vol 81 ◽  
pp. 73-78
Author(s):  
Takasi Sugatani
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Multiplicative Group ◽  
Convergent Power Series ◽  
Weierstrass Preparation ◽  
Preparation Theorem

Let B be a B-ring with a nonarchimedean valuation | |, i.e., B is an integral domain satisfying the following conditions: (i) B is bounded (| a | ≤ 1 for every a ∊ B), (ii) the boundary forms a multiplicative group.

Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order

2021 ◽  
Vol 8 (3) ◽  
pp. 537-548
Author(s):  
S. E. Fadugba ◽  
◽  
F. Ali ◽  
A. B. Abubakar ◽  
◽  
...  
Keyword(s):  
Power Series ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Epidemic Model ◽  
Differential Transform Method ◽  
Host Community ◽  
Convergent Power Series ◽  
Transform Method ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This paper proposes the Caputo Fractional Reduced Differential Transform Method (CFRDTM) for Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model with fractional order in a host community. CFRDTM is the combination of the Caputo Fractional Derivative (CFD) and the well-known Reduced Differential Transform Method (RDTM). CFRDTM demonstrates feasible progress and efficiency of operation. The properties of the model were analyzed and investigated. The fractional SEIR epidemic model has been solved via CFRDTM successfully. Hence, CFRDTM provides the solutions of the model in the form of a convergent power series with easily computable components without any restrictive assumptions.

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