A note on the summation of divergent power series

1969 ◽  
Vol 66 (1) ◽  
pp. 109-114 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Severe Restriction ◽  
Mathematical Problems ◽  
Divergent Power Series ◽  
Euler Transformation ◽  
Direct Use

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

scholarly journals Time-dependent solution of the NIMFA equations around the epidemic threshold

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1299-1355
Author(s):  
Bastian Prasse ◽  
Piet Van Mieghem
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
State Vector ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mean Field Approximation ◽  
Contact Network ◽  
Epidemic Threshold ◽  
Linear Differential

AbstractThe majority of epidemic models are described by non-linear differential equations which do not have a closed-form solution. Due to the absence of a closed-form solution, the understanding of the precise dynamics of a virus is rather limited. We solve the differential equations of the N-intertwined mean-field approximation of the susceptible-infected-susceptible epidemic process with heterogeneous spreading parameters around the epidemic threshold for an arbitrary contact network, provided that the initial viral state vector is small or parallel to the steady-state vector. Numerical simulations demonstrate that the solution around the epidemic threshold is accurate, also above the epidemic threshold and for general initial viral states that are below the steady-state.

scholarly journals Closed-Form Solutions to Differential Equations via Differential Evolution

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
L. Mex ◽  
Carlos A. Cruz-Villar ◽  
F. Peñuñuri
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Closed Form ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Independent Variables ◽  
Effectiveness And Efficiency ◽  
Derivatives Of

We focus on solving ordinary differential equations using the evolutionary algorithm known as differential evolution (DE). The main purpose is to obtain a closed-form solution to differential equations. To solve the problem at hand, three steps are proposed. First, the problem is stated as an optimization problem where the independent variables areelementaryfunctions. Second, as the domain of DE is real numbers, we propose a grammar that assigns numbers to functions. Third, to avoid truncation and subtractive cancellation errors, to increase the efficiency of the calculation of derivatives, the dual numbers are used to obtain derivatives of functions. Some examples validating the effectiveness and efficiency of our method are presented.

scholarly journals On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Panayotis E. Nastou ◽  
Paul Spirakis ◽  
Yannis C. Stamatiou ◽  
Apostolos Tsiakalos
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Closed Form Expression ◽  
Form Expression ◽  
Abel Differential Equations ◽  
Solution Methodology

We investigate the properties of a general class of differential equations described bydy(t)/dt=fk+1(t)y(t)k+1+fk(t)y(t)k+⋯+f2(t)y(t)2+f1(t)y(t)+f0(t),withk>1a positive integer andfi(t), 0≤i≤k+1, withfi(t), real functions oft. Fork=2, these equations reduce to the class ofAbel differential equations of the first kind,for which a standard solution procedure is available. However, fork>2no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values ofkconnects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values ofk, we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology.

Closed-Form Vibration Response of a Special Class of Spinning, Cyclic Symmetric Rotor-Bearing-Housing Systems

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Closed Form ◽  
Coordinate Transformation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Periodic Coefficients ◽  
Closed Form Solutions ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

Vibration of a spinning, cyclic symmetric rotor supported by flexible bearings and housing is governed by a set of ordinary differential equations with periodic coefficients. As a result, analytical solutions of such systems are generally not available. This paper is to prove that closed-form solutions are available for such systems if the following two conditions are met. First, the rotor has a rigid hub and the rest of the rotor is flexible. Second, elastic mode shapes of the rotor's flexible part only present axial displacement. Under these two conditions, the periodic coefficients will only appear between repeated modes of the spinning rotor and vibration modes of the stationary housing. This unique structure enables a coordinate transformation to convert the governing ordinary differential equations with periodic coefficients into a set of ordinary differential equations with constant coefficients, whose closed-form solution is readily available. Moreover, the coordinate transformation can be derived explicitly. Finally, we demonstrate the closed-form solution through a benchmark numerical model that consists of a spinning rotor, a stationary housing, and two elastic bearings. In particular, the rotor is a circular disk with four evenly spaced radial slots and a central rigid hub. The housing is a square plate with a central rigid shaft and is fixed at four corners. The two elastic bearings connect the rotor and the housing between the hub and shaft. Numerical results confirm that the original equation of motion with periodic coefficients and the closed-form solutions predict the same vibration response.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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