Closed‐Form Solution of the Differential Equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P=0 by Normal‐Ordering Exponential Operators

1970 ◽  
Vol 11 (4) ◽  
pp. 1235-1237 ◽  
Author(s):  
Ralph Wilcox
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Normal Ordering

scholarly journals Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 331 ◽  
Author(s):  
Huda Bakodah ◽  
Abdelhalim Ebaid
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Approximate Solutions ◽  
Form Solution ◽  
Proportional Delay ◽  
Delay Term ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Delay Differential

The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced.

Pascal's recurrence relation and even-aged-forest stand dynamics

1992 ◽  
Vol 22 (12) ◽  
pp. 1996-1999
Author(s):  
Rolfe A. Leary ◽  
Hien Phan ◽  
Kevin Nimerfro
Keyword(s):  
Differential Equation ◽  
Relative Density ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Integral Form ◽  
Forest Stand ◽  
Form Solution ◽  
Density Change ◽  
Stand Dynamics

A common method of modelling forest stand dynamics is to use permanent growth plot remeasurements to calibrate a whole-stand growth model expressed as an ordinary differential equation. To obtain an estimate of future conditions, either the differential equation is integrated numerically or, if analytic, the differential equation is solved in closed form. In the latter case, a future condition is obtained simply by evaluating the integral form for the age of interest, subject to appropriate initial conditions. An older method of modelling forest stand dynamics was to use a normal or near-normal yield table as a density standard and calibrate a relative density change equation from permanent plot remeasurements. An estimate of a future stand property could be obtained by iterating from a known initial relative density. In this paper we show that when the relative density change equation has a particular form, the historical method also has a closed form solution, given by a sequence of polynomials with coefficients from successive rows of Pascal's arithmetic triangle.

Fractional Calculus Model of the Semilunar Heart Valve Vibrations

2003 ◽  
Author(s):  
Moustafa El-Shahed
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Numerical Methods ◽  
Fractional Calculus ◽  
Closed Form ◽  
Heart Valve ◽  
Laplace Transformation ◽  
Closed Form Solution ◽  
Equation Of Motion ◽  
Form Solution

The objective of this paper is to solve the equation of motion of semilunar heart valve vibrations. The vibrations of the closed semilunar valves were modeled with a Caputo hactional derivative of order α. With the help of Laplace transformation, closed-form solution is obtained for the equation of motion in terms of Mittag-Leffler function. An alternative Method for Semi-differential equation, when α = 0.5, is examined using MATHEMATICA. The simplicity of these solutions makes them ideal for testing the accuracy of numerical methods. This solution can be of some interest for a better fit of experimental data.

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.

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