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.

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.

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

scholarly journals On Some Properties of the Star Graph

VLSI Design ◽  
1995 ◽  
Vol 2 (4) ◽  
pp. 389-396 ◽  
Author(s):  
Ke Qiu ◽  
Selim G. Akl
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Recursive Formula ◽  
Form Solution ◽  
Star Graph ◽  
Star Graphs ◽  
Different Dimensions ◽  
Homogeneous Linear ◽  
Fixed Node

We derive some properties of the star graph in this paper. In particular, we compute the number of nodes at distance i from a fixed node e in a star graph. To this end, a recursive formula is first obtained. This recursive formula is, in general, hard to solve for a closed form solution. We then study the relations among the number of nodes at distance i to node e in star graphs of different dimensions. This study reveals a very interesting relation among these numbers, which leads to a simple homogeneous linear recursive formula whose characteristic equation is easy to solve. Thus, we get a systematic way to obtain a closed form solution with given initial conditions for any fixed i.

scholarly journals Closed-Form Solutions for Gradient Elastic Beams with Geometric Discontinuities by Laplace Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Laplace Transform ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Elastic Beam ◽  
Form Solution ◽  
Static Deflection ◽  
Closed Form Solutions ◽  
Geometric Discontinuities ◽  
Algebraic Forms

The static bending solution of a gradient elastic beam with external discontinuities is presented by Laplace transform. Its utility lies in the ability to switch differential equations to algebraic forms that are more easily solved. A Laplace transformation is applied to the governing equation which is then solved for the static deflection of the microbeam. The exact static response of the gradient elastic beam with external discontinuities is obtained by applying known initial conditions when the others are derived from boundary conditions. The results are given in a series of figures and compared with their classical counterparts. The main contribution of this paper is to provide a closed-form solution for the static deflection of microbeams under geometric discontinuities.

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.

Sign in / Sign up

Export Citation Format

Share Document