837 research outputs found
An algebraic method to solve the radial Schrödinger equation
AbstractWe propose a method of numerical integration of differential equations of the type x2y″+f(x)y=0 by approximating its solution with solutions of equations of the type x2y″+(ax2+bx+c)y=0. This approximation is performed by segmentary approximation on an interval. We apply the method to obtain approximate solutions of the radial Schrödinger equation on a given interval and test it for two different potentials. We conclude that our method gives a similar accuracy than the Taylor method of higher order
A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages
المشتقة الثانية طريقة رنك-كوتا الفعالة الجديدة من الرتبة الخامسة (TDRK) قد تم تطويرها من أجل الحل العددي للمعادلات التفاضلية الاعتيادية من الرتبة الأولى (ODEs). تم اشتقاق الطريقة الجديدة باستخدام خاصية الأول نفس الأخير (FSAL) . قمنا بتحليل استقرار الطريقة. تم عرض النتائج العددية لتوضيح كفاءة الطريقة الجديدة بالمقارنة مع بعض طرق رنك-كوتا (RK) المعروفة.A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods
The Fokker–Planck equation for a bistable potential
AbstractThe Fokker–Planck equation is studied through its relation to a Schrödinger-type equation. The advantage of this combination is that we can construct the probability distribution of the Fokker–Planck equation by using well-known solutions of the Schrödinger equation. By making use of such a combination, we present the solution of the Fokker–Planck equation for a bistable potential related to a double oscillator. Thus, we can observe the temporal evolution of the system describing its dynamic properties such as the time τ to overcome the barrier. By calculating the rates k=1/τ as a function of the inverse scaled temperature 1/D, where D is the diffusion coefficient, we compare the aspect of the curve k×1/D, with the ones obtained from other studies related to four different kinds of activated process. We notice that there are similarities in some ranges of the scaled temperatures, where the different processes follow the Arrhenius behavior. We propose that the type of bistable potential used in this study may be used, qualitatively, as a simple model, whose rates share common features with the rates of some single rate-limited thermally activated processes
On modified Runge–Kutta trees and methods
AbstractModified Runge–Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge–Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested
A two-step trigonometrically fitted semi-implicit hybrid method for solving special second order oscillatory differential equation
In this paper, we derived a semi-implicit hybrid method (SIHM) which is a two-step method to solve special second order ordinary differential equations (ODEs). The SIHM which is three-stage and fourth-order is then trigonometrically fitted and denoted by TF-SIHM3(4). The method is constructed using trigonometrically fitted properties instead of using phase-lag and amplification properties. Numerical integration show that TF-SIHM3(4) is more accurate in term of accuracy compared to the existing explicit and implicit methods of the same order
Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth
We consider a population dynamics model coupling cell growth to a diffusion
in the space of metabolic phenotypes as it can be obtained from realistic
constraints-based modelling. In the asymptotic regime of slow diffusion, that
coincides with the relevant experimental range, the resulting non-linear
Fokker-Planck equation is solved for the steady state in the WKB approximation
that maps it into the ground state of a quantum particle in an Airy potential
plus a centrifugal term. We retrieve scaling laws for growth rate fluctuations
and time response with respect to the distance from the maximum growth rate
suggesting that suboptimal populations can have a faster response to
perturbations.Comment: 24 pages, 6 figure
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