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Research Papers

# The Dynamical Functional Particle Method: An Approach for Boundary Value Problems

[+] Author and Article Information
Sverker Edvardsson1

Division of Computational Mathematics and Physics, FSCN,  Mid Sweden University, SE-851 70 Sundsvall, Swedensverker.edvardsson@miun.se

Division of Computational Mathematics and Physics, FSCN,  Mid Sweden University, SE-851 70 Sundsvall, Sweden

1

Corresponding author.

J. Appl. Mech 79(2), 021012 (Feb 24, 2012) (9 pages) doi:10.1115/1.4005563 History: Received February 25, 2011; Revised July 05, 2011; Posted January 31, 2012; Published February 13, 2012; Online February 24, 2012

## Abstract

The present work is concerned with new ideas of potential value for solving differential equations. First, a brief introduction to particle methods in mechanics is made by revisiting the vibrating string. The full case of nonlinear motion is studied and the corresponding nonlinear differential equations are derived. It is suggested that the particle origin of these equations is of more general interest than usually considered. A novel possibility to develop particle methods for solving differential equations in a direct way is investigated. The dynamical functional particle method (DFPM) is developed as a solution method for boundary value problems. DFPM is based on the concept of an interaction functional as a dynamical force field acting on quasi particles. The approach is not limited to linear equations. We exemplify by applying DFPM to several linear Schrödinger type of problems as well as a nonlinear case. It is seen that DFPM performs very well in comparison with some standard numerical libraries. In all cases, the convergence rates are exponential in time.

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## Figures

Figure 4

The exact periods Ts(A) as a function of the initial amplitude A by using the particle method of the current work. In the example (Table 1), A = 10 m. Ts (A) was recorded at the position x = 10, see Fig. 3 The nonlinear effect can also be studied approximately under the assumption that the tension T=T0+T1 is constant along the string [20]. The approximate period is then given by Ts(A)=L/(T0/λ01+(π2/2ɛ)(A2/L2)). Given the cross section area S and the Young’s modulus E, the tension at equilibrium is given by T0=ESɛ, and the quasi static contribution is given by T1=(π2/2)ES(A2/L2). The approximate expression for Ts(A) works reasonably well in b) but, as expected, fails in a).

Figure 5

Solution for the harmonic oscillator V=x2 using a particle method where N = 81

Figure 6

The pair functions Φ0000(r1,r2) and Φ0001(r1,r2) for the s-limit Helium ground state  1S and first excited state  3S, respectively. The extrapolated eigenvalues are −2.8790287673(2) and −2.174264856(3) Hartrees, respectively. This is in agreement with the earlier studies [(39),30].

Figure 7

The convergence of the Helium groundstate  1S and first excited state  3S. In this case η0 and η1 are the same as the corresponding damping parameters (see text). The initial values 0.5 and 11 in the figure depend on the initial wavefunctions Ψ0(0) and Ψ1(0). The overall CPU times in the Tables  23 do not depend much on the initial conditions. The mesh was in both cases Δx=0.1/1.14 (i.e., k = 4).

Figure 8

The groundstate solutions to Eq. 29 for λ=0,1,2,…,10. Also displayed are the corresponding calculated values for ω.

Figure 1

Quasi-particle picture of the vibrating string

Figure 2

The tensions Ti+1 and Ti for quasi-particle i. Note that Ti+1 is a vector length, i.e., Δxi+1 and Δyi+1 specify the direction of the vector, and analogously for Ti.

Figure 9

(a) Groundstate and first excited state for λ=1. (b) The error y=|Ψ(t)-Ψ(∞)| is plotted. One good property of DFPM is that the convergence is in general exponential. The lower part of the main figure shows the evolution of ω as the nonlinear parameter λ increases.

Figure 3

The exact particle solution versus the solution of the wave equation (approx.). In the exact solution all the particles were extrapolated. Due to the symmetry of the problem, certain x-locations are still conserved, even for the exact solution. These are at x=10, 20 and 30. The ‘period’ Ts for the exact solution is conveniently followed at, e.g., x=10.

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