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Quadrupole mass filter with fuzzy initial conditions
Advances in Difference Equations volume 2013, Article number: 45 (2013)
The employment of the fuzzy method to solve differential equations has been well studied. In this article, Mathieu differential equations of the quadrupole mass filter (QMF) have been solved by using the fuzzy method. This method has not been yet investigated in the QMF with fuzzy initial conditions. We survey the physical properties of the confined ion. The results of numerical simulations are presented and discussed.
1 Quadrupole mass filter with hyperbolic rods
The quadrupole mass filter (QMF) principle was established by Paul and Steinwedel . It consists of four parallel hyperbolic cross-section rod electrodes as shown in Figure 1. If opposite pairs of electrodes have steady potentials and , it follows that the potential in the space between them is
where r and z are position coordinates from centerline and
where is direct potential, is the zero to peak amplitude of the RF voltage, and Ω is RF angular frequency [1–5]. Equation (1) satisfies the Laplace equation and boundary conditions of the system. In each of the perpendicular directions r and z, the equation of motion for a particle of mass m and charge e in such a time varying field may be written
where , and
Here is the distance from the rod surface to the z axis. These equations are forms of Mathieu’s differential equation [6–9]. The solutions of this equation are complex and are discussed in the literature . The solutions are classified as stable and unstable. If the amplitude of oscillatory motion is less or equal to one, the charged particle does not strike the rods and then its trajectory is stable. When the amplitude of a particle trajectory is larger than one, then the particle motion is mathematically unstable.
2 Preliminaries of the fuzzy sets
Fuzzy sets: According to Zadeh (1965), a fuzzy set is a generalization of a classical set that allows a membership function to take any value in the unit interval . The formal definition of a fuzzy set  is as follows.
Definition 1 Let Ω be a universal set. A fuzzy set A in Ω is defined by a membership function that maps every element in Ω to the unit interval . A fuzzy set A in Ω may also be presented as a set of ordered pairs of a generic element x and its membership value, as shown in the following equation:
Definition 2 Let A be a fuzzy interval defined in R. The α-cut of A is the crisp set that contains all elements in R such that the membership values of A is greater than or equal to a, that is,
For a fuzzy interval A, its α-cuts are closed intervals in R, and we denote them by
Definition 3 A fuzzy interval A is called equipossible fuzzy if it is fully determined by the pair of crisp numbers with and its membership function is given by
Definition 4 A fuzzy interval A is called triangular fuzzy if it is fully determined by the triplet of crisp numbers with and its membership function is given by
Definition 5 Let and . We say that f is strongly generalized H-differentiable at , if there exists an element such that
(1) for all sufficiently near to 0, , and the limits
(2) for all sufficiently near to 0, , and the limits
In the special case when f is a fuzzy-valued function, we have the following result.
Theorem 1 
Let be a function and denote for each . Then
• if f is differentiable in the first form (1) in Definition 5, then and are differentiable functions and ,
• if f is differentiable in the second form (2) in Definition 5, then and are differentiable functions and .
3 The quadrupole mass filter motion equations with fuzzy initial values
In this section we use the presented new algorithm for solving the equations of motion for a singly charged positive ion in the QMF by fuzzy initial values.
We consider the equations of motion in the QMF with fuzzy initial values as follows:
where z, r are fuzzy functions of ξ and , and , are trapezoidal or trapezoidal shaped fuzzy numbers.
We denote the fuzzy functions z and r by and , respectively. It means that the α-level sets of and for are
4 Fuzzy initial value problems
where y is a fuzzy function of t, is a fuzzy function of the crisp variable t and the fuzzy variable y, ; is the second fuzzy derivative of y and , are trapezoidal or trapezoidal shaped fuzzy numbers.
We denote the fuzzy function y by . It means that the α-level sets of for are
5 Formulation of a new algorithm for solving second-order ordinary differential equations
Consider a second-order initial value problem of the form
Here, , are called the initial conditions, while K is constant.
This was turned into an iterative scheme by substituting a guess () for the value of on the right, and the result as the definition of , which substitutes for the value of on the left. The new algorithm was defined to compute the sequence as follows:
The answer of Eq. (21), , is defined as approximated by the following series:
Here, M is the order of approximation and is the approximate answer for Eq. (21).
6 To solve quadrupole mass filter systems
With this fuzzy initial values the exact solutions (see Appendix 1) are as follows:
Here MathieuC and MathieuS are even and odd functions of ξ, respectively; and , are initial conditions and , are constant. Now, by using Eqs. (22) up to (24), the m th-order solution of Eqs. (11) and (12) with initials conditions (26) up to (29) is as follows:
which can be easily solved by symbolic computation software such as Matlab, Maple and Mathematica. After finding , and , , we have
Here, M is the order of approximation. By Maple software (see Appendix 2), we now successively obtain
7 Numerical results
In this section we bring some numerical results for illustration (see Appendix 3). Initial parameters for these numerical results have been used as follows:
Figure 2(a) and (b) shows the approximate solutions with , (a): , , (b): , , gold color: , and green color: , . Figure 2(a) and (b) shows that for we have and . Figure 3 shows the α-discretization of a fuzzy interval, blue color: , and red color: , . Figure 4(a) and (b) shows the ion trajectories in real time for , and , (a): , , (b): , , blue line: , and red line: , . Figure 5 shows the ion trajectory in and plans when , and , blue color (solid line): and red color (dash line): .
8 Discussion and conclusion
We have solved fuzzy differential equations of a quadrupole mass filter with fuzzy initial conditions by a proposed new algorithm. The results obtained show that fuzzy solution is compatible to an exact solution. For the illustration, the numerical results with order 20 () have been presented. Figure 2(a) and (b) is the approximate solution with for z and r, respectively. In this figure, for we have found that and , respectively. Therefore, probability gives the normal answer for a quadrupole mass filter without fuzzy initial conditions. The method showed a potential application to solve complicated linear and nonlinear differential equations in a quadrupole field, especially in fine tuning accelerators and, generally speaking, in physics of high energy.
Maple code for deriving the analytical solutions:
Maple code for deriving the analytical solutions by the new proposed algorithm:
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The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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