Quadrupole mass filter with fuzzy initial conditions
© Seddighi Chaharborj et al.; licensee Springer 2013
Received: 15 April 2012
Accepted: 12 February 2013
Published: 1 March 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
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 5 Let and . We say that f is strongly generalized H-differentiable at , if there exists an element such that
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.
where z, r are fuzzy functions of ξ and , and , are trapezoidal or trapezoidal shaped fuzzy numbers.
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.
5 Formulation of a new algorithm for solving second-order ordinary differential equations
Here, , are called the initial conditions, while K is constant.
Here, M is the order of approximation and is the approximate answer for Eq. (21).
6 To solve quadrupole mass filter systems
7 Numerical results
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.
The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
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