- Open Access
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.
- quadrupole mass filter
- Mathieu differential equations
- ordinary differential equation
- fuzzy initial conditions
- ion trajectory
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 .
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.
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.
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).
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.
- Paul W, Steinwedel H, Naturforch Z: J. Chem. Educ.. 1953, A8: 448.Google Scholar
- Kashanian F, Nouri S, Chaharborj SS, Rizam ABM: Int. J. Mass Spectrom.. 2011, 303: 199. 10.1016/j.ijms.2011.02.001View ArticleGoogle Scholar
- Kiai SMS, Chaharborj SS, Bakar MRA, Fudziah I: J. Anal. At. Spectrom.. 2011, 26: 2247. 10.1039/c1ja10170fView ArticleGoogle Scholar
- Chaharborj SS, Kiai SMS, Bakar MRA, Ziaeian I, Fudziah I: Int. J. Mass Spectrom.. 2012, 309: 63.View ArticleGoogle Scholar
- Chaharborj SS, Kiai SMS: J. Mass Spectrom.. 2010, 45: 111.Google Scholar
- March RE, Todd JFJ: Practical Aspects of Ion Trap Mass Spectrometry: Chemical, Environmental, and Biomedical Application. CRC Press, New York; 1995.Google Scholar
- March RE: J. Mass Spectrom.. 1997, 32: 263. 10.1002/(SICI)1096-9888(199703)32:3<263::AID-JMS501>3.0.CO;2-1View ArticleGoogle Scholar
- March RE: Int. J. Mass Spectrom.. 2000, 200: 285. 10.1016/S1387-3806(00)00345-6View ArticleGoogle Scholar
- Philip EM, Denton MB: J. Chem. Educ.. 1986, 63: 617. 10.1021/ed063p617View ArticleGoogle Scholar
- McLachlan NW Environmental, and Biomedical Application. In Theory and Application of Mathieu Functions. Clarendon, Oxford; 1947.Google Scholar
- Abbasbandy S, Allahviranloo T, Darabi P: Math. Comput. Appl.. 2011, 16: 935-946.MathSciNetGoogle Scholar
- Ghanaie ZA, Moghadam MM: J. Math. Comput. Sci.. 2011, 2: 208-221.Google Scholar
- Allahviranloo T, Ahmady E, Ahmady N: Int. J. Comput. Math.. 2009, 86: 730-742. 10.1080/00207160701704564MathSciNetView ArticleGoogle Scholar
- Allahviranloo T, Ahmady E, Ahmady N: Inf. Sci.. 2008, 178: 1309-1324. 10.1016/j.ins.2007.10.013MathSciNetView ArticleGoogle Scholar
- Barnabas B, Imre JR, Attila LB: Inf. Sci.. 2007, 177: 1648-1662. 10.1016/j.ins.2006.08.021View ArticleGoogle Scholar
- Diamond P, Kloeden P: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, Singapore; 1994.View ArticleGoogle Scholar
- James JB, Thomas F: Fuzzy Sets Syst.. 2001, 121: 247-255. 10.1016/S0165-0114(00)00028-2View ArticleGoogle Scholar
- Georgiou DN, Juan JN, Rosana RL: Nonlinear Anal., Theory Methods Appl.. 2005, 63: 587-600. 10.1016/j.na.2005.05.020View ArticleGoogle Scholar
- Marek TM: Nonlinear Anal., Real World Appl.. 2012, 13: 860-881. 10.1016/j.nonrwa.2011.08.022MathSciNetView ArticleGoogle Scholar
- Khastan A, Nieto JJ, Rosana RL: Fuzzy Sets Syst.. 2011, 177: 20-33. 10.1016/j.fss.2011.02.020View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.