Associated codes the BDO-RFQ and the LIDOS RFQ for optimization parameters of RFQ channel are used. As an example, beam dynamics simulation results for 47.2 MHz RFQ accelerator are discussed.RFQ accelerator is used as the initial part of large linacs for industrial and medical purposes. There are many methods and codes to design a RFQ channel. Below we show how mathematical optimization methods can be used in the practical design of a RFQ accelerator [1], [2], [3], [4].Accelerating and focusing processes in RFQ accelerators are controlled by the smooth changing of the four parameters: vane modulation, intervane voltage, synchronous phase and aperture. These parameters define the length of the modulation period.At the same time today we have a well-developed mathematical theory of the multiparameter optimization which can be applied to beam dynamics and plasma problems and formally give us a possibility to optimize any accelerating or focusing channel [5].To create directional optimization methods based on the analytical approaches we use the following mathematical modelHere W n and W are the initial and the current beam energy, ω = 2π/λ is the RF-field frequency; k = 2π/l; L is the acceleration period length;; U is the intervane voltage, T is the acceleration efficiency, depending on the vane modulation and the ratio of the aperture to the acceleration period length; η = U T (U T )max ; the maximum value (U T ) max is the initial data. The last term in the phase motion equation corresponds to a space charge field. Constraint (3) concerns the defocusing parameter, usually 0.01 ≤ A ≤ 0.02. Constraint (4) concerns with the monotonic decrease of the phase length rms value ∆ϕ. It is necessary for space charge dominated beam acceleration. So we need to find plots η(ζ) and ϕ(ζ) to provide bunching and acceleration of a beam. To solve this problem we need to introduce some functional [1, 2], describing the capture of an accelerated particles and other needed beam parameters at the output of the channel. Constraints (3), (4) are included also.Optimization of the longitudinal motion of particles was carried out using the BDO-RFQ code developed at the SaintPetersburg State University [7]. As a result of the optimization the functions η(ζ) and ϕ(ζ) have been identified, providing desired characteristics of the output beam.After optimization of the phase motion, given the specified intensity values E = U/R 0 and focusing parameter Q = U/R 2 0 , the average aperture radius R 0 and the intervane voltage can be determined, then the efficiency T is determined, which unambiguously corresponds to the modulation of the electrodes m. The correspondence between the variable ζ and the cell number n is set by n = ζ/κ where κ = Ω/ω. Thus all the necessary dependencies are defined U (n), m(n), R 0 (n), ϕ s (n) for the initial approximation.The proposed technique is implemented in the BDO-RFQ code. The initial data array [U (n), m(n), R 0 (n), ϕ s (n)], obtained on the basis of the found optimized laws and then used to convert them into t...