Research Article Bessel Equation in the Semiunbounded Interval x [x 0, ]: Solving in the Neighbourhood of an Irregular Singular Point

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1 International Mathematics and Mathematical Sciences Volume 2016, Article ID , 7 pages Research Article Bessel Equation in the Semiunbounded Interval x [x 0, ]: Solving in the Neighbourhood of an Irregular Singular Point Qing-Hua Zhang, 1 Jian Ma, 2 and Yuanyuan Qu 1 1 The First Institute of Oceanography, State Oceanic Administration, Qingdao , China 2 College of Marine Sciences, Shanghai Ocean University, Shanghai 20106, China Correspondence should be addressed to Jian Ma; majian@shou.edu.cn Received 28 April 2016; Accepted 28 June 2016 Academic Editor: Theodore E. Simos Copyright 2016 Qing-Hua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study expresses the solution of the Bessel equation in the neighbourhood of x= as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval [x 0, ]. In order to guarantee the series uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution at x (not to mention a so-called formal solution), but a solution in the interval [x 0, ]with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal. 1. Introduction The general form of the Bessel equation is x 2 y (x) +λ 1 xy (x) +(λ 2 x 2 +λ )y(x) =0, (1) λ j (j = 1, 2, ) are arbitrary complex constants, with λ 1 =0and λ 2 =0,andx [0, ]. The Bessel equation only has two singular points on the real axis: x=0and x=. For convenience, set division point at x 0 > 0, to separate the entire solution interval into two parts, [0, x 0 ] and [x 0, ],thelatterofwhichis the semiunbounded interval that this study will solve the equationin.itiseasytoverifythatx = 0 is a regular singular point, but x = is irregular. Previous studies showed that the solution in the neighbourhood of the isolated singular point could be written as a combination of a knownform singular divisor and a specific nonsingular function. To find the solution in the neighbourhood of the regular singular point, one often uses the power series extension to reconstruct the nonsingular function. In the neighbourhood of the irregular singular point, attempts have been made with power series traditionally, but already existing efforts cannot find a uniformly convergent solution, which can only give an asymptotic solution at x.this so-called regular formal solution can only indicate the variation tendency of the solution [1 4]. However, the failure to obtain a solution with certain regularities of distribution using power series method does not necessarily mean inexistence of this kind of solution. As is known, the power series with uniform convergence in the complex number space converges to a complex analytic function, so that it is full differentiable. Yet, the Bessel equation is only a second-order differential equation, which demands limited smoothness (e.g., second-order derivative) only, but not full smoothness. Overdemanded smoothness could be the reason for the inability to find the solution with the uniformly convergent power series. This study adopts the corrected Fourier series [5 10] with limited smoothness to construct the nonsingular function and insert it into the derived equations from the Bessel equation. By introducing the constraint and compatibility conditions at the same time, all coefficients of the corrected

2 2 International Mathematics and Mathematical Sciences Fourier series can be determined completely. Thus, one can obtain the solution with certain regularities of distribution in the interval x [x 0, ],butnotanasymptoticorformal solution. There is no limitation on the equation parameter {λ j } in the whole solution procedure, which could be any arbitrary complex constant, so that the solution method given in this study is universal. 2. The Solution of the Bessel Equation in the Semiunbounded Interval [x 0, ] 2.1. Obtaining the Derived Equation. For convenience, we introduce coordinate transformation y (x) =y(x 0 ξ 1 )=y e (ξ), ξ = x 0 x, (2) which transforms the semiunbounded interval x [x 0, ]to aboundedoneξ [0,1]. Equation (1) turns into the following form with the new coordinate (Appendix A): ξ 4 y e (ξ) +(2 λ 1)ξ y e (ξ) +(λ 20 +λ ξ 2 )y e (ξ) =0, ξ [0, 1], λ 20 =λ 2 x 2 0. () It is easy to find that ξ=0is an irregular singular point of the equation, and then the irregular solution can be written as y e (ξ) =R(ξ) V (ξ), R (ξ) =ξ ρ e σξ 1 =ξ ρ exp (σξ 1 ), ρ, σ in R(ξ) are constants to be determined. V(ξ) is a nonsingular function with V(0) = 0,which satisfies the following equation: (4) ξ 4 V (ξ) +P 1 (ξ) V (ξ) +P 2 (ξ) V (ξ) =0, (5) P 1 (ξ) =2ξ 4 R P 2 (ξ) =ξ 4 R R +(2 λ 1)ξ, R +(2 λ 1)ξ R Using the differential relationship, one can obtain R +λ 20 +λ ξ 2. R R =ρξ 1 σξ 2, R R =ρ(ρ 1)ξ 2 +(2 2ρ)σξ +σ 2 ξ 4, P 1 (ξ) =(q 11 +q 12 ξ) ξ 2, P 2 (ξ) =q 21 +q 22 ξ+q 2 ξ 2. Hence,(5)canbewrittenasthefollowingform: L V ξ 4 V +(q 11 +q 12 ξ) ξ 2 V +(q 21 +q 22 ξ+q 2 ξ 2 )V=0, (6) (7) (8) (9) Let ξ 0,and,from one can obtain that is, which determines that q 11 = 2σ, q 12 =2+2ρ λ 1, q 21 =σ 2 +λ 20, q 22 =(λ 1 2ρ)σ, q 2 =ρ(ρ+1 λ 1 )+λ. L {V} =0, [L {V}] =0, (10) (11) q 21 =q 22 =0; (12) σ 2 +λ 20 =0, λ 1 2ρ=0, σ= { σ 1 =σ 0, σ 0 = λ 20 = λ 2 x 0, { σ { 2 = σ 0, σ 0 = λ 20 = λ 2 x 0, (1) (14a) ρ= λ 1 2, (14b) q 11 = 2σ 1, q 12 =2, q 2 = λ 1 2 [1 λ 1 2 ]+λ. So, the derived equation is obtained: (15) L 1 {V, σ} ξ 2 V +q 1 (ξ, σ) V +q 0 V=0, (16) q 1 (ξ, σ) =2(ξ σ), q 0 =q 2. (17) 2.2. Nonsingular Classic Solution of the Derived Solution. The form of the derived equation (see (16)) still seems to be a singular equation, but it can be guaranteed to have nonsingular classic solution after introducing certain constraint conditions (Appendix B). That is, ξ [L 1 {V, σ}] ξ=0 =0; (18a)

3 International Mathematics and Mathematical Sciences that is, 2σV (0, σ) +(2+q 0 )V (0, σ) =0. (18b) Because V (0, σ) <, V(ξ) can be expanded to the following corrected Fourier series: V (ξ) = n N V (k) (ξ) = n N A n exp (iα n ξ) + a l ξ l l!, A n (iα n ) k exp (iα n ξ) + a l l=k ξ l k (l k)!, k=1,2, (19) which is uniformly convergent and α n =2nπ.Atthe same time, we introduce the Fourier projection: 1 F 1 n = exp ( iα n ξ) dξ. (20) 0 Inserting expansion equation (19) to the derived equation (see (16)), one can obtain (Appendix C) n A n GR (n, ξ, σ) exp (iα n ξ) + After the Fourier projection, A n R(n,n 0,σ)= n N a l S(l,n 0,σ), a l GS (l, ξ, σ) =0. (21) R(n,n 0,σ)=F 1 GR (n, ξ, σ) exp (iα n ξ) n0 =F 1 GR (n, ξ, σ) n0 n, S(l,n 0,σ)= F 1 GS (l, ξ, σ) n0. It can be derived that A n = n 0 N, (22) (2) a l A e (l, n, σ), (24) A e (l,n,σ)is the solution of the following equation: A e (l, n, σ) R(n,n 0,σ)=S(l,n 0,σ). (25) n Inserting solution (24) into extension (19), one can obtain V (ξ, σ) = V (k) (ξ, σ) = a l Z 0 (l, ξ, σ), a l Z k (l, ξ, σ), k=1,2, (26) Z 0 (l, ξ, σ) = A e (l, n, σ) exp (iα n ξ) + ξl n l!, Z k (l, ξ, σ) =Z (k) 0 (l, ξ, σ), k=1,2. (27) The compatibility condition is required [5, 6] in Fourier projection of (16) (Appendix B): that is, L 1 {V (ξ, σ)} 1 0 =0; (28a) V (1, σ) +q 1 (1, σ) V (1, σ) q 1 (0, σ) V (0, σ) (28b) +q 0 [V (1, σ) V(0, σ)] =0. Insert solution (26) into constraint (18a) and (18b) and consistency (28a) and (28b) conditions to obtain the equation determining {a l } (Appendix D): Itcanbesolvedthat hence, Note that V (ξ, σ) = a l β 1l (σ) =0, a l β 2l (σ) =0. (29) a l =γ(l, σ) ; (0) γ (l, σ) Z 0 (l, ξ, σ). (1) V 1 (ξ) =V(ξ,σ 1 )=V(ξ,σ 0 ), V 2 (ξ) =V(ξ,σ 2 )=V(ξ, σ 0 ). Finally, the solution of () is obtained: y e1 (ξ) =ξ λ 1/2 exp (σ 0 ξ 1 )V 1 (ξ), y e2 (ξ) =ξ λ 1/2 exp ( σ 0 ξ 1 )V 2 (ξ).. Examples and Discussion (2) () As mentioned before, the solution method given here satisfies the equation with each parameter being any arbitrary complex constant, so that the first set of examples we will show aretoprovidetheresultsofthefunctionsv 1 (ξ) and V 2 (ξ) according to several sets of different complex constants. Their accuracy of approximation to satisfy the derived equation will also be estimated. The second set of examples aim to solve V 1 (ξ), V 2 (ξ) and, in turn, the solution of the Bessel equation, y e1 (ξ) and y e2 (ξ), with several sets of certain common real constants. Noting their linear combination as y 1 (x) and y 2 (x), we will compare them with already known Bessel functions and modified Bessel functions.

4 4 International Mathematics and Mathematical Sciences Table 1: Solution and relative error for Example 1. ξ V 1 (ξ) V 2 (ξ) Error(ξ, 1) Error(ξ, 2) 0 ( 1.512, ) (2.6727, ) /5 ( 1.57, ) (2.6668, ) /5 ( 1.65, 2.791) (2.6608, ) /5 ( 1.698, 2.792) (2.6549, ) /5 ( 1.760, 2.792) (2.6488, ) ( 1.822, 2.799) (2.6428,1.6141) Table 2: Solution and relative error for Example 2. ξ V 1 (ξ) V 2 (ξ) Error(ξ, 1) Error(ξ, 2) 0 (2.652, 0.074) ( , 1.684) /5 (2.627, 0.118) ( , 1.668) /5 (2.601, 0.160) ( , 1.650) /5 (2.6275, 0.202) ( ,1.62) /5 (2.6248, 0.242) ( ,1.612) (2.6220, 0.281) ( , 1.592) First Set of Examples Example 1. Let λ 1 = (1 + i)/2, λ 2 = 1, λ = ] 2 (] = 0.), x 0 = 10.0, andn=50,andcalculatev 1 (ξ) and V 2 (ξ). To examine the accuracy of the solution to approximate the derived equation, note Table : Solution and relative error for Example. ξ V 1 (ξ) V 2 (ξ) Error(ξ, 1) Error(ξ, 2) 0 ( , ) (1.0966, ) /5 ( , ) (1.0916, ) /5 ( , ) (1.0865, ) /5 ( 1.09, 0.407) (1.0814, ) /5 ( 1.099, 0.402) (1.0762, ) ( , ) (1.0711, ) it is hard to imagine to approximate them with uniform convergence by using the power series. On the other hand, when letting x 0 = 10.0 and N=50, they satisfy the derived equation at relative errors smaller than ,meaning accuracy is significant enough..2. Second Set of Examples Example 1. To compare with Bessel function of the first kind, let λ 1 =λ 2 =1, λ = ] 2 (] = 0., 2] = integer), x 0 = 20.0, and N = 200. For these parameters, the Bessel equation has solutions of the Bessel function of the first kind, J ] (x) and J ] (x), intheneighbourhoodofx = 0, which has infinite points of zero on the real axis. And when x 1, they have the following Asymptotic Formula to determine the distribution of the zero points: g 1 (ξ, j) = ξ 2 V j (ξ), g 2 (ξ, j) = q 1 (ξ, σ j )V j (ξ), g (ξ, j) = q 0 V j (ξ), (4) x 1 (m) =(m+ ] )π, x 2 (m) =(m ] 2 1 (7) 4 )π. g 0 (ξ, j) = g 1 (ξ, j) + g 2 (ξ, j) + g (ξ, j), with complex module E l (ξ, j) = g l (ξ, j), l=0,1,2,. (5) And then, we can define the relative accuracy of the solution V j (ξ) to satisfy the equation Error (ξ, j) = E 0 (ξ, j) Max {E 1 (ξ, j), E 2 (ξ, j), E (ξ, j)}. (6) Values of the solution and relative error are shown in Table 1. Example 2. Let λ 1 =1, λ 2 =(1+i)/2, λ = ] 2 (] = 0.), x 0 = 10.0, andn = 50,andtheresultsarepresentedin Table 2. Example. Let λ 1 =1, λ 2 =1, λ =( ] 2 +i)/2(] = 0.), x 0 = 10.0,andN=50,andtheresultsareinTable. From the above three examples, it is easy to see that functions V 1 (ξ) and V 2 (ξ) vary very slowly with ξ; thus, The solutions obtained in this study y e1 (ξ) and y e2 (ξ) are conjugate complex ( σ 0 = i λ 2 x 0 ; Appendix E); however, real solution compatible with J ] (x) and J ] (x) can be obtained by linear combination of certain form (Appendix F): y 1 (x) =b 11 y e1 (ξ) +b 12 y e2 (ξ), y 2 (x) =b 21 y e1 (ξ) +b 22 y e2 (ξ), with the zero points noted as x 1 (m) and x 2 (m). (8) Table 4 compares the zero points x 1 (m) of J ] (x) and x 1 (m) of y 1 (x) and the zero points x 2 (m) of J ] (x) and x 2 (m) of y 2 (x).itiseasytofindthatthesolutionobtainedhere(after rearranging) has consistent distribution of zero points with Bessel function of the first kind, indicating the credibility of the solution method provided by the present study [1, 2]. Example 2. Let λ 1 = 1, λ 2 = 1, λ = 0, x 0 =.0, and N = 200, which represent the Bessel equation with modified Bessel functions I 0 (x) and K 0 (x) in the neighbourhood of x=0. With this set of parameters, the solutions obtained by this paper are y e1 (ξ) and y e2 (ξ) in the interval x [x 0, ]or

5 International Mathematics and Mathematical Sciences 5 Table 4: Comparisons of the zero points of the Bessel function in Example 1 between the asymptotic formula and the corrected Fourier series in this study. m x 1 (m) x 1 (m) x 2 (m) x 2 (m) Relative errors: Error 1 (m)= ( x 1 (m) x 1 (m))/x 1 (m) < , Error 2 (m)= ( x 2 (m) x 2 (m))/x 2 (m) < Table 5: Comparisons between the solution of Example 2 obtained with known modified Bessel functions and that calculated in this study. x y 1 (x) I 0 (x) y 2 (x) K 0 (x) Relative errors: Error 1 (x) = (y 1 (x) I 0 (x))/i 0 (x) < , Error 2 (x) = (y 2 (x) K 0 (x))/k 0 (x) < ξ [0,1], with their linear combination also being solution of the equation y 1 (x) =b 11 y e1 (ξ) +b 12 y e2 (ξ), y 2 (x) =b 21 y e1 (ξ) +b 22 y e2 (ξ). (9) From the table of Bessel functions, one can look up the values of functions I 0 (x) and K 0 (x) in the interval x [0,10].Atthe two arbitrarily selected points in the interval x [,10],we linearly fit y 1 (x) and I 0 (x) and y 2 (x) and K 0 (x), to determine constant b ij.theobtaineddistributionsoffunctionsy 1 (x) and y 2 (x) are compared with those of functions I 0 (x) and K 0 (x) in Table 5, which shows apparent consistency between the solution obtained with known modified Bessel functions and that calculated in this study. Example. Let λ 1 = 1, λ 2 = 1, λ = 1, x 0 =.0, and N = 200, and this set of parameters corresponds to Bessel equation with modified Bessel functions I 1 (x) and K 1 (x) as the solution in the neighbourhood of x = 0. The corresponding solution obtained here is y e1 (ξ) and y e2 (ξ), whose linear combination can construct new solution functions, after linear fit with functions I 1 (x) and K 1 (x): y 1 (x) =b 11 y e1 (ξ) +b 12 y e2 (ξ), y 2 (x) =b 21 y e1 (ξ) +b 22 y e2 (ξ). (40) Comparisons between their distributions and those of I 1 (x) and K 1 (x) are illustrated in Table 6. It can be easily seen that the solution obtained here is completely consistent with the known solution with modified Bessel functions [1, 2]. Table 6: Comparisons between the solution of Example obtained with known modified Bessel functions and that calculated in this study. x y 1 (x) I 1 (x) y 2 (x) K 1 (x) Relative errors: Error 1 (x) = (y 1 (x) I 1 (x))/i 1 (x) < , Error 2 (x) = (y 2 (x) K 1 (x))/k 1 (x) < Conclusions The present study proves that the solution of the Bessel equation in the interval with irregular singular point can be expressed as the product of a singular divisor and a nonsingular function, the latter of which can be approximated with uniform convergence by corrected Fourier series with limited smoothness in the interval. Thus, we are able to obtain

6 6 International Mathematics and Mathematical Sciences the solution with certain regularities of distribution of the Bessel equation in the semiunbounded interval with irregular singular point, but no more an asymptotic solution at x, or a regular formal solution. This indicates that it is incorrect to conclude existence of only asymptotic or regular formal solution in the neighbourhood of an irregular singular point only because of the inadequacy of certain solution method. Appendix A. Coordinate Transformation There is a relationship between the old and new coordinates as follows: ξ =( x 0 x ) = 1 ξ 2, x 0 ξ = 2 x 0 ξξ = 2 x 2 0 ξ, y (x) =y e (ξ) ξ = 1 x 0 ξ 2 y e (ξ), y (x) =y e (ξ) ξ 2 +y e (ξ) ξ = 1 x0 2 ξ 4 y e (ξ) + 2 x0 2 ξ y e (ξ). (A.1) Inserting these relations into (1), one can derive () using the new coordinate after rearrangements. B. Constraint and Compatibility Based on Appendix B of [4], although the differential equation L 1 {V(ξ)} = f(ξ) stillhasaformofsingularequation, its nonsingular solution can exist with certain constraint condition; that is, ξ [L 1 {V (ξ)}] ξ=0 =f (ξ) ξ=0. (B.1) On the other hand, the classic solution of the equation not only demands the existence of its Fourier projected equation F 1 L 1 {V} n =F 1 f (ξ) n, but also requires the following compatibility condition: L 1 {V} 1 0 =f(ξ) 1 0. C. Parameters for (21) and (22) Consider GR (n, ξ, σ) = 2 (iα n ) 2 ξ2 2! +2iα nξ+(q 0 2iα n σ), GS (l, ξ, σ) =(l 2 +l+q 0 ) ξl l! 2σ ξl 1 (l 1)!, (B.2) (B.) R(n,n 0,σ) = 2 (iα n ) 2 I e0 (2, n 0 n)+2iα n I e0 (1, n 0 n) S(l,n 0,σ) +(q 0 2iα n σ) δ (n 0 n), = {(l 2 +l+q 0 )I e0 (l, n 0 ) 2σI e0 (l 1,n 0 )}, (C.1) I e0 (l, n) = F 1 ξ l /l! n. D. Determining Coefficients of (26) In (29), β 1l (σ) = 2σZ 2 (l, 0, σ) +(2+q 0 )Z 1 (l, 0, σ), β 2l (σ) =[ξ 2 Z 2 (l, ξ, σ) +q 1 (ξ, σ) Z 1 (l, ξ, σ) 1 +q 0 Z 0 (l, ξ, σ)]. 0 Let a =1, which can derive a 1 β 11 (σ) +a 2 β 12 (σ) = β 1 (σ), a 1 β 21 (σ) +a 2 β 22 (σ) = β 2 (σ). Itcanbesolvedthata l = γ(l, σ), β 1 (σ) β 12 (σ) γ (1, σ) = Δ, β 2 (σ) β 22 (σ) β 11 (σ) β 1 (σ) γ (2, σ) = Δ, β 21 (σ) β 2 (σ) β 11 (σ) β 12 (σ) Δ= ; β 21 (σ) β 22 (σ) atthesametime,letγ(, σ) = 1. E. Proof of Conjugate Relation (D.1) (D.2) (D.) Because parameters λ 1 =λ 2 =1and λ = ] 2 are all real numbers, σ 0 = λ 2 x 0 =ix 0, (E.1) σ 0 = ix 0 = σ 0, and the parameters in (16) satisfy the following relation: [q 1 (ξ, σ)] =q 1 (ξ, σ), (E.2) [q 0 ] =q 0. It is known that V 1 (ξ) = V(ξ, σ 0 ) and V 2 (ξ) = V(ξ, σ 0 ) satisfy (16), noted as ξ 2 V 1 +q 1 (ξ, σ 0 )V 1 +q 0V 1 =0, (E.) ξ 2 V 2 +q 1 (ξ, σ 0 )V 2 +q 0V 2 =0. (E.4)

7 International Mathematics and Mathematical Sciences 7 Solving the conjugate complex for (E.4), 0=[ξ 2 V 2 +q 1 (ξ, σ 0 )V 2 +q 0V 2 ] =ξ 2 [V 2 ] +q 1 (ξ, σ 0)[V 2 ] +q 0 [V 2] =ξ 2 (V 2 ) +q 1 (ξ, σ 0 )(V 2 ) +q 0 V 2. (E.5) Comparing (E.5) and (E.), a conjugate relation is obtained: It is known that V 2 (ξ) =V 1 (ξ). y e1 (ξ) =ξ λ 1/2 exp (σ 0 ξ 1 )V 1 (ξ), y e2 (ξ) =ξ λ 1/2 exp (σ 0 ξ 1 )V 2 (ξ), so the conjugate relation is obtained: [y e2 (ξ)] =ξ λ 1/2 exp ( σ 0 ξ 1 )V 2 (ξ) =ξ λ 1/2 exp (σ 0 ξ 1 )V 1 (ξ) =y e1 (ξ). F. Linear Combination for Real Solution (E.6) (E.7) (E.8) Taking x 1,x 2 large enough, x 1 = 20, x 2 =60, for example, the corresponding ξ 1 =1/x 1, ξ 2 =1/x 2. Demanding y 1 (x 1 )=J ] (x 1 ), y 1 (x 2 )=J ] (x 2 ), y 2 (x 1 )=J ] (x 1 ), (F.1) References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, [2] L. C. Andrews, Special Functions of Mathematics for Engineers, McGraw-Hill, New York, NY, USA, [] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Springer, Berlin, Germany, 199. [4] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II,vol.14ofSpringerSeriesinComputational Mathematics, Springer, Berlin, Germany, 2nd edition, [5] Q.-H. Zhang, S. Chen, and Y. Qu, Corrected Fourier series and its application to function approximation, International Journal of Mathematics and Mathematical Sciences, vol.2005,no.1,pp. 42, [6] Q.-H. Zhang, S. Chen, J. Ma, and Y. Qu, Solutions of linear ordinary differential equations with non-singular varying coefficients by using the corrected Fourier series, Applied Mathematics and Computation,vol.187,no.2,pp ,2007. [7] Q.-H. Zhang, J. Ma, and Y. Qu, Unified solution for the Legendre equation in the interval [-1, 1] -an example of solving linear singular-ordinary differential equations, Applied Mathematics and Computation,vol.289,pp.11 2,2016. [8] C. M. Verrelli, Fourier series expansion for synchronization of permanentmagnetelectricmotors, Applied Mathematics and Computation,vol.217,no.9,pp ,2011. [9] C. M. Verrelli, Synchronization of permanent magnet electric motors: new nonlinear advanced results, Nonlinear Analysis: Real World Applications,vol.1,no.1,pp ,2012. [10] N. H. B. Zainal and A. Kilicman, Comparison between Fourier and corrected Fourier series methods, Malaysian Mathematical Sciences,vol.7,no.2,pp ,201. y 2 (x 2 )=J ] (x 2 ) or b 11 y e1 (ξ 1 )+b 12 y e2 (ξ 1 )=J ] (x 1 ), b 11 y e1 (ξ 2 )+b 12 y e2 (ξ 2 )=J ] (x 2 ), b 21 y e1 (ξ 1 )+b 22 y e2 (ξ 1 )=J ] (x 1 ), (F.2) b 21 y e1 (ξ 2 )+b 22 y e2 (ξ 2 )=J ] (x 2 ), (b 11,b 12 ) and (b 21,b 22 ) can be determined. Competing Interests The authors declare that there are no competing interests regarding the publication of this paper. Acknowledgments The authors acknowledge the support from the National Natural Science Foundation of China (no ) and Shanghai Chair Professor-Eastern Scholar Program, China.

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