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22
Approximate analytical method for finding eigenvalues of Sturm-Liouville problem with generalized boundary condition of the third kind
V.D. Lukyanov1, D. A. Bulekbaev2, A. V. Morozov2, L. V. Nosova2 Joint-Stock Company "Avangard", Kondrat'evsky, 72, St. Petersburg, 195271, Russia 2 Mozhaisky Military Space Academy, Zhdanovskaya, 13, St. Petersburg, 197198, Russia lukyanovvd@rambler.ru, atiman@mail.ru, alex.morozof@gmail.com, lvn1201@gmail.com
PACS 02.60.Lj, 47.61.Fg, 62.25.-g DOI 10.17586/2220-8054-2020-11-3-275-284
The Sturm-Liouville problem is solved for a linear differential second-order equation with generalized boundary conditions of the third kind Generalized boundary conditions consist of a linear combination of the boundary values of a function and its derivative. The coefficients of the linear combination are polynomials of the boundary problem eigenvalue. A method of approximate analytical calculation of boundary problem eigenvalues is proposed The calculation error of an eigenvalue is estimated.
Keywords: Sturm-Liouville problem, boundary conditions of the third kind, eigenfunctions, eigenvalues, approximation. Received: 21 June 2020
1. Introduction
Advances of nanotechnologies make it possible to design and create microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). These systems find use as primary converters in miniature sensors of physical quantities [1].
Characteristic dimensions of used systems elements: hundreds of nanometers for MEMS with operating frequencies of sensors up to 10 gHz and tens of nanometers for NEMS with operating frequencies of sensors up to tens of gHz [2,3].
High-quality electromechanical resonators are used as primary converters to provide sensor sensitivities and measurement accuracy of physical quantities. Various aspects of these devices research, technological developments and their various applications are presented in the reviews [4-12].
The simplest resonators in MEMS and NEMS are one-dimensional distributed elastic structures (strings, rods, beams loaded with sensitive elements - masses). Technologies for making such resonators are presented, for example, in [13,14].
The mathematical model of time harmonic elastic oscillations of resonators is the Sturm-Liouville boundary problem. The Sturm-Liouville problem is a boundary problem on a segment for an ordinary linear homogeneous differential equation with homogeneous boundary conditions at the ends of the segment [15].
To describe the motion of an MEMS or NEMS element, a generalized boundary condition of the third kind is used, containing eigenvalues [15-17]. A generalized boundary condition is a linear combination of the sought function and its first-order derivative, wherein the coefficients of the linear combination are polynomials with respect to the sought eigenvalue of the problem. The mathematical theory such of problems is built in the works [18,19], where a generalization of the classical Sturm-Liouville problem theory is given [20,21].
2. Statement of the Sturm-Liouville problem with generalized boundary conditions of the third kind
We consider the Sturm-Liouville boundary problem for a linear homogeneous ordinary second-order differential equation
v" = -mv, (1)
where y = y(x) at -1 < x < 1, m is an eigenvalue of the problem, m > 0. Taking into account the positivity m we will accept m = A2, where A > 0.
As boundary conditions for the differential equation, we have generalized homogeneous conditions of the third kind [15-17]:
ai (M)y(1)+ £i (m)v'(1)=0, (2)
a2(M)v(-1)+ £2 (m)v'(-1) = 0. (3)
Multipliers ak = ak (m) and £k = £k (m) (k=1,2) are polynomials with respect to m. Polynomials coefficients are parameters of the problem. Parameters depend on characteristics of physical models in application problems. If the
total number of such parameters under conditions (2) and (3) is equal to the number S, then we will call the boundary problem S-parametric. We will denote problem parameters by Yi,Y2,-Ys.
If the homogeneous boundary problem (1)-(3) has a non-zero solution y0 (x) at a value ^ = then the value is called an eigenvalue, and the corresponding solution y0(x) is called an eigenfunction of the boundary problem.
3. Sturm-Liouville problem eigenvalues calculation algorithm
To solve the Sturm-Liouville problem means to find all the problem eigenvalues and eigenfunctions. Eigenvalues are analytically calculated for the simplest boundary conditions of the first and second kind. In cases of boundary conditions of the third kind, eigenvalues are calculated as the roots of a transcendental equation lacking an exact analytical solution. Therefore, various approximate methods are used to find eigenvalues. These methods produce only an asymptotic estimate of an eigenvalue without discussing the accuracy of the result obtained, which is important in applied research.
The proposed new method makes it possible to calculate a set of boundary problem eigenvalues and to evaluate effectively the accuracy of their calculation.
Consider the problem of finding all eigenvalues and eigenfunctions of the Sturm-Liouville problem (1)-(3). To solve the problem, we find the general solution of the differential equation (1)
y = Ci cos Ax + C2 sin Ax, (4)
where C1 and C2 depend on the value of A.
Satisfying the boundary conditions (2) and (3), we obtain a homogeneous system of linear algebraic equations for finding quantities C1 and C2 in the general solution (4):
iCip-(A) + C2q+(A)=0,
\Cip+(A)+ C2q-(A)=0, ()
where
(A) = afc(A2)cos A ± A^fc(A2) sin A,
q± (A) = ±ak (A2) sin A + A^fc (A2) cos A
at k=1,2. We take upper or lower symbol in formulas for p± (A) and q± (A) simultaneously.
The principal determinant of the homogeneous system of algebraic equations (5) is in the form of
A(A, yi, 72, ...,ys )= P-(A) q-(A) - (A) q+ (A). (6)
The arguments of the determinant are a sought value A and parameters y1, Y2, ... ,Ys included in the conditions (2) and (3).
In order to find a non-zero solution to the homogeneous system of linear algebraic equations (5), we will require that the principal determinant of this system to be zero:
A(A, yi, Y2, ... , Ys) = 0. (7)
Equation (7) is called a characteristic equation. To find an eigenvalues ^ of the Sturm-Liouville problem, we look for the positive roots A of characteristic equation (7), and then look for the eigenvalue ^ = A2 > 0.
As can be seen from equality (6), the principal determinant is the sum of the components consisting of the works of the polynomial and trigonometric functions that depend on a sought A value. Thus, equation (7) is transcendental and does not allow an analytic solution of the form of
A = <^(yi, Y2, ... , Ys), (8)
which would allow to calculate an eigenvalue from known problem parameter values.
The proposed approximate method provides an approximate representation of the view (8) and estimates the accuracy of this representation.
Let us give an algorithm of the proposed method of obtaining an approximate solution of the characteristic equation.
In the first step we will make a table of function values A = ^(y1, y2, ... , YS) on some set of variable Y1, y2, ... , YS values. To do this, we record the determinant A(A, y1, y2, ... , YS) as a linear combination of trigonometric functions with multipliers representing some polynomials with respect to A value. Coefficients of these polynomials are calculated through parameter y1, Y2, ... , YS values.
The dependence of the determinant on each parameter is not more than quadratic, since the problem parameters are coefficients of the polynomials and enter in the polynomials linearly, and the polynomials themselves are only multiplied according to formula (6).
Using this circumstance, we solve equation (7) with respect to some arbitrarily chosen parameter 7i with the i number. We get a functional dependence of the form of
Yi = ^¿(A, Yi, 72, ..., Yi-i, Yi+i, ... , Ys). (9)
We now arbitrarily set argument values of the function and calculate the value Yi. For example, we can select arguments values of the function on a uniform grid of the A, y1, y2, ..., Yi-1, Yi+1, ... , ys arguments. Values of the parameter 7i are then calculated. If necessary, if the values obtained for the parameter 7i are not sufficient, the relation (9) can be recorded with another selection of the parameter on the left hand side of the equality (9). We can now choose additional values for the 7i parameter. As a result, we create a table of function values.
In the second step of solving characteristic equation we have a sufficient number of function values. We find an approximating function ri = ^¿(A, y1, y2, ..., 7i-1, 7i+1, ... , ys) using the obtained set of the function values (fi given by formula (9). We will select the approximation function so that it has an inverse function with respect to the variable A at fixed values of the y2, ..., 7i-1, 7i+1, ... , ys arguments:
A = ^-1(71, 72, ..., 7i-1, ri, 7i+1, ... , Ys). In order to calculate the approximate value of the sought A* value, we set in this formula ri = 7i:
A* = $j-1(71, 72, ..., 7i-1, 7i, 7i+1, ... , Ys). (10)
Then we calculate our eigenvalue m* = A2.
As an example of the implementation of the method, we consider the problem of finding eigen frequencies of a longitudinally oscillating rod loaded with masses at the ends.
4. Statement of eigen frequencies problem for a longitudinally oscillating rod loaded at the ends with masses
We have an elastic uniform rod: the rod length is 21, the constant cross-sectional area is F, the rod material has Young's modulus E, linear density is p. Body 1 with mass M1 and body 2 with mass M2 are fixed at the rod ends. The placement of rod (R) and bodies 1 and 2 in the coordinate system is shown in Fig. 1.
FIG. 1. Longitudinally oscillating rod (R) loaded at the ends with bodies 1 and 2. Choice of the coordinate system is shown
Small longitudinal displacements U = U (X, t) of rod cross section with coordinate X from equilibrium position at moment of time t satisfy to the equation
pFd2u (X,t) EFd2u (X,t)
pF dt2 = EF dX 2 (11)
at -1 < X < 1.
Boundary conditions describe masses oscillations under the action of the elastic rod: we have on the right end of the rod if X = 1
MdMu)= dUM (12)
dt2 = (12)
we have on the left end of the rod if X = -1
M2 ^ = ef^ . ,13,
In order to find eigen frequencies of rod oscillations it is considered that dependence of longitudinal displacement of rod cross-section on time is harmonic with circular frequency w: U(X, t) = Y(X) e-iWi, Y = Y(X) is the amplitude of longitudinal displacement of rod cross-section at the point with coordinate X.
Let us pass in equations (11)-(13) to dimensionless values. We introduce the dimensionless coordinate x = X/1, then -1 < x < 1, and introduce the dimensionless amplitude of the longitudinal displacement of rod cross-section y(x) = Y (X )/1.
We introduce also dimensionless quantities: dimensionless eigenvalue u = pw2l2/E, dimensionless eigen frequency A = ^u, and two dimensionless parameters of the problem: y1 = M1/M0 and y2 = M2/M1, where the mass M0 is equal to half the mass of the rod: M0 = pFl. Note that problem parameters have the properties: y1 > 0,
Y2 > 0.
Taking into account the introduced dimensionless quantities problem (11)-(13) takes the form of boundary problem (1)-(3)
y" = -A2y, (14)
at -1 < x < 1, with edge conditions of type (2) and (3) where a1(A2) = y1A2, a2(A2) = y1Y2A2, here pk = pk(A2) = ( —1)k at k=1,2:
Y1A2y(1) - y'(1) = 0, (15)
Y1Y2A2 y(-1)+ y'(-1)=0. (16)
We have a two-parametric boundary problem with parameters y1 and y2.
A simpler single-parameter problem is discussed in [22], when the masses attached to the rod ends are the same:
M1 = M2 or y2 = 1.
The general solution of equation (14) has the form (4). We obtain from the boundary conditions (15) and (16) a system of linear algebraic equations of the form of (5), where
p±(A) = ak (A2) cos A ± A sin A, q±(A) = ±ak (A2) sin A ^ A cos A
at k = 1, 2.
The characteristic equation (7) for finding non-zero eigen frequencies is converted to the form of
y2 Y2 A2 sin2A - Y1(1+ Y2) Acos2A - sin2A = 0. (17)
The obtained equation defines the implicit dependence of the dimensionless frequency A on the problem parameters Y1 and y2. As shown in item 3 and seen from equation (17), it is not possible to obtain an analytical solution of the characteristic equation with respect to a sought value A.
The proposed method of solving the characteristic equation makes it possible to obtain an approximate analytical A = A(y1, y2) dependence.
Note that the characteristic equation (17) was obtained in 6.2 [23] in solving the problem of a substance diffusion through a permeable wall. The equation was solved graphically for the particular y2 = 1 case, when this equation is converted to the form of cot 2A = ( y2 A2 - 1) /2y1 A. The graphs of functions from the right and left hand sides of the equation were drawn at a fixed value of the y1 parameter. The sought frequency values A were obtained as abscissas of the intersections points of these graphs.
As a result, in [23] there is a table of values of five first sought eigen frequency values As for three parameter y1 values. It is difficult to use the obtained results in practice: first, the number of frequencies found is insufficient for the qualitative numerical simulation of a physical problem, second, modeling may require a different parameter value y1 than the table value, third, the case y2 = 1 will be interesting.
5. Approximate solution of characteristic equation for the problem of longitudinal oscillations of elastic rod loaded with masses
In computer simulation of practical problems it is convenient to have an approximate analytical solution of the characteristic equation for all roots of this equation and at any values of problem parameters.
Approximate methods used to solve equations of type (17) give analytical estimates for eigenvalues sufficient for large eigenvalues [24]. Obtaining approximate formulas for the first eigenvalues near zero causes difficulties. Estimates of the accuracy of approximate formulas for eigenvalues near zero are not usually considered. The proposed method of solving the characteristic equation allows for solve these problems.
5.1. Finding dependencies of eigen frequency A on Sturm-Liouville problem parameters 71 and 72 in tabular and graphical view
Let's take advantage of the fact that the characteristic equation (17) cannot be solved in elementary functions relative to the eigen frequency A, but can be resolved relative to the parameters y1 or y2.
Consider the values A and y2 in the characteristic equation as independent variables (0 < A < y2 > 0), and consider the parameter y1 > 0 as dependent variable. The left hand side of equation (17) is a quadratic function relative to the y1 variable. We find the dependence of the value y1 on the variables A and y2 , if we calculate the roots of quadratic equation (17):
-± = Y±(A,Y2) = ^-^ (w(A,Y2) ± VW2(A,Y2) + 4Y2A2) , (18)
where the function W(A, -2) = (1 + -2)A cot 2A is used. In formula (18), either the upper or lower signs are taken at the same time.
Denote the functions —± = - ± (A, -2) by —± = —± (A, -2) if argument A belongs to the interval (1, vn] with the number n:
—±(A —2) = ——2 ) at A e (vn- 1, vn].
5.2. Properties of (A, y2 ) functions
1. Functions - ±(A, -2) are defined for all A > 0 but values vn = ±nn/2 at n = 1, 2, .... Direct substitution of values A = vn into equation (17) proves that relations (17) and (18) are equivalent.
2. Functions —±(A, —2) are even relative to the A variable, since the function W(A, —2) in equality (18) is even, and the degrees of A are even.
3. Functions —±n(A, —2) are continuous on each (vn- 1, vn] interval.
4. Functions —±n(A, —2) are of constant signs: —+„(A, —2) > 0, — ^(A, —2) < 0. Taking into account the positivity of possible parameter — 1 values we will study further only the functions —+„(A, —2).
5. We have the equality —+n(vn+ 1, —2) = 0 and the limit value —+„(A, —2) ^ as A ^ vn + 0.
6. The graphs of functions — +(A) = — +n(A, —2 at n = 1, 2, 3 are shown in Fig. 2(a) for three fixed values of the —2 parameter. One can see that functions — +n(A, —2) decrease monotonically. Function graphs with different values of the parameter —2 do not intersect. A function graph with a larger parameter —2 value is below a function graph with a smaller parameter —2 value. This follows from the comparison of function values — + = —¿(A, —2) in the middle of the n interval at An = vn-1 + n/4. We have according to formula (18) W(An, —2) = 0, and we get the value of the function —+(An, —2) = 1/^^-2). Comparing two such values for two curves with different parameter values
(1) ^ (2) • —2 ) > —() gives
—+n(An,—( 1 )) —+n(An,—(2))
(2)
-7ty < 1. (19)
—2 )
7. Functions — + = —¿(A, —2) are monotonic and continuous in the domain of their definition A e (vn- 1, vn] at n = 1, 2, ..., and have inverse functions for a fixed parameter value —2 = —(0). We denote these inverse functions by An = A+(—1, —2). Functions An = A+ (—1, —2) are monotonic, continuous in the domain of their —1 > 0 definition. The inverse function graphs An = A+(—1, —2) are shown in Fig. 2(b) for the same values of the —2 parameter. The graphs of these functions are symmetrical to the graphs of the functions — + = —+„(A, —2) with respect to the bisector of the first quadrant.
Function graphs An = A+(—1, —2) for n = 1, 2, 3 are graphs of the exact solution of the problem at fixed values of the —2 parameter. We find the sought values An of the Sturm-Liouville problem at the value of the parameter —1 = —(0) as ordinates of intersection points of the straight line (vertical straight line in Fig. 2(b)) with function graphs
A = A+(—1, —2) at n = 1, 2, ....
5.3. Approximation functional dependencies between eigenvalues and parameters of the Sturm-Liouville problem
The result of the first step of the proposed method was obtaining functional dependencies —+ = —+„(A, —2) and graphs of the exact An = A+(—1, —2) solution. The disadvantage of the graphical solution of the characteristic equation is the difficulty of using its results in computer modeling of a physical problem.
In the second step, we approximate the functions —+ = —+„(A, —2) with elementary functions r 1n = r 1 n(A, —2), where the first index 1 coincides with the index of the parameter — 1, and the second index n is the number of interval (vn- 1, vn] at which the functions are defined.
Let's choose functions r1n (A, —2) so that they have the properties 1-7 of the — + (A, —2) functions. Let interpolationapproximation functions r1n(A, —2) interpolate the values of the functions — + = —+„(A, —2) at A = vn and provide their limit values as A ^ vn-1 + 0; let r 1n(A, —2) approximate the values of the functions — + = — +n(A, —2) at internal points of interval (vn-1, vn) [25].
Taking into account properties 5 and 7 of the — + = — ^(A, —2) functions, select functions r1n = r1n(A, —2) in the form of
2
- A2 \ rn
—+ « Tm = r„(A, — 2)= An—j , (20)
where parameters An, sn and rn are introduced. The additional condition rn > 0 provides the required function values at the ends of the (vn-1, vn] interval. By verifying directly, one can find out that the functions selected in the form (20) satisfy the 1-4 properties. The additional condition sn < 0 ensures that property 6 is met.
Fig. 2. Mutually inverse functions: 7+ = 7 + (A, 72) - (a); A = A+(yi, 72) - (b). Curves 1, 2, 3 are constructed for functions with indices n = 1, 2, 3 at three values of the y2 parameter: y2 = 0.5 - dashed lines; y2 = 1.0 - solid lines; y2 =2.0 - chain-dotted lines
The parameters An, sn and rn introduced in the right side of formula (20) allow one to approximate the functions Y+ = Yin (A, Y2) at the internal points of the (vn-1, vn ] interval using the least squares method. After calculating parameters, we get elementary functions r1n = r1n(A, y2). Further consideration is carried out for an arbitrary fixed number n of the (vn-1, vn] interval. The function r1n = r1n(A, y2) has the inverse function at interval (vn-1, vn], and the inverse function is also elementary:
X ^ X = IVn+1(AnYSn )qn + vn+1(r1n)qn~ (21)
An ~An =v ^ , (21)
where qn = 1/rn.
Note that when solving the single-parameter problem in [22], when choosing the form of approximation r = rn(A) function, evenness of the function y+(A) relative to variable A was not taken into account. Accounting for the evenness of the function y+(A, y2) relative to variable A in formula (20) increases the approximation's accuracy.
We find the approximation parameters rn, sn and An by the least squares method, which provides the best approximation of the y+ = Y+x(A, Y2) function.
After calculating approximation parameters, the values for An are obtained using formula (21) at the given values of the problem parameters y1 and y2. Then, we find the eigenvalue = A2n of the Sturm-Liouville problem.
5.4. Finding of approximation parameters
(j)
To find approximation parameters An, sn and rn we set J of values for the problem parameter y2 : Y2 , where j = 1, 2, ..., J. We will also select Mn of values for the argument Anm, where m = 1, 2, ..., Mn on each interval (vn-1, vn). The values of the arguments y2j) ^ (0, T) and Anm G (vn-1, vn] must cover uniformly the domain of the function y+x = Y+x(A, Y2) definition, the quantity T being determined by the largest value the parameter y2 can take.
Let's calculate M J of values of the y+x = Y+i(A, Y2) function: Y1nmj = Y+x(Anm, y2j) ). To apply the least squares method, take the logarithm of both sides of equation (20)
ln(r 1 n(A, Y2)) = qn + Sn lnY2 + rnRn(A), (22)
where
qn = ln(An), (23)
R.w = ln () ■ (24)
We will look for the qn, sn and rn values from the condition of ensuring the minimum value of the function Fn(qn, sn, rn). The function Fn(qn, sn, rn) is the average value of the sum of the difference squares for values
ln(7+nmj) andln(rin(A
Yj))):
Fn(qn7 rn) J Mn
ln Y+nmj - ln(rin(Anm, y2^))
1
^ ^ (ln Yl+nmj - in - sn In Y(j) - TnÄn(Anm^ .
(25)
Mn J^ ^ V ,1nmj
n j=1 m=1
Here and elsewhere, we use the symbol of averaging to calculate the average value of some variable value Z if Z takes
the z11; z12, ..., zMJ values:
1 M J Z = .
m=1 j=1
The condition of the minimum for function Fn(qn, sn, rn) is the equality to zero of its partial derivatives regarding the arguments qn, sn and rn. This condition results in a system of linear algebraic equations regarding the quantities
qn, rn, and sn
A X
Bn
(26)
where the elements of the third order matrix are: A11 = 1, A12 = A21 = ln —2, A13 = A31 = Rn, A22 = (ln —2)2, A23 = A32 = Rn ln —2, A33 = (Rn)2; Xn = (qn, sn, rn)T is matrix column of the required values, sign T means transposition of matrix;
B„ = (ln — +mj, ln —+nmJ. ln —2, ln —+„mj R„)T
is the free member column.
The solution of system (26) gives values
Xn
A- 1B
A n Bn.
(27)
(28)
We find the approximation parameter An from the relation (23)
An = eqn.
5.5. Results of numerical calculations of eigen frequencies
To calculate the values of the function r 1n = r 1 n(A, 72) at each n interval (vn- 1, vn], n =1, 2, ..., N, we select the values of the first argument as Anm = An0 + AA (m - 1), m =1, 2, ..., M, where An0 is the initial value, and A A is the step of changing the argument A. We select the values of the second argument Y2j = y0 + Ay (j - 1), j = 1, 2, ..., J, where yo is the initial value, and Ay is the step of changing the argument y2. It was accepted for calculations: M = 8, J = 5, An0 = vn + 0.001, AA = v1/(M +1) - 0.01, y0 = 0.5, Ay = 0.5.
The results for calculation of the approximation constants An and rn by formulas (27) and (28) for the first ten intervals (vn-1, vn], n =1,2,..., 10 are shown by points in Figs. 3,4.
Fig. 3. Dependencies yn = An and yn = An on interval number n are shown by points and crosses respectively
Fig. 4. Dependencies yn
and yn
on interval number
n are shown by points and crosses respectively
Calculations showed also that the third approximation constant sn does not depend on the problem parameters —1 and —2. The value sn is equal to -0.5 for any n with relative error not more than 0.1%, which is in full compliance with the property 6 and the relation (19), of item 5.2.
2
= r
n
n
For numerical calculations of eigen frequency with any number n according to formula (21), we use the found numerical values for parameters An and rn at n =1, 2, ..., N to obtain their interpolation-approximation values An and rn [25].
We find dependencies of the parameters An and rn on interval number n in the form:
A„ = ^n(pA) = An + (A1 - An) e_PA (n-1); (29)
Xn = ) = rN + (r1 - rN) e_pr (n-1), (30)
where we distinguish interpolation nodes n = 1, n = N and approximation nodes 2 < n < N - 1. You have introduced parameters pA and pr to approximate dependencies.
We find the parameter pA so that approximate equalities An « An at n = 1, 2,..., N are performed in the best
way.
According to equation (29) we have the exact equality A1 = A1 at n = 1, and we have the approximate equality An « An, where N is large enough.
We find values pn providing the equalities An = ^n(pn) for other values n = 2,..., N - 1:
-h A \ 1 i (a1 - An
Pn = (An) = -7 ln '
n - 1'" V An - An
Using the least squares method, we calculate the optimal value of the parameter pA, which provides the minimum of the value d. This value is average squared displacement of the value pA from all calculated values pn at n = 2, ...,N - 1
1 Nd = d(pA) = E (Pn - PA)2 .
n=2
From the minimum condition of the function d = d(pA) we get the optimal value of the approximation parameter as the average value of all calculated pn
1 N-
PA = P^ (31)
n=2
where averaging is performed without taking into account edge values A1 and AN. Similar reasoning for parameter pr leads to the formula
1 N-1 / x
Pr = ^E . (32)
N - 2 n=2 n - 1 Vrn -
The calculation of parameters pA and pr by formulas (31) and (32) using the data given in Figs. 3,4 gave values pA = 0.48 and pr = 0.42. Having obtained the optimal value of approximation parameters pA and pr we calculate values An and fn for n = 1, 2,..., N by formulas (29) and (30).
The calculation results An and rn at N = 10 are shown by crosses in Figs. 3,4.
Fig. 5 shows the graphs of the exact dependencies of the first two eigen frequencies A1 = A1(y1, y2) and A2 = A2(y1, y2) on the parameter y1 for three fixed values of the parameter y2. These dependencies are represented by solid lines. The exact frequencies values were plotted using the technique shown in Fig. 2(b). The dependencies of approximate values of eigen frequencies A1 = A1(y1, y2) and A2 = A2(y1, y2) on the parameter y1 are shown in Fig. 5 by dashed lines. Approximate frequencies values were calculated using formulas (21), (31) and (32).
5.6. About error of eigen frequencies calculation
We calculate relative error of found eigen frequency value on n interval for estimation of accuracy of approximate formula (21)
An
1 An
An
(33)
We have the previously obtained analytical dependence y+1 = Y+i(A, 72) given by formula (18). Approximation of this dependence r1n = r1n(A, 72) is introduced by formula (20). The function r1n = r1n(A, y2) has the inverse function An = r_n1(Y1, 72).
An _ An
On
Fig. 5. Dependencies of exact eigen frequencies Ai and A2 (solid curves 1,2) and approximate eigen frequencies Ai and A2 (dashed lines 1,2) on parameter 71 at three fixed values of parameter 72: curves (a) - for 72 = 2,0; curves (b) - for 72 = 1,0; curves (c) - for 72 = 0, 5
We get a parametric functional dependence 5n = #n(7i, 72) if we consider the quantity A as a parameter. Indeed, if the value An is set, we have the following values: 7+n(An) = 7+n(An, 72) and An = r-n(7 + , 72) = r-1 (7+n(An, 72), 72). Then, taking into account equality (33), we give the function dn = ^n(7i, 72) parametrically:
5n(An)= 1 - £ r-1 (Ti+n(An, 72), 72)
7i+n(An) = 7+n(An Y2).
(34)
Relative errors of eigen frequency calculations at the first ten intervals were obtained by formulas (34) for case 72 = 1. The table shows estimates An of the largest values of relative errors at the first ten intervals: 5n < An at
A e (vn- 1, Vn].
Table 1. Estimates An of relative error of eigen frequency calculations on intervals with number n
n 1 2 3 4 5 6 7 8 9 10
An 0.01% 1.0% 0.9% 0.5% 0.3% 0.2% 0.1% 0.1% 0.05% 0.01%
It can be seen from the table that relative errors of eigen frequency calculations, starting from the third, monoton-ically decrease.
6. Conclusion
The presence of approximate analytical dependencies for the eigenvalues of the Sturm-Liouville problem on the problem parameters makes it possible to carry out a comprehensive mathematical study of the physical phenomenon from which the Sturm-Liouville problem is derived.
For example, it may be recommended to use the proposed method of solving characteristic equations to solve inverse problems, such as identification of boundary conditions of spectral problems by eigenvalues [26], identification of local defects in mechanical objects (rods, beams, pipelines) [27].
Use of the proposed method in solving inverse problems of oscillations of micro- and nano-objects, in particular MEMS and NEMS, will allow expanding the possibilities of nanometrology.
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