ARTIFICIAL SMALL PARAMETER METHOD — SOLVING MIXED BOUNDARY VALUE PROBLEMS

The phrase quoted from Nayfeh’s book applies also to the case of an artificial small parameter. Note that the introduction of the artificial small parameter is usually motivated either by a lack of a real physical small parameter or by a rather narrow application zone for the natural small parameter [1, 2, 11, 14, 24]. In general, the expression “small parameter” can be used in a different manner. Namely, the following key question occurs: is it possible to obtain a useful information directly through either a natural small parameter or an introduction of an artificial one (or through the application of a useful summation procedure)? This problem has been addressed in references [1, 12, 13, 26, 35]. In this respect it is worthwhile to speak rather directly on the “methods devoted to the development using a parameter” than to speak only on a “small parameter” [26]. From this point of view there is no difference between a real and an artificial small parameter. However, following tradition, the phrase “artificial small parameter” will be used. It is worth noticing that the idea of introducing a small parameter has been used in different branches of mathematics. For example, Dorodnitzyn [18] proposed the method of introduction of the parameter ε into the input equations and the boundary conditions in the way that for ε = 0 a simplified problem is obtained, whereas for ε = 1 the original problem is described. In other words, Dorodnitzyn has applied the continuation method [25, 38, 47, 48] widely known in numerical mathematics. A serious problem occurred due to divergent series occurrence appearing for ε = 1. In order to overcome the occurring difficulties, the so-called methods of analytical continuation have been proposed, but they did not work satisfactorily [33, 34, 40, 41, 42]. Some authors used the artificial parameter approach in a special way. Namely, they observed that a transition from ε = 0 to ε = 1 represents a homotopy transformation yielding today’s accepted term the “homotopy perturbation technique” [19, 20, 22, 23, 24, 28, 29, 30, 31]. However, this mentioned technique can be satisfactorily applied only in connection with an effective method of summation.


Introduction
The phrase quoted from Nayfeh's book applies also to the case of an artificial small parameter.Note that the introduction of the artificial small parameter is usually motivated either by a lack of a real physical small parameter or by a rather narrow application zone for the natural small parameter [1,2,11,14,24].In general, the expression "small parameter" can be used in a different manner.Namely, the following key question occurs: is it possible to obtain a useful information directly through either a natural small parameter or an introduction of an artificial one (or through the application of a useful summation procedure)?This problem has been addressed in references [1,12,13,26,35].
In this respect it is worthwhile to speak rather directly on the "methods devoted to the development using a parameter" than to speak only on a "small parameter" [26].
From this point of view there is no difference between a real and an artificial small parameter.However, following tradition, the phrase "artificial small parameter" will be used.It is worth noticing that the idea of introducing a small parameter has been used in different branches of mathematics.For example, Dorodnitzyn [18] proposed the method of introduction of the parameter ε into the input equations and the boundary conditions in the way that for ε = 0 a simplified problem is obtained, whereas for ε = 1 the original problem is described.In other words, Dorodnitzyn has applied the continuation method [25,38,47,48] widely known in numerical mathematics.A serious problem occurred due to divergent series occurrence appearing for ε = 1.In order to overcome the occurring difficulties, the so-called methods of analytical continuation have been proposed, but they did not work satisfactorily [33,34,40,41,42].
Some authors used the artificial parameter approach in a special way.Namely, they observed that a transition from ε = 0 to ε = 1 represents a homotopy transformation yielding today's accepted term the "homotopy perturbation technique" [19,20,22,23,24,28,29,30,31].However, this mentioned technique can be satisfactorily applied only in connection with an effective method of summation.
This work is devoted to the description of the method presented in [3,4,5,6,7,8,9,10,19,20] with an emphasis on its advantages.The paper is organized as follows.In Section 2 an introductory simple example is analyzed.Vibrations of clamped plate are discussed in detail in Section 3. In Section 4 the asymptotic method is applied to a static problem for a clamped plate.Here also the problem of solving an infinite system of algebraic equations is studied.Buckling of a rectangular plate is analyzed in Section 5.In Section 6 the results which have been obtained are summarized.

Simple example
In our first example a problem which has an exact solution, that is, frequencies of the clamped beam (−0.5l < x < 0.5l), is analyzed.Note that in a static case the Padé approach yields the exact solution directly.The equation being analyzed reads where E is the Young modulus; F, I are area and statics moment of the beam cross-section; ρ is density; ω is frequency; w is normal displacement; X is a spatial coordinate.
The following nondimensional equation governs the vibrations of the beam: where x = X/l, λ 4 = ω 2 ρFl 4 /EI.The boundary conditions have the following form: Let us introduce ε in a way to get the following boundary conditions: (2.4) Note that for ε = 0 simple support is realized, whereas for ε = 1 rigid clamping (2.3) occurs.If 0 < ε < 1 elastic clamping appears.
A general solution to (2.2) reads I. V. Andrianov et al. 327 and after satisfying the boundary conditions (2.4) it yields the following transcendental equations for both symmetric and antisymmetric vibrations with respect to the point x = 0: ) Let us search for an eigenvalue λ in the following series form: Substituting (2.8) into (2.6)(or (2.7)), and applying the classical splitting procedure with respect to the powers of ε, one gets ) Observe that since the series (2.9), (2.10) are divergent for ε = 1, a summation procedure can be applied in order to receive a useful information.In our case a Padé approximation is applied [12,13].
In what follows we are going to briefly describe the Padé approximation [12,13].Let where c 0 = λ 0 , We are now going to compare the eigenvalues of the problem (2.2), (2.3), and (2.4) yielded from the truncated series (2.9) (or (2.10)), and the Padé approximation (2.12) with a known exact value.A comparison is carried out only for the first eigenvalue, since the boundary conditions influence significantly only first eigenvalues.
Consider now the problem where a transition to transcendental equations can be omitted.
Beam vibrations are sought in the form After a substitution of the series (2.13) into (2.2) and into the boundary conditions (2.4), and after a splitting with respect to ε, the following boundary value problems are obtained: ) Padé approximation of the series (2.16) (or (2.17)) gives where c 0 = w 0 , We compare the amplitude of the first harmonics of the clamped beam governed by (2.16) (or (2.17)) with the known exact value w 0 = 1.133, see [50].
Taking account of the three first terms of the series (2.16) for ε = 1 gives w 0 = 1.082 (error 1.77%).

Vibrations of a clamped plate
We are going to analyze the free vibrations of a clamped rectangular plate.The governing equation reads where The input equation can be transformed to the following equivalent form: where The following boundary conditions are applied: where k = a/b.Observe that the parameter ε is introduced to modify the boundary conditions (3.3) in the following way: (3.4) For ε = 0, simply supported boundary conditions are realized, whereas for ε = 1, the contour of plate is clamped.Finally, for 0 < ε < 1, an elastic clamping of the plate edges is realized.
As a result, the following perturbation series is obtained: where , Note that the Padé transformation of the series (3.12) has the form where The numerical simulation [50] yields λ = 5.998 for the squared plate, the series part (3.12) gives λ = 4.854 for ε = 1 (error is 19.08%), whereas formula (3.14) yields λ = 5.742 (the error obtained is of 4.22%).
The plate bending equation reads where q = Qb 4 D −1 and Q denotes normal load.
The following boundary conditions are applied: 3) The perturbed boundary conditions (4.2) read )

.5)
An equation governing the behavior of the bended plate is sought in the form where α m = πm/2, E m are unknown coefficients.The boundary conditions (4.4) yield the following system of linear algebraic equations: where 332 Artificial small parameter method Applying and substituting (4.9) into (4.7),one gets Note that the Padé transformation of the perturbation series (4.9) for E i has the form where A deflection of the plate center computed through formula (4.6) with the application of E i represented by the five terms of the series (4.9) gives 1.797 • 10 −3 q for ε = 1 (error of 42.60%).A deflection computed with the use of E i through the Padé transformation (4.11) yields 1.275 • 10 −3 q (error of 1.20%).A deflection obtained numerically yields 1.260 • 10 −3 q, see [49].
The solution obtained emphasizes an important role of the introduction of an artificial small parameter while solving an infinite system of linear algebraic equations.Observe that the often used classical truncation method does not allow us to obtain a solution structure for complex mixed boundary value problems [36].Although one may expect I. V. Andrianov et al. 333 estimations of the solution coefficient asymptotics, this belongs rather to a separate task (see [36]).A method of introduction of the artificial small parameter allows us to account often for all system coefficients x.It plays an alternative role to the method of reduction and can be applied in order to estimate its accuracy.
It is worth noticing that in a similar application (see [17,32]) the number of operations required to obtain a solution of the system of equations with a finite number of unknowns dramatically decreases.
Finally, note that the artificial method of perturbation is effective also in the case of solution to the system of nonlinear equations [3,4,5].

Stability of a rectangular plate with mixed boundary conditions
Consider the plate shown in Figure 5.1.It is assumed that the boundary conditions in the plate plane secure a homogeneous prebuckling plate state.
The input equation governing plate stability reads where P is a compressed load.The equivalent nondimensional equation reads where x = X/b, y = Y/b, N = Pb 2 /D.The boundary conditions have the form w = 0, w xx = 0 for x = ±0.5k,w = 0, w yy = H(x) w yy ∓ w y for y = ±0.5, ( where We now introduce the parameter ε according to the scheme used earlier, namely, w = 0, w xx = 0 for x = ±0.5k,w = 0, w yy = H(x) w yy ∓ w y for y = ±0.5. (5.4) The critical force N and defection w read (5.5) Substituting (5.5) into (5.2),(5.4), the following series of boundary conditions is obtained: w iy for y = ±0.5. ( A solution to the boundary value problems (5.6), for j = 1,2,3, yields (5.7) Note that the Padé approximation for the series (5.7) has the form where In what follows we are going to estimate an error of the solution which is obtained in a limiting case corresponding to full clamping of the plate sides y = ±0.5.The exact solution gives (for the squared plate) N = 8.6044π 2 [15], for ε = 1 formula (5.8) gives I. V. Andrianov et al. 335 N = 8.7136π 2 (error of 1.27%); the series (5.7) yields N = 4.7757π 2 (error of 44.5%).Numerical solution of the transcendental equation for m = 2 yields N = 7.6913π 2 , see [15]; formula (5.8) for ε = 1 gives N = 7.7156π 2 (error of 0.31%), whereas the series (5.7) gives N = 6.4456π 2 (error of 16.13%).
It is worth noticing that in the places where boundary conditions are changing rapidly one may expect singularities.Since these singularities are known [43,44], they can be introduced into a solution using known methods for matching singular solutions and known asymptotic solutions [39].
The critical force N versus geometrical ratios of the types of mixed boundary conditions µ is reported in Figure 5.1.The results obtained with the help of the described method and the R-function method [46] are represented by curves 1 and 2, respectively.In the given graphs one may distinguish two zones: the first one is for µ ∈ [0.0,0.15],where the plate buckling is associated with an occurrence of two half-waves in direction x; the second one is for µ ∈ [0.15,0.5],where the plate buckling loss is characterized through an occurrence of one half-wave in x-direction.Therefore, for µ = 0.15 one may expect buckling through either the first or second buckling form.
The described method allows for an investigation of the influence of clamping stiffness ε on the buckling force N. The dependence of the buckling force N versus the parameter ε is shown in Figure 5.2 for some values of µ.It should be emphasized that in the case of an elastic clamping of the edges y = ±0.5 with no mixed boundary conditions, the additional equilibrium forms appear for ε = 0.96.Buckling with an occurrence of one (two) halfwave in the x-direction may appear for ε < 0.96 (ε > 0.96).A simultaneous occurrence of the first and second equilibrium forms, depending on the parameter ε, takes place for mixed boundary conditions.Furthermore, the additional equilibrium forms for µ → 0 appear for ε → 1.0.A threshold value of µ = 0.25 corresponds to plate stability loss one buckled half-wave in x-direction for ε = 1.
For the plate shown in Figure 5.3, the solutions (5.7), (5.8) still hold.However, in the latter case the coefficients γ im should be sought in the form isin(πγi)cos(πγm) − msin(πγm)cos(πγi) for i = m. (5.9) The dependence of N versus µ is reported in Figure 5.3, whereas the dependence N(ε) is shown in Figure 5.4 for the same values of µ.Curve 1 corresponds to formulas (5.7), (5.8); curve 2 corresponds to the R-function method [46]; curve 3 (2.5)represents the results reported in [27] (see [21]), and the bullets represent the experimental data shown in [21].To conclude, a good coincidence is achieved for both computational and experimental data for the whole variation of the parameter µ.

Concluding remarks
The proposed computational method possesses advantages in comparison with the known methods devoted to solving the problems associated with mixed boundary conditions, that is, the methods of Bubnov-Galerkin et al. (see [36]).Namely, it does not require a priori knowledge of the shapes of deformed surfaces.Furthermore, the proposed approach does not lead also to a high-order system of transcendental equations, as in the case of dynamical edge effect method [16].The proposed asymptotic method allows for solution representation in an analytic form, which is important when applying any optimal design for solution of direct problems.Applying FEM or BEM for initial ε and µ, one may also solve a boundary value problem.
It should be emphasized, however, that FEM method is universal with respect to a space fulfilled by a plate.It is rather difficult to apply an asymptotic method to complex form spaces, since they require knowledge of the analytical solution of zero-order approximation.Besides, applying an asymptotic method does not provide an easier way to introduce higher accuracy, since the construction of higher approximations is rather difficult.However, one may require a solution obtained through two methods in order to control the reliability of the obtained approximate solution.In the case of complex plate forms, the results obtained through the asymptotic method can serve either as the initial values for FEM or as tests for FEM, if a transition from a complex to simple geometry is possible.
The proposed approach can also be applied to solve 3D problems of elasticity, hydromechanics, and so forth.It allows for a significant extension of the classical method of variables separation and the Fourier method.
One may also match the asymptotic method with the Bubnov-Galerkin-type methods.Indeed, after obtaining an infinite set of linear algebraic equations [36], the method of artificial small parameter can be applied easily.
Finally, in the context of the approach which has been introduced and described, the domain decomposition method can be modified [45].

8 N/π 2 Figure 5 . 1 .
Figure 5.1.Comparison of the stability estimation results of the plate with mixed boundary conditions using conditions either from this paper or from those based on R-function application.

8 N/π 2 Figure 5 . 2 .
Figure 5.2.Investigation of the influence of clamping space dimension on buckling load of the plate.

I 8 N/π 2 Figure 5 . 3 . 8 N/π 2 Figure 5 . 4 .
Figure 5.3.Comparison of the computational results of plate stability estimation with mixed boundary conditions obtained through our proposed method, R-function approach, and experimental results.