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  • smad pathway where is some parameter eigenvalue A B C


    where β is some parameter (eigenvalue); A, B, C and D are the unknown coefficients. The general solution of the problem is expressed by the sum
    The coefficients A, B, C and D in formula (16) are determined from the boundary conditions at the longitudinal edges: y=0 and y=1. Satisfying the conditions for =0, we obtain the =0, and then solution (16) can be written in the following form:
    Let us write the expressions for the bending moment and the lateral force at the edge y=1:
    Next, if we impose the requirement that these values must become zero for any values of x, we shall obtain a system of two equations for determining the coefficients A and D:
    For the system to have a non-trivial solution, its determinant must become zero; this leads to a transcendental equation with respect to the parameter β: where
    The transcendental Eq. (21) was obtained through similar arguments in one form or another by a number of authors [9–11]. A detailed investigation of this equation was carried out in Ref. [11] where it was proved that it had an infinite number of complex roots. Moreover, it is easy to see that if is a root of Eq. (21), then any combination of the form will also be a root. Substituting (23) into Eq. (21) and separating the real and the imaginary components, we arrive at the two equations relating and :
    The smad pathway of this system are such [11] that the positive numbers are within the following limits: When searching for the numbers and , as a first approximation we can accept that
    The imaginary part is found from the transcendental equation
    The following approximations are found by Newton\'s method: where
    Kitover [11] gives the values of the first five quarters of complex roots for ν=0.25, but in this case, Eq. (21) also has two real roots which differ only in their signs. The real roots were also omitted in Ref. [9] for ν=0.3. Substituting (23) into Eq. (28) and separating the real and the imaginary components, we obtain:
    The following notations have been introduced here:
    The first nine complex roots of transcendental Eq. (21) that we computer-calculated for ν=0.3 by formulae (30) are listed in Table 1. Returning to system (20), we obtain from its first equation:
    Let us note that the second equation of system (20) can also be used to express D through A, but this form of solution will ultimately produce the same result as the first one, as they are related by Eq. (21). Now homogenous solutions (18) will be written as an infinite series where the eigenfunctions have the form and, in turn,
    Within solution (33) which satisfies the biharmonic equation and the boundary conditions at the longitudinal edges y=0 and y=1, the coefficients must be determined from boundary conditions (12) and (13) on the transverse edges ; these boundary conditions, taking into account formula (15), will take the form where the concentrated force at the angular point ;
    Thus, to find the coefficients (or ) of series (33), the function g(y) (see Eq. (15)) should be expanded into a series in terms of non-orthogonal functions of the form (36); additionally, conditions (37) and (38) should be satisfied. Since it does not seem possible to strictly meet the above requirements, let us turn to the approximate method based on minimizing a certain functional. Ustinov and Yudovich [12,13] have proved that the elementary homogenous solutions of the biharmonic equation form a complete system of functions (for example, see Ref. [14] on the completeness of a system of functions). In particular, from this follows that there does exist a unique solution for the infinite system obtained by minimizing the functional which is a positive definite function and is represented as a sum of the quadratic (Φ2) and the linear (Φ1) forms of the coefficients of the homogenous solutions and the constant term. From this also follows the convergence of the reduction method for determining and the convergence of the approximate solutions at k → ∞.