Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations

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Introduction
When investigating the nonlinear equations of the form   The theory of branching solutions of nonlinear equations arose in close connection with applied problems and development of its ever-regulated by the new applied problems.
Some of these problems is reflected in the monographs [3,4,20], as well as in several articles (see, eg, [5,6] and references therein) Application of the cited above approach to the nonlinear integral operator arising at synthesis of the antenna systems according to the given amplitude directivity pattern, brings to the nonlinear two-parameter eigenvalue problem (,) Tf f

 
with an integral operator (,)   T   analytically depending on two spectral parameters  and  .
The essential difference of the two-parameter problems from the one-parameter ones is that the two-parameter problem can not have at all the solutions or, on the contrary, to have them as a continuum set, which in the case of real parameters are the curves of eigenvalues.Such problems are still not investigated because there are still many open questions connected with this problem such as, for example, the existence of solutions and their number, and also the development of numerical methods of solving such spectral problems for algebraic, differential and integral equations.
In the given work an algorithm of finding the branching lines of the integral equation arising in the variational statement of the synthesis problem of antenna array according to the given amplitude directivity pattern as, for example, in [2] is proposed.

Preliminary. Nonlinear synthesis problem
We consider the radiating system, which consists of identical and identically oriented radiators of the same for all radiators directivity pattern (DP), in which the phase centers are located on the plane XOY (grid plane) of Cartesian coordinate system.We believe that the coordinates of radiators ( , ) nm xy form a rectangular equidistant grid, focused on the axes and symmetric with respect to these axes.Then the function that describes the DP (plane array factor) of equidistant plane system of radiators (plane array) has the form [2]. where nm I are the complex currents on the radiators, ,   are the angular coordinates of a spherical coordinate system (,,) R   whose center coincides with the center of the Cartesian coordinate system XOY , 2 () M n is the integer function that sets the number of elements 22 () 2 () 1 Nn Mn  in the n  th row of the array.Thus, the number of elements N in this array is equal to and is described by the function that is continuous and nonnegative in  and is equal to zero outside.
We must find such currents nm I on radiators that created by them directivity pattern will approach by the amplitude to the given directivity pattern 12 (,)  in the best way.To this end, we consider the variational statement of the problem as, for example, in [2] or [18].

Variational statement of the synthesis problem
on the space () m i n , , which characterizes the magnitude of mean-square deviation of modules of the given directivity pattern and the synthesized one in the region  .
From the necessary condition of the functional () I  minimum, we obtain a nonlinear system of equations for the optimum currents on radiators and equation ( 4) for optimal currents takes the form Equivalence of equations ( 7) and (8) means that between the solutions of these equations one-to-one correspondence exists, i.e., to each solution of equation ( 7) corresponds the solution of equation ( 8) and vice versa.This means that if a solution of equation (8), the corresponding to it solution of equation ( 7) is determined by the formula f   is a solution of equation (7), the corresponding to it solution of equation ( 8) is determined by the relation

(,)
. ( 2) Since equations ( 7) and ( 8) are nonlinear equations (Hammerstein type), they may have nonunique solutions, the number and properties of which depend on the number of elements in the antenna array and their placement, and also on the properties of the given amplitude directivity pattern 12 (,) It is easy to see that one of possible solutions of equation (7) .  In Fig. 1 shows the trivial solution, which creates a symmetrical inphase current distribution on the radiators of array (Fig. 2).The amplitude of the synthesized directivity pattern, which branches off from 01212 (, ,,) f cc


, is shown in Fig. 3, and the optimal current on the radiators that it creates, is asymmetric and is shifted to the first quadrant relatively of the center of array (Fig. 4)., which is shown in Fig. 5, numerical examples of the trivial and the branching solutions are shown in Fig. 6 -Fig.9. From these figures we see that the branching solution (the amplitude directivity pattern of which is shown in Fig. 8, and the optimal distribution of the current on the radiators that it generates is shown in Fig. 9) more accurately than the trivial solution (11) (amplitude directivity pattern is shown in Fig. 6, which is created by symmetric inphase current (Fig. 7)) approximates the given directivity pattern not only in the mean square approximation (in terms of the values of functional (3) -0.050109 in comparison with 0.185960) to about 73%, but also with respect to the form.
Thus, in most cases from the practical point of view the nontrivial solution, that branches off from 01212 (, ,,) f cc


with growth of parameters 1 c and 2 c is interesting.(, ,,) obtained by linearization of equation ( 7), has solutions distinct from identical zero [20].Thus, we have obtained the nonlinear (with respect to parameters 1 c and 2 c ) two- , .cc   (13) It is easy to be convinced, that at arbitrary finite values 1 0 c  , 2 0 c  , the function 01212 (, ,,)


is the eigenfunction of equation (12).From this it follows, that the operator The study of such equations is carried out in [2,18], and it is possible to apply , for example, the algorithms of the work [11,13,15] to solution of such equations.
In the given work the numerical algorithms to solve more complicated problem when the variables are not separated, are proposed.

Basic equations
First we will show that the kernel 121212 (, ,, ,,) Kc c


will not be the eigenfunction of equation ( 16) anymore.That is, we have eliminated a continuum set of eigenvalues from a spectrum of the operator (16), which coincides with the first quadrant of the plane 2  R that corresponds to the function 01212 ( with symmetric matrix ( , ) Thus, the problem of finding lines the branching of solutions of equation ( 7) is reduced to finding the eigenvalues curves of nonlinear two-parameter spectral problem (19).
Obviously, in order the problem (19) to have a nonzero solution it is necessary that (,) d e t (,) 0 , i.e. the eigenvalues of problem ( 19) are zeros of function (,)    .

Algorithm of finding the eigenvalue curves
The main calculational part of algorithm proposed is the implementation method proposed in [14,15] to compute all eigenvalues of the nonlinear matrix spectral problem belonging to some given range of the spectral parameter  at the given value of parameter  .

In the problem (21)
n n u   , and ( , ) n   T is the real () nn  matrix whose elements depend nonlinearly on the parameters  and  .In order to detail how the method [15] is applied to the problem under consideration in this paper, we present the necessary results from [15].
Thus, we replace in the problem (21), for example, the parameter  by the expression   

The argument principle of meromorphic function
In the basis of algorithm of finding number of zeros and their approximations, which are in some areas G , is the statement that follows from the argument principle of meromorphic functions.
Statement.Let the meromorphic function ()  f  have in the region G m zeros 12 , ,, m    (with regard for their multiplicity) and no zeros on the boundary  of region G , then the number m is determined in accordance with the principle of the argument 0 () 1 2( ) and relations are true, where  The found eigenvalues can be refined, using them as initial approximations for Newton's method 1 () , 0,1,2, () or for one of bilateral analogies of Newton's method [15] The argument principle (23) and formula of the argument principle type (24), (25) were repeatedly applied in solving various spectral problems (see, for example, [1, 7, 9] and references therein), but the peculiarity of the proposed algorithm is to compute the values of function () f  and its derivatives basing on LU-decomposition of the matrix ( ) n  T .

Numerical procedure of calculating the derivatives (the first and the second) for the matrix determinant
.


If some of the principal minors of the matrix of order 1 jn   are zero, then the decomposition (32) may not exist or, if it exists, it is ambiguous.
In practice, the best way to establish the possibility of LU-decomposition is to try to calculate it.It may happen that 0 rr u  ( r is the order of the main minor of the matrix, which is zero).To avoid this, in the process of decomposition one may use a series of permutations of rows (and/or columns) of matrix D with a choice of principal element.In this case the decomposition (37) can be written as

PC = NU + 2MV + LW
where P is a permutation matrix, moreover det ( 1) q   P , where q i s a n u m b e r o f permutations (for example, rows ).In this case the relations (38) take the form 1 ()( 1 ) , .

  
Since in the relations (38) the value of function and its derivative is calculated only on the boundary region G , i.e. at the given points , 1, ... , j jN

 
, then for their calculation we use the same numerical procedure of decomposition of matrices (37


and their approximate values we find by solving the system of equations ( 24), after calculating the right part of the formula (42).
Step 4. Using the decomposition (37) for real values  , we refine all eigenvalues that fall in the area G , using the Newton method or bilateral analogue of Newton's method (44).As initial approximation we take the approximate values obtained in Step 3.
Step 6.If necessary, we correct the area G by changing it center and / or radius and go to Step 2, otherwise go to Step 7.
Step 7. The end.
Application of modification of algorithm for linear two-parameter problems was considered in [16].

Algorithm of finding the bifurcation points of eigenvalue curves
Note that if two eigenvalue curves intersect at some point, this point is called the point of bifurcation (or branch the point).Sufficient criterion for the existence of such points have been known long ago (see, for example, [8]) and consists in that the point ( , are satisfied, and the second order partial derivatives are different from zero.But this criterion was not often used in practical calculations, because it requires calculation of derivatives of the determinant of matrix. Using the algorithm of computing derivatives of the determinant of the matrix proposed in section 2 this criterion can be effectively used to calculate the bifurcation points of equation (45).
Thus, the problem consists in determining such parameters  and  which are the solutions of two nonlinear algebraic equations (,) det ( , ) Note that, with some approximation to solution of (46), for its solution the iterative process of Newton's method can be applied as in [12] 1 1 1 (, ) (, ) , (, ) Further we assume that the determinant of matrix of the second derivatives (48) whose elements are calculated at point different ( , ) mm   from zero.
Thus, at each step iterative process to compute the function (,) d e t (,) f     Τ and its partial derivatives (first and second) only for fixed values of the parameters  and  .This can be realized in a numerical procedure using the LU-decomposition of matrix (,)   T , namely: , , , ,

  
Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 303 , , ii ii vu ii ii wv ii ii wv ii ii wv ii ii wv which are obtained from decomposition (,) (,)(,) .
From this it follows that to calculate ( , ) ,  Thus, the algorithm of finding of the bifurcation points of eigenvalue curves of twoparametric spectral problem consists of the following steps.

Algorithm 2.
Step 1. Step 6. Compute the next approximation to  and  by the formula (47) Step 7. end for m Step 8. if (, )   mm f f    then go to Step 10.
Step 9. else Initialize a different initial approximation to the bifurcation point and go to Step 3.
Step 10.The end

Analysis of numerical results
In conducting a series of numerical experiments on the synthesis of antenna arrays, Algorithms 1 and 2 were used to find the curves of eigenvalues for two-parameter eigenvalue problem, which are the branching lines of solutions of nonlinear synthesis equation (7) and their bifurcation points.Numerical calculations were carried out as for the problems in which in the function  f   are separated, the results obtained by different approaches (reduction of one-parameter problems to the transcendental equations and solving them [2], by methods of descent [18] and also by bilateral methods proposed in [11,13,15]) are the same.
Note that the bifurcation points (at least their rough estimates) may be obtained graphically from Fig. 10 -Fig.13, and may be clarified by the Algorithm 2.

Concluding remarks
Numerical experiments with the calculation of eigenvalues and eigenvectors, realized for certain specified types of directivity pattern by the proposed algorithms, and comparison them with existing results obtained by other methods shows their efficiency (in terms of bilateral approximations and convergence rate).Developed and implemented algorithms for numerical finding of the branching lines of nonlinear integral equations, the kernels of which nonlinearly depend on two spectral parameters and their bifurcation points, yielded the new results, namely:


We have found all valid solutions (the curves of eigenvalues) of the problem (19), which fall in the interval of modified parameters  and  , that we are interested in.


For the problems in which the function Since the spectral parameters are the geometric and electromagnetic characteristics of radiating systems, the solution of this problem makes it possible to obtain the necessary information at the design stage, choosing the optimal ones with respect to the size and electrodynamic characteristics of the radiating system.
Note that such two-dimensional problem was studied also in the works [10,17,19], but there numerical results obtained for some directivity patterns 12 (,) F   are not reliable.
To complete we shall mark, that the offered algorithm of calculation of derivatives of matrix determinant can be used and in the approach in which basis the implicit function theorem is.In such approach it is necessary to solve the Cauchy problem for which the right part of equation ( 51) can be calculated by the algorithm of calculation of derivatives of matrix determinant.Besides by the algorithm, given in this paper, it is possible numerically to define a number of eigenvalues, and, therefore, the eigenvalue curves, which are in the given range of spectral parameters and to calculate the initial value for Cauchy problem for each curve.


nonlinearly depends both on the parameter  and the function f , the formalistic approach, which is based on linearization, is applied.The application of this approach shows, that the branching points of equation can be only those values of parameter  , for which unit ( 1   ) is the eigenvalue of the corresponding linearized equation (see, eg,[20])() A ff  with the operator-valued function :( )A CX H  ( () XH is a set of linear operators, C  is the spectral parameter), nonlinearly depending on the parameter  .If the linearized equation linearly depends on the parameter  , i.e.Aff   , then its eigenvalues will be the branching points of initial equation.In a general case the curves of eigenvalues ()   appears and then the branching points will be those values of parameter  of the problem

Figure 6 .Figure 7 .Figure 8 .Figure 9 .
Figure 6.Amplitude directivity pattern of trivial solution 01212 (, ,,) fc c  Tc c has a spectrum, which coincides with the first quadrant of the plane 2 R .The problem consists in finding such range of real parameters 11 c   and 22 c   of the problem (13), for which there appear the solutions different from 01212 (, ,,) f cc  .It should be noted that in a special case, when it is possible to separate variables in the

 and consider the appropriate
fixed values  and  .Then, obviously, the eigenvalues of problem (() nn  matrix whose elements depend nonlinearly on the parameter  .One should determine how many zeros of the function ()f  , and, therefore, the eigenvalues of the problem are in some given range of change of parameter , calculate each of them.

Thus
the system (24) we can find the zeros of functions () f  that are in the region G .By putting the interval , , and applying the above statement to the meromorphic function ( ) det ( ) find all the eigenvalues of problem (22), belonging to the given region G , i.e. to the given interval , tt cd   [] .The integrals in (23) and (25) we can replace by some approximate quadrature formulas, such as rectangles at N points on  , and since  is a circle, then to calculate quantities , 0

F
  that describes the given directivity pattern of the array, the variables are separated and are not separated.In Fig. 10 -Fig.13 are shown the curves of eigenvalues for four problems, in which the given directivity patterns are defined by the formulas 12 (,)1

F 12 (
  admits separation of variables, another solution to the problem(19) has been found (for example, for what is shown in Fig.10and Fig.11, respectively), which corresponds to the synthesized directivity patterns in which the variables are not separated.
[18], the synthesis problem we formulate as a problem of minimizing the functional[18] Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 285 nm InM M mM M   is , ,,) cc Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 295   Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 297 Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 299 Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 301 , and also the number of points of partition N of the boundary  of the region G , i.e. the circle.Step 2. Determine the value of parameter kk kk uv are the elements of matrices U, V in decomposition (37) at fixed j    .N o w , i f w e k n o w s o m e a p p r o x i m a t i o n to the eigenvalue, then the correction () / () cd   [] , where we find the eigenvalues of problem (21).This can be a single-spaced interval or a sequence of intervals , , Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 305 , To set the accuracy of calculations: with respect of the parametersp

Table 1 .
Numerical Algorithms of Finding the Branching Lines and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 309 Bifurcation points for different types of the given directivity pattern