Numerical Analysis of the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Conditions

Received:2009-01-23 Published online:2009-03-10 Abstract. In this paper the eigenvalue problem for one-dimensional di fferential operator with nonlocal integral conditions is investigated numeric ally. The special cases of general problem are analyzed and hypothesis about the dependence of th spectral structure of that problem on the coefficient of differential operator and the parameters of nonlocal conditions are formulated.


Introduction. Statement of the problem
We consider the eigenvalue problem for one-dimensional differential operator with given nonlocal integral boundary conditions, where p(x) > p 0 ≥ 0, γ 0 , γ 1 ∈ R, and the corresponding eigenvalue problem for difference operator.Such values of λ that problem (1)-( 3) has the non-trivial solution u(x) ≡ 0 are called eigenvalues and set of all eigenvalues is called spectrum of the problem.
The analysis of the spectrum of a difference operator with a nonlocal conditions permits us to analyze the stability of difference schemes [1,2], justify the convergence of iterative methods for finite-difference equations [3,4] and is also of interest in itself.
When p(x) ≡ 1, γ 0 = γ 1 = 0, we get the well-known problem with classical boundary conditions (see, e.g., [5]).The eigenvalues of such problem are real, positive and simple: When, for example, γ 1 = 0 and instead of classical boundary condition at the point x = 1 we have nonlocal Bitsadze-Samarskii type boundary condition the spectral structure of the problem is more complicated: subject to parameters γ 1 and ξ, the eigenvalues can be both real (positive or non-positive) and complex numbers (see [6]).
In papers [7,8] the eigenvalue problems for one-dimensional differential operator with p(x) ≡ 1 and various nonlocal integral conditions are investigated analytically.However, such problems with a non-constant coefficient p(x) are met in the literature quite rarely and are considerably less investigated.
The main aim of the present paper is to investigate special cases of problem ( 1)-( 3) numerically and to formulate the hypothesis about the dependence of qualitative structure of the spectrum of the problem on coefficient p(x) and parameters γ 0 , γ 1 .
The paper is organized as follows.In Section 2 we describe the finite-difference scheme and formulate equivalent eigenvalue problem for the transition matrix.Technical details and results of numerical analysis of the special cases of the problem are presented and discussed in Section 3. Some remarks in Section 4 conclude the paper.
Let us define the uniform discrete grid on the interval (0, 1): We approximate the differential problem (1)-( 3) by the following finite-difference scheme: where ).We rewrite integral conditions ( 5) and ( 6) in the form of a system of two linear algebraic equations with unknowns U 0 and U N : This system has a unique solution if its determinant is not equal to zero, i.e., then We can write the solution of system (7) in the form Now we define the square matrix of order (N − 1) , where One can see, that the difference eigenvalue problem ( 4)-( 6) is equivalent to the eigenvalue problem for the matrix A.
3 Numerical results and discussion

Technical details of numerical analysis
In order to analyze the qualitative structure of the spectrum of the differential eigenvalue problem (1)-( 3) numerically, the eigenvalue problem (9) for the matrix A was solved.For numerical analysis MATLAB (The MathWorks, Inc.) software package [9] was used.Numerical experiments were executed with different types of coefficient p(x) and various values of parameters γ 0 , γ 1 .We executed series of numerical solution procedures for problem (9) with concrete coefficients p(x) and points (γ 0 , γ 1 ) from discrete set selected in the rectangle with restrictions In order to ensure the validity of inequality (8), in all numerical experiments we used Note that opportunities of symbolic calculations give us quite efficient way to avoid some undesirable effects of numerical calculations when observing the behaviour of the function where E is the identity matrix.Now let us analyze a couple of cases of the coefficient p(x).
These results coincide with analytical results in case p(x) ≡ 1 (see [7]).Note that when λ = 0, the general solution of equation ( 1) in Case 1 is and the condition γ 0 + γ 1 = 2 for λ = 0 being an eigenvalue of the differential problem can be easily derived analytically.Indeed, by substituting solution (10) into boundary conditions ( 2) and (3), we get the system of two linear algebraic equations with unknowns c 1 and c 2 : Thus, this system has a unique solution if its determinant is equal to zero, i.e. γ 1 +γ 2 = 2.
When λ = 0, the general solution of equation ( 1) in Case 2 is As marked in all graphs of Fig. 1, in this case the problem has eigenvalue λ = 0 when parameters γ 0 and γ 1 belongs to the straight line, which is denoted as dashed line.Using the same technique as in Case 1, it is easy to make sure, that for every α, −2 < α < 2, α = 0, the equation of the certain straight line is When γ 0 and γ 1 are located somewhere above the straight line in the northeast direction from the origin of coordinates, the considered problem has exactly one negative eigenvalue.
The parameters γ 0 and γ 1 when exists non-negative multiple eigenvalue are located on the hyperbola-like curves.As we can see from Fig. 1, the directions of branches of hyperbolas are different in cases of decreasing and increasing functions p(x).Each hyperbola divide the coordinate plane (γ 0 , γ 1 ) into three unbounded regions.All the eigenvalues are real, when point (γ 0 , γ 1 ) belongs to the region between two branches of the hyperbola.However, when (γ 0 , γ 1 ) belongs to one of two other regions, the considered problem has complex eigenvalue.
The structure of the spectrum is rather complicated when the parameters γ 0 and γ 1 are located near one of the half-branches of the hyperbola (see quadrants in the southeast and the northwest directions from the origin of coordinates in graphs Fig. 1(a), 1(b), 1(c) and Fig. 1(d), 1(e), 1(f), respectively).When (γ 0 , γ 1 ) belongs to the region between the dashed straight line and the ray which start point lies in that straight line and which is denoted as dash-dot line with circles in the graphs of Fig. 1, there exist two negative eigenvalues.Moreover, when γ 0 and γ 1 are located somewhere between two rays which are denoted as dotted line with crosses and dash-dot line with circles, the considered problem has conjugate complex eigenvalues with negative real parts.
The considered problem has complex eigenvalues with real parts equal to zero, two negative eigenvalues, non-negative multiple eigenvalue or eigenvalue equal to zero when parameters γ0 and γ1 are located on dotted curves with crosses ( and dashed curves (-----), respectively.Now let us generalize the results of numerical analysis of the problem in Case 2: Subject to nonlocal boundary conditions parameters γ 0 and γ 1 , both real and complex numbers can be eigenvalues of the problem in Case 2. The problem has multiple nonnegative eigenvalues when γ 0 and γ 1 belongs to the hyperbola-like curves.Parameters γ 0 and γ 1 , when λ = 0 is an eigenvalue of the problem, are located on the straight line (see equation (11)).In addition, there exist such values of γ 0 and γ 1 that considered problem has one or two negative eigenvalues or complex eigenvalues with non-positive real parts.

Conclusions
The results of numerical analysis allow us to make the following conclusions: • We can investigate the general problem (9) in particular cases and formulate hypothesis about the qualitative structure of the spectrum of this problem when using the technique of numerical analysis.
• Despite the fact that in particular cases the qualitative structure of the spectrum is simple, in general case this structure can be rather complicated as well as the qualitative behaviour of eigenvalues: depending on the parameters γ 0 and γ 1 both real numbers (positive or non-positive) and complex numbers (with positive or nonpositive real parts) can be eigenvalues of the problem.