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Burkhan T. Kalimbetov, Marat A. Temirbekov, Zhanibek O. Khabibullayev, "Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator", Abstract and Applied Analysis, vol. 2012, Article ID 120192, 16 pages, 2012. https://doi.org/10.1155/2012/120192
Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator
The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.
The investigation of singular perturbed systems for ordinary and partial differential equations occurring in systems with slow and fast variables, chemical kinetics, the mathematical theory of boundary layer, control with application of geoinformational technologies, quantum mechanics, and plasma physics (the Samarsky-Ionkin problem) has been studied by many researchers (see, e.g., [1–19]).
In this work, the algorithm for construction of an asymptotical solution for linear singular perturbed systems with the irreversible limit operator is given—the regularization method . The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to the study of systems of first-order partial differential equations with incomplete (more exactly, point) initial data.
In this paper, we consider linear singular perturbed systems in the form where is a matrix of order is a known function, is a constant vector, and is a small parameter, in the case of violation of stability of a spectrum of the limiting operator .
Difference of such type problems from similar problems with a stable spectrum (i.e., in the case of ) is that the limiting system at violation of stability of the spectrum can have either no solutions or uncountable set of them. In the last case, presence of discontinuous on the segment solutions of the limiting system is not excluded. Under conditions, one can prove (see, e.g., [1, 6]) that the exact solution of problem (1.1) tends (at ) to a smooth solution of the limiting system. However, there is a problematic problem about construction of an asymptotic solution of problem (1.1). When the spectrum is instable, essentially special singularities are arising in the solution of system (1.1). These singularities are not selected by the spectrum of the limiting operator . As it was shown in [3–7], they were induced by instability points of the spectrum.
In the present work, the algorithm of regularization method  is generalized on singular perturbed systems of the form (1.1), the limiting operator of which has some instable points of the spectrum. In order to construct the spectrum, we use the new algorithm requiring more constructive theory of solvability of iterative problems. These problems arose in application of the algorithm.
We will consider the problem (1.1) at the following conditions. Assume that(i); for any , the spectrum of the operator satisfies the conditions:(ii) (here - are even natural numbers),(iii),(iv),(v).
2. Regularization of the Problem
We introduce basic regularized variables by the spectrum of the limiting operator Instable points of the spectrum induce additional regularized variables described by the formulas We consider a vector function instead of the solution to be found for problem (1.1). This vector function is such that
For , it is natural to set the following problem: We determine the solution of problem (2.4) in the form of a series with coefficients .
3. Resolvability of Iterative Problems
We solve each of the iterative problems in the following space of functions: where are eigenvectors of the operator corresponding eigenvalues . We represent in the form of where It is easy to note that each of the systems () can be written in the form where are the corresponding right hand side. Using representations of space , we can write system (3.3) in the equivalent form where .
We have the following result.
Theorem 3.1. Let and satisfy conditions (i)–(iv). Then, system (3.4) is solvable in the if and only if where are basic elements of the kernel of the operator
Proof. Let .
Determine solutions of system (3.4) in the form Substituting (3.8) in (3.4) and equating separately coefficients at and , we obtain the equations
One can see from this that obtained equations are solvable if and only if and these conditions coincide with conditions (3.6). Theorem 3.1 is proved.
Remark 3.2. Equations (1.1) imply that under conditions (3.6), system (3.4) has a solution in representable in the form
where are arbitrary functions.
Consider now system (3.5). As in points , , this system does not always have a solution in . Introduce the space consisting of vector functions having the properties where are eigenvectors of the operator with regard to eigenvalues . Let , that is, Determine a solution of system (3.5) in the Substituting this function in (3.5), we obtain Since is a basis in , we get It is easy to see that (3.18) has the unique solution By virtue of conditions (3.14), the function can be represented in the form where is the certain scalar function, , and we see that where are arbitrary constants, . However, the solution of system (3.5) should belong to the space , and it means that . Therefore, constants in (3.21) and functions are determined uniquely in the form Thus, under conditions (3.14), system (3.5) has the solution in of where (in points , this equality is understood in the limiting sense). We summarize received outcome in the form of the following assertion.
Theorem 3.3. Let the operator satisfy condition (i), and let its spectrum satisfy conditions (ii)–(iv). Then, for any vector function , system (3.5) has the unique solution in space .
For uniquely determination of functions , consider system (3.4) with additional conditions: where is a constant vector.
We have the following result.
Theorem 3.4. Let conditions of Theorem 3.1 hold. Then, the system (3.4) with additional conditions (3.24)-(3.25) has solutions of the form (3.11) in which all summands are uniquely determinate except for . Functions in the last summand are determined by the formula where are known functions, and arbitrary constants.
Proof. Denote in (3.11) that
Using (3.11) and condition (3.24), we obtain the equality
Multiplying this equality scalarly by , we get
By (3.11) and conditions (3.25), we have
Considering these equations with initial conditions (3.30), we can uniquely obtain functions .
Now, using (3.11) and conditions (3.26), we get This implies that have the form (3.27) where Theorem 3.4 is proved.
Remark 3.5. If conditions (3.6) hold for and , then system (3.3) has a solution in the space , representable in the form of where is a function in the form of (3.11), and is a function in the form of (3.23); moreover, functions are found uniquely in (3.11), and functions are determined up to arbitrary constants in the form of (3.27).
Let us give the following result.
Theorem 3.6. Let and conditions (i)–(iv), (3.6), (3.24)–(3.26) hold. Then, there exist unique numbers involved in (3.27), such that the function (3.34) satisfies the condition where is a fixed vector function.
Proof. To determine functions uniquely, calculate
Denote by the known function
and write the conditions (3.13) for Taking into account expression (3.27) for , we get
Using the Leibnitz formula, we obtain that
Therefore, previous equalities are written in the form of where
We obtain from here sequentially the numbers Theorem 3.6 is proved.
So, if and conditions (3.6), (3.24)–(3.26), and (3.35) are valid, then the systems (3.4), (3.5) (and (3.3) together with them) are solvable uniquely in the class Two sequential problems and () are connected uniquely by conditions (3.23)–(3.25), (3.30); therefore, by virtue of Theorems 3.1–3.6, they are solvable uniquely in the space .
4. Asymptotical Character of Formal Solutions
Let be solutions of formal problems (), in the space respectively. Compose the partial sum for series (2.4): and take its restriction .
We have the following result.
Theorem 4.1. Let conditions (i)–(v) hold. Then, for sufficiently small , the estimates hold. Here, is the exact solution of problem (1.1), and is the states above restriction of the th partial sum of series (2.4).
Proof. The restriction of series (2.4) satisfies the initial condition and system (1.1) up to terms containing , that is, where is a known function satisfying the estimate Under conditions of Theorem 4.1 on the spectrum of the operator for the fundamental matrix of the system , the estimate is valid. Here, is sufficiently small. Now, write the equation for the remainder term We obtain from this equation that whence we get the estimate where . So, we obtain the estimate Taking instead of the partial sum we get which implies the estimates (4.2). Theorem 4.1 is proved.
Let it be required to construct the asymptotical solution for the Cauchy problem where is a small parameter. Eigenvalues of the matrix are Eigenvectors of matrices and , are, respectively, We get . Therefore,
Hence, we can apply to problem (5.1) the above developed algorithm of the regularization method.
At first, obtain the basic Lagrange-Silvestre polynomials Since there will be two such polynomials: and .
Take the arbitrary numbers and and set the interpolation conditions for the polynomial , Expand onto partial fractions from where Use the interpolation polynomial (5.4). We get . Hence, (5.6) takes the form Since numbers and are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,
Introduce the regularizing variables
Construct the extended problem corresponding to problem (5.1): where .
Determining solutions of problem (5.10) in the form of a series we obtain the following iteration problems:
Directly calculating, we obtain the solution of system (5.12) in the form of where are for now arbitrary functions.
For solvability of problem (5.13) in the space , it is necessary and sufficient to fulfill the conditions
Using solution (5.15) and the initial condition we obtain the equation
Multiplying it (scalar) on and , we obtain the values
Producing here restriction on the functions we obtain the principal term of the asymptotics for the solution of problem (5.1):
The zero-order asymptotical solution is obtained: it satisfies the estimate where is an exact solution of problem (1.1), and is a constant independent of at sufficiently small .
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