Development of the Lomov Regularization Method for a Singularly Perturbed Cauchy Problem and a Boundary Value Problem on the Half-Line for Parabolic Equations with a “Simple” Rational Turning Point

Eliseev, A. G.; Ratnikova, T. A.; Shaposhnikova, D. A.

doi:10.1134/s0012266122030041

Cited by 2 publications

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“…Moreover, there is still no complete mathematical theory for singularly perturbed problems with an unstable spectrum, although they began to be studied from a general mathematical standpoint about 50 years ago. Of particular interest among such problems are those in which the spectral features are expressed in the form of point instability (see, for example, [9][10][11][12]). In works devoted to singularly perturbed problems, some of the features of this type are called turning points, and their classification is as follows:…”

Section: Introductionmentioning

confidence: 99%

“…Here, we give links to several recent studies in the framework of the method of regularization of singularly perturbed problems with singularities in the spectrum of the limit operator of the indicated form: for a simple turning point, see papers [9,10], for a weak turning point, see [11], and for a strong turning point, see [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, it contains an inhomogeneous Schrödinger equation, which, as will become clear in the main text of the article, significantly complicates the process of constructing a regularized asymptotic series . In many ways, our studies on the asymptotic integration of a mixed problem on a semiaxis for a nonstationary and inhomogeneous Schrödinger equation with the above-mentioned Hamiltonian at h → 0 represent the development of ideas from [12,13], where the Cauchy problem for a parabolic equation with "strong" turning point.…”

Section: Introductionmentioning

confidence: 99%

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Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Yeliseev,

Ratnikova,

Shaposhnikova

2023

Mathematics

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In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a “strong” turning point for the limit operator on time interval that do not contain focal points. Based on the ideas of the regularization method for asymptotic integration of problems with an unstable spectrum, it is shown how regularizing functions should be constructed for this type of singularity. The paper describes in detail the formalism of the regularization method, justifies the algorithm, constructs an asymptotic solution of any order in a small parameter, and proves a theorem on the asymptotic convergence of the resulting series.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Yeliseev,

Ratnikova,

Shaposhnikova

2023

Mathematics

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Algorithm for constructing a polynomial solution of a program control problem for a dynamic system in partial derivatives

Zubova

2023

Modeling of Systems and Processes

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The completely controlled dynamic system in partial derivatives is considered. The problem of constructing state and control functions in an analytical form is solved. The basic method is the cascade decomposition method, which is algorithmically implemented in three stages: forward cascade decomposition, central stage and reverse. The method is based on the properties of the matrix coefficient at the derivative of the control function. Decomposition means a p-step transition from the original system to a reduced system that is quite similar in form to the original one, but with respect to functions from subspaces. The given conditions are reduced in the process of decomposition. When passing to the p-th step system, additional conditions appear on the partial derivatives of the components of the state function. The number of extra conditions at each point is equal to the number of decomposition steps. The matrix coefficient at the derivative of the control function of the reduced system of the last step is surjective. It is this property that determines the presence of the property of complete controllability of the system under consideration. The first stage of decomposition - the stage of the direct move ends with the detection of the number of decomposition steps and the identification of the property of complete controllability. The task of the central stage of decomposition is to construct the state function of the reduced system of the last step in an analytical form. The state function of the reduced system is the basis function that determines the form of the state function of the original system. Necessary and sufficient conditions for the existence of a basis function in polynomial form are established. The minimum degree of the polynomial is also set, which is determined by the number of decomposition steps. Formulas for constructing vector functions - coefficients of the basis function polynomial are given. Formulas for constructing the control function of the reduced system are given in polynomial form. During the last stage of decomposition, the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system in polynomial form as well. While the last stage of decomposition the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system also in polynomial form. A step-by-step algorithm for solving the program control problem for a dynamic system in partial derivatives has been developed. Formulas for constructing state and control functions in polynomial form are given. An example of a three-dimensional dynamical partial differential system with a surjective matrix coefficient in the first-step splitting system is given. The implementation of the proposed algorithm is demonstrated. The state and control functions are constructed in the form of a polynomial of minimum degree.

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Development of the Lomov Regularization Method for a Singularly Perturbed Cauchy Problem and a Boundary Value Problem on the Half-Line for Parabolic Equations with a “Simple” Rational Turning Point

Cited by 2 publications

References 5 publications

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

Algorithm for constructing a polynomial solution of a program control problem for a dynamic system in partial derivatives

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