The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator

Eliseev, Aleksandr Georgievich

doi:10.1070/sm9444

Sb. Math.

2021

DOI: 10.1070/sm9444

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The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator

Aleksandr Georgievich Eliseev¹

Abstract: An asymptotic solution of the linear Cauchy problem in the presence of a ‘weak’ turning point of the limit operator is built using Lomov’s regularization method. The major singularities of the problem are written out in an explicit form. Estimates are given with respect to , which characterise the behaviour of the singularities as . The asymptotic convergence of the regularized series i… Show more

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Cited by 2 publications

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Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

Bibulova

Kalimbetov

Safonov

2022

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This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.

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Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

Bibulova

Kalimbetov

Safonov

2022

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The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

Lomovtsev,

Kukharev

2023

Wseas Transactions on Mathematics

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The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general telegraph equation with variable coefficients under Cauchy conditions on a curved line in the plane. The global correctness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the definition of its classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the righthand side of general telegraph equation depends only on one of two independent variables, then it is necessary and sufficient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two variables and is continuous, then in its integral smoothness requirements it is necessary and sufficient the continuity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary and sufficient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In the works of other authors, there is no necessary (minimally sufficient) smoothness on the right-hand sides of the hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

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The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator

Cited by 2 publications

References 4 publications

Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

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