Abstract:
A finite difference analogue of the wave equation with potential perturbation is investigated, which simulates the behaviour of an infinite rod under the action of an external longitudinal force field. For a homogeneous rod, describing solutions of travelling wave type is equivalent to describing the full space of classical solutions to an induced one-parameter family of functional differential equations of point type, with the characteristic of the travelling wave as parameter. For an inhomogeneous rod, the space of solutions of travelling wave type is trivial, and their ‘proper’ extension is defined as solutions of ‘quasitravelling’ wave type. By contrast to the case of a homogeneous rod, describing the solutions of quasitravelling wave type is equivalent to describing the quotient of the full space of impulsive solutions to an induced one-parameter family of point-type functional differential equations by an equivalence relation connected with the definition of solutions of quasitravelling wave
type. Stability of stationary solutions is analyzed.
Bibliography: 9 titles.
Keywords:
functional differential equations, scale of function spaces, impulsive solutions, wave equation, travelling waves.
This publication is cited in the following 10 articles:
L. A. Beklaryan, A. L. Beklaryan, “Dualism in the Theory of Soliton Solutions”, Comput. Math. and Math. Phys., 64:7 (2024), 1472
L. A. Beklaryan, A. L. Beklaryan, “Dualism in the theory of soliton solutions”, Comput. Math. Math. Phys., 64:7 (2024), 1472–1490
L. A. Beklaryan, “Dualism in the Theory of Soliton Solutions II”, Comput. Math. and Math. Phys., 64:11 (2024), 2588
A. L. Beklaryan, L. A. Beklaryan, “Existence of bounded soliton solutions in the problem of longitudinal oscillations of an elastic infinite rod in a field with a nonlinear potential of general form”, Comput. Math. Math. Phys., 62:6 (2022), 904–919
L. A. Beklaryan, A. L. Beklaryan, “Existence of bounded soliton solutions in the problem of longitudinal vibrations of an infinite elastic rod in a field with a strongly nonlinear potential”, Comput. Math. Math. Phys., 61:12 (2021), 1980–1994
Levon A. Beklaryan, Armen L. Beklaryan, Lecture Notes in Computer Science, 13078, Optimization and Applications, 2021, 165
L. A. Beklaryan, “A new approach to the question of the existence of bounded solutions of functional differential equations of
point type”, Izv. Math., 84:2 (2020), 209–245
Levon A. Beklaryan, Armen L. Beklaryan, Alexander Yu. Gornov, Communications in Computer and Information Science, 974, Optimization and Applications, 2019, 291
L. A. Beklaryan, “A new approach to the question of existence of periodic solutions
for functional differential equations of point type”, Izv. Math., 82:6 (2018), 1077–1107
L. A. Vlasenko, A. G. Rutkas, “On a Class of Impulsive Functional-Differential Equations with Nonatomic Difference Operator”, Math. Notes, 95:1 (2014), 32–42
Citation in format AMSBIB
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