Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and Rapidly Changing Kernels
Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly...
Kalimbetov, Burkhan;Safonov, Valeriy
axioms Article Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefﬁcients and Rapidly Changing Kernels 1, 2 Burkhan Kalimbetov * and Valeriy Safonov Department of Mathematics, Akhmed Yassawi University, B. Sattarkhanov 29, Turkestan 161200, Kazakhstan Department of Higher Mathematics, National Research University «MPEI», Krasnokazarmennaya 14, 111250 Moscow, Russia; Singsaf@yandex.ru * Correspondence: email@example.com Received: 30 September 2020; Accepted: 10 November 2020; Published: 13 November 2020 Abstract: In this paper, we consider a system with rapidly oscillating coefﬁcients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the inﬂuence of the integral term on the asymptotic behavior of the solution of the original problem. Keywords: singular perturbation; integro-differential equation; rapidly oscillating coefﬁcient; regularization; asymptotic convergence; resonant exhibitors MSC: 34K26; 45J05 1. Introduction In the study of various issues related to dynamic stability, with the properties of media with a periodic structure, in the study of other applied problems, one has to deal with differential equations with rapidly oscillating coefﬁcients. Equations of this kind can describe some mechanical or electrical systems that are under the inﬂuence of high-frequency external forces, automatic control systems with a linear adjustable object, etc. As an example, we can cite the principle of operation of an oscillator with a small mass and a nonlinear restoring force, in which a high-frequency periodic force with a large amplitude acts. The presence of high-frequency terms creates serious problems for their direct numerical solutions. Therefore, asymptotic methods are usually applied to such equations ﬁrst, the most famous of which are the Feshchenko–Shkil–Nikolenko splitting method [1–5] and the Lomov’s regularization method [6–8]. It should also be noted that singularly perturbed equations are the object of study by several Russian researchers, as well as other scientists (see, for example [9–22]). In this paper, the Lomov’s regularization method is generalized to previously unexplored integro-differential equations with rapidly oscillating coefﬁcients and with rapidly decreasing kernels of the form t R dz b(t) 1 m(q)dq 0 # a(t)z #g(t) cos z e K(t, s)z(s, #)ds = h(t), z(t , #) = z , t 2 [t , T] (1) 0 0 dt # where z = z(t, #), h(t), b (t) > 0, a(t) > 0, m(t) < 0, a(t) 6= m(t) (8t 2 [t , T]) , g(t) are scalar functions, z is a constant, # > 0 is a small parameter. In the case b (t) = 2g (t) , and of the absence of an integral term, such a system was considered in [6–8]. Axioms 2020, 9, 131; doi:10.3390/axioms9040131 www.mdpi.com/journal/axioms Axioms 2020, 9, 131 2 of 12 The limit operator a(t) has a spectrum l t = a(t), functions l t = ib t and ( ) ( ) ( ) 1 2 b(t) l (t) = +ib (t) are associated with the presence in Equation (1) of a rapidly oscillating cos , and the function l (t) = m(t) characterizes the rapid change in the kernel of the integral operator. We introduce the following notations: l (t) = (l (t) , ..., l (t)) , 1 4 m = m , ..., m is multi-index with non-negative components m , j = 1, 4, ( ) 1 4 j jmj = m is multi-index height m, j=1 (m, l (t)) = m l (t) . j j j=1 Assume that the following conditions are met: ¥ ¥ (1) a(t), b(t), m(t) 2 C ([t , T] , R) , g(t), h(t) 2 C ([t , T] , C) , 0 0 K(t, s) 2 C ft s t T,Cg ; (2) the relations (m, l (t)) = 0, (m, l (t)) = l (t) , j 2 f1, ..., 4g for all multi-indices m with jmj 2 or are not fulﬁlled for any t 2 [t , T] , or are fulﬁlled identically on the whole segment t 2 [t , T] . In other words, resonant multi-indices are exhausted by the following sets G = m : (m, l (t)) 0, m 2,8t 2 [t , T] , f j j g 0 0 G = m : (m, l (t)) l (t) , jmj 2,8t 2 [t , T] , j = 1, 4. j j 0 Under these conditions, we will develop an algorithm for constructing a regularized  asymptotic solution of the problem (1). 2. Regularization of the Problem (1) i i b(t ) + b(t ) 0 0 # # Denote by s = s # independent of the t quantities s = e , s = e , and rewrite ( ) j j 1 2 the Equation (1) in the form R R t t i 0 i 0 b (q)dq + b (q)dq g(t) dz # t # t 0 0 L z(t, #) # a(t)z # e s + e s z 1 2 dt 2 (2) m(q)dq 0 e K(t, s)z(s, #)ds = h(t), z(t , #) = z , t 2 [t , T]. 0 0 We introduce regularizing variables y (t) t = l (q)dq , j = 1, 4 (3) j j # # and instead of problem (2) we consider the problem g(t) ¶z ˜ 4 ¶z ˜ t t 2 3 L z ˜ (t, t, #) # + å l (t) l (t)z ˜