EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6488 ISSN 1307-5543 – ejpam.com Published by New York Business Global Asymptotic Solutions to a Singularly Perturbed Partial Integro-Differential Equation with a Rapidly Oscillating Right-Hand Side Muminbek Begaidarov1, Dana Bibulova1, Burkhan Kalimbetov1,2,∗ 1 Department of Mathematics, M. Auezov South Kazakhstan Research University, Shymkent, Kazakhstan 2 Department of Mathematics, A. Kuatbekov Peoples’ Friendship University, Shymkent, Kazakhstan Abstract. The paper considers the Cauchy problem for a singularly perturbed integro-differential partial differential equation with a rapidly oscillating right-hand side. When considering such prob- lems, it turned out that the existing technique for regularizing singularly perturbed equations is not effective and requires significant rethinking. The development of a new technique for constructing regularized asymptotic solutions for integro-differential equations with partial derivatives and ex- ponential inhomogeneity constitutes the main content of this work. The problem was regularized and the normal and unique solvability of general iterative problems was proved. The asymptotic convergence of formal solutions is proved and a solution to the first iterative problem is constructed. 2020 Mathematics Subject Classifications: 45K05 Key Words and Phrases: Singular perturbation, partial integro-differential equation, rapidly oscillating right-hand side, solvability of iterative problems, regularization of an integral 1. Introduction In the paper, we consider the Cauchy problem for the singularly perturbed integro- differential equation with partial derivatives: ε∂y(x,t,ε)∂x = a(x)y(x, t, ε) + x∫ x0 K(x, t, s)y(s, t, ε)ds+ h1(x, t)+ +h2(x, t)e iβ(x) ε , y(x0, t, ε) = y0(t) ((x, t) ∈ [x0, X]× [0, T ]) (1.1) where β′(x) > 0, a(x) is a scalar functions, y0(t) constant, ε > 0 is a small parameter. The purpose of the work is to generalize the algorithm of the regularization method [1, 2] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6488 Email addresses: mr.0909@mail.ru (M. Begaidarov), danass86@mail.ru (D. Bibulova), bkalimbetov@mail.ru (B. Kalimbetov) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 2 of 20 on problems of type (1.1) and analysis of singularities in the solution y(t, ε), introduced by the integral operator t∫ 0 K(x, t, s)y(s, t, ε)ds and the rapidly oscillating inhomogeneity h2(x, t)e iβ(x) ε . For the sake of simplicity, a scalar version of this problem is studied. Lomov’s regularization method [1, 2] was developed to construct regularized asymp- totic solutions of ordinary differential equations in the case of stability of the spectrum of the limit operator. Problems devoted to the construction of regularized asymptotic solutions of Cauchy problems in the presence of weak turning points of the limit operator are considered in the works of [3–5], initialization in the work of [6]. The works of [7] considered the problems of constructing a regularized asymptotic solution to a nonlin- ear differential equation in a Banach space and the analytical aspects of the theory of Tikhonov systems [8]. Singularly perturbed ordinary differential equations with rapidly oscillating coefficients from the perspective of the regularization method were carried out in the work of [9]. The justification of the regularization method for linear and nonlinear integro-differential equations with a zero operator of the differential part was studied in the works of [10, 11]. Singularly perturbed integro-differential equations with rapidly oscillating coefficients and rapidly changing kernels in the case of a multiple spectrum were considered in the studies of [12–14], with rapidly oscillating coefficients and with rapidly oscillating inho- mogeneities in the works of [15–21]. The Fredholm integro-differential equation with a rapidly decreasing kernel and an exponentially oscillating inhomogeneity was studied in the work of [22]. The integro-differential Cauchy problem with exponential inhomogeneity and with a spectral value that vanishes at an isolated point on a segment of an inde- pendent variable is considered in the work of [23]. The problem belongs to the class of singularly perturbed equations with an unstable spectrum and has not been considered previously in the presence of an integral operator. It is especially difficult to study it in the vicinity of zero spectral value of the inhomogeneity. In this case, it is not possible to apply the well-known procedure of the Lomov regularization method, so the researchers chose a method for constructing the asymptotics of the solution to the original problem, based on the use of the regularized asymptotics of the fundamental solution of the cor- responding homogeneous equation, the construction of which from the standpoint of the regularization method has not been considered until now. It should be noted that singularly perturbed differential and integro-differential equa- tions with fractional derivatives in the absence and presence of rapidly oscillating com- ponents were considered in works [24–28]. In these works, the ideas of the regularization method were generalized for equations with fractional derivatives, regularized asymptotic solutions of problems were constructed, and the influence of rapidly oscillating coefficients on the leading term of the asymptotics was studied. For the first time, singularly perturbed partial integro-differential equations from the standpoint of the Lomov regularization method [1] were studied in the works of [29, 30]. First, a system of integro-differential partial differential equations with slowly varying kernels is considered. It turned out that the regularization procedure significantly depends M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 3 of 20 on the type of integral operator. It turns out that the most difficult case is when the upper limit of the integral is not a differentiation variable. The case is studied when the upper limit of the integral operator coincides with the differentiation variable. Next, we consider a system of integro-differential partial differential equations with rapidly changing kernels. The study revealed that the type of upper limit of the integral operator in such equations leads to two fundamentally different situations. The most difficult situation arises when the upper bound of the integration operator does not coincide with the differentiation variable. As studies have shown, in this case the integral operator can have characteristic values, and to construct the asymptotics, more stringent conditions on the initial data of the problem will be required. A singularly perturbed partial integro-differential equation with rapidly oscillating coefficients in the absence of resonance was studied in the works of [31–33]. Thus, in this paper, S. A. Lomov’s regularization method [1] is generalized to integro- differential partial differential equations with exponentially oscillating right-hand side. The influence of oscillating components on the structure of the asymptotics of the solution to the original problem will be revealed. Denote by λ1(x) = −a(x), β′(x) is a frequency of rapidly oscillating inhomogeneity. In the function λ2(x) = β′(x) will be called the spectrum of a rapidly oscillating inhomo- geneity. We assume that the conditions are fulfilled: (i) a(x), β(x) ∈ C∞ ([x0, X],R) , hj(x, t) ∈ C∞ ([x0, X]× [0, T ],C2 ) , j = 1, 2, K(x, t, s) ∈ C∞ ({x0 ≤ x ≤ s ≤ X, 0 ≤ t ≤ T},C2 ) ; (ii) λ1(x) ̸= λ2(x), λj(x) ̸= 0 (∀x ∈ [x0, X]), j = 1, 2; (iii) λ1(x) < 0 (∀x ∈ [x0, X]). We will develop an algorithm for constructing a regularized [1] asymptotic solution of problem (1.1). 2. Regularization of the problem Based on the spectrum λ1(x) = −a(x), the frequency of the rapidly oscillating inho- mogeneity λ2(x) = iβ′(x), we introduce regularizing functions of the form: τj = 1 ε x∫ x0 λj(θ)dθ ≡ ψj(x) ε , j = 1, 2. We introduce the notation τ = (τ1, τ2), ψj(x, ε) = (ψ1(x, ε), ψ2(x, ε)) and instead of the desired solution y(x, t, ε) to problem (1.1), we will study some extended function u(x, t, τ, ε) such that its restriction identically u(x, t, τ, ε)| τ= ψ(x) ε ≡ y(x, t, ε) coincides with the desired solution to problem (1.1). We find the total derivative for the M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 4 of 20 function u ( x, t, ψ(x)ε , ε ) , and instead of problem (1.1), consider the problem ε∂u∂x + 2∑ j=1 λj(x) ∂u ∂τj − λ1(x)u− x∫ x0 K(x, t, s)u(s, t, ψ(s)ε , ε)ds = h1(x, t)+ +h2(x, t)e τ2σ, u(x0, t, 0, ε) = y0(t), ( (x, t) ∈ [x0, X]× [0, T ] ) , σ = e iβ(x0) ε . (2.1) However, it cannot be considered fully regularized, since it does not regularize the integral Ju ≡ J ( u(x, t, τ, σ, ε)|x=s,τ=ψ(s)/ε ) = x∫ x0 K (x, t, s)u ( s, t, ψ(s) ε , σ, ε ) ds (2.2) has not been regularized in it. To regularize it, we introduce a class Mε that is asymptot- ically invariant with respect to the operator (see [1], p. 62). Definition 1. We will say that a function u(x, t, τ, σ) belongs to the class U , if it can be represented as a sum U = { u(x, t, τ, σ) : u = u0(x, t, σ) + 2∑ j=1 uj(x, t, σ)e τj , uj(x, t, σ) ∈ C∞ ([0, X]× [0, T ],C2 ) , j = 0, 2 } . (2.3) As a class Mε we take restrictions of the class Mε for τ = ψ(x) ε . We prove that U | τ= ψ(x) ε is invariant with respect to the integral operator J . Theorem 1. Let conditions (i) - and (iii) be satisfied. Then the class Mε = U | τ= ψ(x) ε is asymptotically invariant with respect to the integral operator J . Proof. Substituting (2.2) into Ju, we have: Ju(x, t, τ) = x∫ x0 K(x, t, s)u0(s, t)ds+ 2∑ j=1 x∫ x0 K(x, t, s)uj(s, t)e 1 ε s∫ x0 λj(θ)dθ ds. It is necessary to show that integrals containing exponentials are expanded into an asymp- totic series in powers ε (for ε→ +0 ). Applying the operation of integration by parts; we will have Jj(x, t, ε) = ε x∫ x0 K(x, t, s)uj(s, t) λj(s) d e 1 ε s∫ x0 λj(θ)dθ  = = ε K(x, t, s)uj(s, t) λj(s) e 1 ε s∫ x0 λj(θ)dθ ∣∣∣∣∣∣ s=x s=x0 − M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 5 of 20 − x∫ x0 ( ∂ ∂s K(x, t, s)uj(s, t) λj(s) ) e 1 ε s∫ x0 λj(θ)dθ ds  = = ε K(x, t, x)uj(x, t) λj(x) e 1 ε x∫ x0 λj(θ)dθ − K(x, t, x0)uj(x0, t) λj(x0) − −ε x∫ x0 ( ∂ ∂s K(x, t, s)uj(s, t) λj(s) ) e 1 ε s∫ x0 λj(θ)dθ ds. Thus, after a single integration by parts, the terms outside the integral are distinguished, which for τ = ψ have the form of summands (2.2), and the integral term is again an integral of the type Jj(t, ε). Multiple integration by parts leads to a formal series Jj(x, t, ε) = ∞∑ ν=0 εν+1 (Iνj (K(x, t, s)uj(s, t)) ) s=x e 1 ε x∫ x0 λj(θ)dθ − − ( Iνj (K(x, t, s)uj(s, t)) ) s=x0 ] (2.4) where I0j = 1 λj(s) , Iν1 = 1 λj(s) ∂ ∂s Iν−1 j , (ν ≥ 1), j = 1, 2. The series (2.4) converge asymptotically as ε→ +0 (uniformly in (x, t) ∈ [0, X]×[0, T ]). The partial sum of this series SN (x, t, ε) = N−1∑ ν=0 (−1)νεν+1 (Iνj (K(x, t, s)uj(s, t)) ) s=x e 1 ε x∫ x0 λj(θ)dθ − − ( Iνj (K(x, t, s)uj(s, t)) ) s=x0 ] satisfies the equality Jj(x, t, ε)− SN (x, t, ε) = (−ε)N x∫ x0 e 1 ε s∫ x0 λj(θ)dθ ∂ ∂s ( IN−1 j (K(x, t, s)uj(s, t)) ) ds. Let us integrate by parts the integral standing here, selecting one more power ε on the right-hand side: Jj(x, t, ε)− SN (x, t, ε) = εN+1 [ (−1)N {( INj (K(x, t, s)uj(s, t)) ) s=x e 1 ε x∫ x0 λj(θ)dθ − M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 6 of 20 − ( INj (K(x, t, s)uj(s, t)) ) s=x0 } + +(−1)N+1 x∫ x0 e 1 ε s∫ x0 λj(θ)dθ IN+1 j (K(x, t, s)uj(s, t))ds  . Due to the infinite differentiability of the function K(x, t, s) on (x, t, τ) ∈ [x0, X]× [0, T ]× ×(0, ε0] and λj(x) on [x0, X], and also due to the uniform boundedness of the function, the equality in the square brackets of the last one is uniformly bounded for (x, t, τ) ∈ ∈ [x0, X]× [0, T ]× {τ : Re τj , j = 1, 2}}). Consequently, ∥Jj(x, t, ε)− SN (x, t, ε)∥C[x0,X]×[0,T ] ≤ CNε N+1 where CN > 0 is a constant independent of ε for ε ∈ (0, ε0] (ε0 > 0 is sufficiently small). This means that the series (2.4)) converges to the integral asymptotically for ε → +0. The Theorem 1 is proved. Thus, the image Ju(t, ε) is expanded into an asymptotic series Ju(t, τ) = x∫ x0 K(x, t, s)u0(s, t)ds+ + 2∑ j=1 ∞∑ ν=0 (−1)νεν+1 [( Iνj (K(x, t, s)uj(s)) ) s=x eτj − ( Iνj (K(t, s)uj(s)) ) s=x0 ] where τ = ψ(x) ε . This proves that the class Mε is asymptotically invariant with respect to the integral operator J . Let us now construct an extension of the operator (2.2). If now the series u(x, t, τ, ε) = ∞∑ k=0 εkuk(x, t, τ), uk(x, t, τ) ∈ U (2.5) on the narrowing τ = ψ(x) ε is an asymptotic series for ε → +0 (uniformly in (x, t) ∈ ∈ [x0, X]×[0, T ]), then the image of Ju(x, t, τ, ε), obviously, will be the same. To construct the series Ju(t, τ, ε), we introduce the so-called order operators. Let uk(x, t, τ) ∈ U be an arbitrary element (2.3). Then we have the expansion (2.5), which can be written as Ju(x, t, τ) = R0u(x, t, τ) + ∞∑ ν=0 εk+1Rν+1uν(x, t, τ) (2.6) where τ = ψ(x) ε , and the order operators Rν have the form: R0u(x, t, τ) = x∫ 0 K(x, t, s)u0 (s, t) ds, (2.70) M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 7 of 20 R1u(x, t, τ) = 2∑ j=1 [( I0j (K(x, t, s)uj(s, t)) ) s=x eτj − ( I0j (K(x, t, s)uj(s, t)) ) s=x0 ] , (2.71) Rν+1u(x, t, τ) = 2∑ j=1 [ ( Iνj (K(x, t, s)uj(s, t)) ) s=x eτj − ( Iνj (K(x, t, s)uj(s, t)) ) s=x0 ], ν ≥ 1. (2.7ν+1) The Rν : U → U operators are called order operators because they extract the sum of the terms of the order ν with respect to the parameter ε in the expression Ju(t, τ). Apply- ing the operator J to the series (2.5), and then using formulas (2.6),(2.70), . . . , (2.7ν+1), and collecting the coefficients at the same powers of ε, we arrive at the following expression for Ju(x, t, τ, ε): Ju(x, t, τ, ε) = ∞∑ k=0 εkJuk(x, t, τ) = ∞∑ r=0 εr r∑ s=0 Rr−sus(x, t, τ)|τ=ψ(x)/ε. This equality is the basis for defining the extension of the operator J . Definition 2. The formal extension of the integral operator (2.2) is the operator J̃ , which acts on each function of the form (2.6) according to the law J̃u(x, t, τ, ε) ≡ J̃ ( ∞∑ k=0 εkuk(x, t, τ) ) def = ∞∑ r=0 εr r∑ s=0 Rr−sus(x, t, τ). (2.8) The operator J̃ is defined at least in the class of series (2.5) (with coefficients uk(x, t, τ) ∈ ∈ U) converging asymptotically at ε → +0 (uniformly in (x, t, τ) ∈ [x0, X] × [0, T ]× ×{τ : Reτj ≤ 0, j = 1, 2}. Now we can write the system, completely regularized with respect to the original problem (1.1): Lεu(x, t, τ, ε) ≡ ε∂u∂x + 2∑ j=1 λj(x) ∂u ∂τi − λ1(x)ũ− J̃u = h1(x, t)+ +h2(x, t)e τ2σ, ũ(x0, t, 0, ε) = y0(t), ((x, t) ∈ [x0, X]× [0, T ]) (2.9) where the operator J̃ has the form (2.8). 3. Solvability of iterative problems We will define the solution of system (2.9) in the form of series (2.5). Substituting (2.5) into (2.9) and equating (2.70), . . . , (2.7ν+1) the coefficients ε at the same powers (taking into account the formulas (2.70), . . . , (2.7ν+1)), we obtain the following systems of M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 8 of 20 equations: Lu0(x, t, τ) ≡ 2∑ j=1 λj(x) ∂u0 ∂τj − λ1(x)u0 −R0u0 = = h1(x, t) + h2(x, t)e τ2σ, u0(x0, t, 0) = y0(t); (3.10) Lu1(x, t, τ) = −∂u0 ∂x +R1u0, u1(x0, t, 0) = 0; (3.11) Lu2(x, t, τ) = −∂u1 ∂x +R1u1 +R2u0, u2(x0, t, 0) = 0; (3.12) ............................................................ Luk(x, t, τ) = −∂uk−1 ∂x +Rku0 + . . .+R1uk−1, uk(x0, t, 0) = 0, k ≥ 1. (3.1k) To calculate solutions to iterative problems of the (3.10),…, (3.1k) we need to study the solvability of the general iterative system Lu(x, t, τ) ≡ 2∑ j=1 λj(x) ∂u ∂τj − λ1(x)u−R0u = H(x, t, τ) u(x0, t, 0) = y∗(t) (3.2) where H(x, t, τ) = H0(x, t) + 2∑ j=1 Hj(x, t)e τj ∈ U is the known vector function of space U, y∗ is the known constant vector of the complex space C, and the operator R0 has the form (see (2.70)): R0u(x, t, τ) ≡ R0 u0(x, t) + 2∑ j=1 uj(x, t)e τj  ∆ = x∫ x0 K(x, t, s)u0(s, t)ds. We introduce scalar (for each x ∈ [x0, X], t ∈ [0, T ]) product in space U : ⟨u,w⟩ ≡ 〈 u0(x, t) + 2∑ j=1 uj(x, t)e τj , w0(x, t) + 2∑ j=1 wj(x, t)e τj 〉 ∆ = 2∑ j=0 (uj(x, t), wj(x, t)) where we denote by (∗, ∗) the usual scalar product in the complex space C. Let us prove the following statement. Theorem 2. Let conditions (i)-(ii) be fulfilled and the right-hand side H(x, t, τ) = H0(x, t)+ + 2∑ j=1 Hj(x, t)e τj of equation (3.2) belongs to the space U . Then the equa- tion(3.2) is solvable in U, if and only if H1(x, t, τ) ≡ 0 ∀(x, t) ∈ [x0, X]× [0, T ]. (3.3) M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 9 of 20 Proof. We will determine the solution of equation (3.2) as an element (2.3) of the space U . Substituting (2.3) into equation (3.2), we will have 2∑ j=1 [λj(x)− λ1(x)]uj(x, t)e τj− −λ1(x)u0(x, t)− x∫ x0 K(x, t, s)u0(s, t)ds = H0(x, t) + 2∑ j=1 Hj(x, t)e τj . Equating here the free terms and coefficients separately for identical exponents, we obtain the following equations: −λ1(x)u0(x, t)− x∫ x0 K(x, t, s)u0(s, t)ds = H0(x, t), (3.40) [λj(x)− λ1(x)]uj(x, t) = Hj(x, t), j = 1, 2. (3.4j) The equation (110) can be written as u0(x, t) = − x∫ x0 λ−1 1 (x)K(x, t, s)u0(s, t)ds− λ−1 1 (x)(x)H0(x, t). (3.5) Due to the smoothness of the kernel −λ−1 1 (x)K(x, t, s) and heterogeneity −λ−1 1 (x)H0(x, t), this Volterra integral equation has a unique solution u0(x, t) ∈ C∞[x0, X]× [0, T ]. The equations (3.42) have unique solutions u2(x, t) = [λ2(x)− λ1(x)] −1H2(x, t) ∈ C∞[x0, X]× [0, T ]. Equation (3.41) are solvable in space C∞[x0, X]× [0, T ] if and only if there are identities H1(x, t, τ) ≡ 0 ∀(x, t) ∈ [x0, X] × [0, T ]. Thus, condition (3.3) is necessary and sufficient for the solvability of equation (3.2) in the space U . The Theorem 2 is proved. Remark 1. If identity (3.3) holds, then under conditions (i)-(ii), equation (3.2) has the following solution in the space U : u(x, t, τ) = u0(x, t) + α1(x, t)e τ1 + [λ2(x)− λ1(x)] −1H2(x, t)e τ2 (3.6) where α1(x, t) ∈ C∞[x0, X]× [0, T ] are arbitrary function, u0(x, t) is the solution of an integral equation (3.5). M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 10 of 20 4. Unique solvability of the general iterative problem in the space U. Remainder theorem As seen from (3.6), the solution of the equation (3.2) is determined ambiguously. However, if its solution satisfies to the additional conditions u(x0, t, 0) = y∗(t),〈 −∂u ∂x +R1u+Q (x0, t, τ) , e τ1 〉 ≡ 0, ∀(x, t) ∈ [x0, X]× [0, T ] (4.1) where Q(x, t, τ) = H0(x, t) + 2∑ j=1 Qj(x, t)e τj is a known function of the space U, y∗ is a constant number of the complex space C, then equation (3.2) will be uniquely solvable in the space U. More precisely, the following result holds. Theorem 3. Let conditions (i)–(ii) be satisfied, the right-hand side H (x, t, τ) of the equation (3.2) belongs to the space U and satisfies the orthogonality condition (3.3). Then equation (3.2) under additional conditions (4.1) is uniquely solvable in U. Proof. Under condition (3.3), equation (3.2) has a solution (3.6) in the space U, where the function α1 (x, t) ∈ C∞[x0, X]× [0, T ], are still arbitrary. Subordinating (3.6) to the first condition (4.1), i.e. u (x0, t, 0) = y∗(t), we obtain the equation −a−1(x0, t)H0(x0, t) + α1(x0, t) + [λ2(x0)− λ1(x0)] −1H2(x0, t) = y∗(t) and we find the values α1(x0, t) = y∗(t) + a−1(x0, t)H0(x0, t)− [λ2(x0)− λ1(x0)] −1H2(x0, t). (4.2) Let us now subordinate solution (3.6) to the second condition (4.1). The right-hand side of this equation has the form −∂u0 ∂x +R1u0 +Q (x, t, τ) = − ∂ ∂x (u0(x, t))− ∂ ∂x (α1(x, t)) e τ1 + ( H2(x, t) λ2(x)− λ1(x) )• eτ2+ + 2∑ j=1 [ K(x, t, x)uj(x, t) λj(x) eτj −K(x, t, x0)uj(x0, t) λj(x0) ] +Q (x, t, τ) . we obtain equations α̇1(x, t)− K(x, t, x) λ1(x) α1(x, t)−Q1(x, t) = 0. Adding the initial conditions (4.2) to them, we can uniquely find the function α1(x, t) : α1(x, t) = α1(x0, t)e − x∫ x0 K(s,t,s) λ1(s) dx + x∫ x0 e − s∫ x0 K(s,t,s) λ1(s) ds Q1(s, t)ds M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 11 of 20 and hence, we define the solution (3.6) of the equation (3.2) in the space in a unique way. The Theorem 3 is proved. Applying Theorems 2 and 3 to iterative problems (3.1k), we find uniquely their solu- tions in the space U and construct series (2.5). Let uεN (x, t) = N∑ k=0 εkuk ( x, t, φ(x)ε ) is the restriction of the N−th partial sum of series (2.5) for τ = φ(x) ε . Same as in [1, 34], it is easy to prove the following statement. Lemma 1. Let conditions (i) - (iii) be satisfied. Then the partial sum uεN (x, t) satisfies problem (1.1) up to O(εN+1) (ε→ +0), i.e. ε duεN (x, t) dt ≡ a(x)uεN (x, t) + x∫ x0 K(x, t, s)uεN (s, t)ds+ h2(x, t)e iβ(x) ε + +h1(x, t) + εN+1RN (x, t, ε), uεN (x0, t) = y0(t), ∀(x, t) ∈ [x0, X]× [0, T ] (4.3) where ||RN (x, t, ε)||C[x0,X]×[0,T ] ≤ R̄N for all (x, t) ∈ [x0, X]× [0, T ] and for all ε ∈ (0, εN ] . Consider now the following problem: ε∂u(x,t,ε)∂x = a(x)u(x, t, ε) + x∫ x0 K(x, t, s)u(s, t, ε)ds+ +Φ(x, t, ε), u(x0, t, ε) = 0, (x, t) ∈ [x0, X]× [0, T ]. (4.4) Let us show that this problem is solvable in the space C1 [x0, X] × [0, T ] (i.e. it has a solution for any right-hand side Φ(t, ε) ∈ C ( [x0, X]× [0, T ],C2 ) and that in this case there is an estimate ||u(x, t, ε)||C[x0,X]×[0,T ] ≤ ν0 ε ||Φ(x, t, ε)||C[x0,X]×[0,T ]. (4.5) Theorem 4. Let conditions (i) - (iii) be satisfied. Then, for sufficiently small ε∈ (0, ε0] problem (4.4) for any right-hand side Φ(x, t, ε) ∈ C [x0, X] × [0, T ] has a unique solution u(x, t, ε) in the space C1 [x0, X] × [0, T ] and estimate (4.5) holds, where ν0 is a constant independent of ε > 0. Proof. Introduce an additional unknown function z(x, t, ε) = x∫ x0 K(x, t, s)z(s, t, ε)ds. Differentiating it with respect to x, we will have ε ∂z (x, t, ε) ∂x = εK(x, t, x)z(x, t, ε) + ε x∫ x0 ∂K(x, t, s) ∂x z(s, t, ε)ds. M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 12 of 20 From this and (4.3) it follows that the vector function w = {z, u} satisfies the following system: ε ∂z∂x = ( a(x) 1 0 0 ) w + ε  0 K(x, t, x)z + x∫ x0 ∂K(x,t,s) ∂x z(x, s, ε)ds + + ( Φ(x, t, ε) 0 ) , w(x0, t, ε) = 0. (4.6) Denote by Y (x, t, s, ε) the normal fundamental matrix of the homogeneous system ε ∂w ∂x = ( a(x) 1 0 0 ) w i.e., the matrix satisfying the equation ε ∂Y (x, η, ε) ∂x = ( a(x) 1 0 0 ) Y (x, η, ε), Y (x, x, ε) = I, x0 ≤ η ≤ x ≤ X. Since the matrix ( a(x) 1 0 0 ) is a matrix of simple structure and its spectrum {λ1(x), 0} lies in the half-plane Reλ1(x) ≤ 0, then the Cauchy matrix Y (x, η, ε) is uniformly bounded, i.e., ∥Y (x, η, ε)∥ ≤ c0 ∀(x, η, ε) : x0 ≤ η ≤ x ≤ X, ε > 0 where the constant c0 > 0 does not depend on ε > 0 (see, for example, [1], pp. 119-120). We now write down an integral system equivalent to system (4.6): w(x, t, ε) = x∫ x0 Y (x, η, ε)  0 K(η, t, η)z(η, t, ε) + η∫ x0 ∂K(η,t,,s) ∂η z(s, t, ε)ds  dη+ + 1 ε x∫ x0 Y (x, η, ε) ( Φ(η, t, ε) 0 ) dη. (4.60) Since for each ε > 0 there exists the solution w(x, t, ε) of the system (4.6) in the space C1 [x0, X] × [0, T ] then substituting it into (4.60) , we obtain the identity. Let’s move on to the norms: ||w(x, t, ε)|| ≤ x∫ x0 ||Y (x, ζ, ε)|| · ||K(ζ, ζ)|| · ||z(ζ, ε)||dζ + x∫ x0 ||Y (x, ζ, ε)||× × ζ∫ x0 ||∂K(ζ, s) ∂x || · ||z(s, ε)||dsdζ + 1 ε x∫ x0 ||Y (x, ζ, ε)|| · ||Φ(ζ, ε)||dζ ≤ M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 13 of 20 ≤ c0k0 x∫ x0 ||w(ζ, ε)||dζ + c0k1 x∫ x0 ζ∫ x0 ||w(s, ε)||dsdζ+ + X0 ε c0||Φ(x, ε)||C[x0,X]×[0,T ] ≤ c0k0 x∫ x0 ||w(s, ε)||ds+ +c0k1 x∫ x0 x∫ x0 ||w(s, ε)||dsdζ + c0X0 ε ||Φ(x, ε)||C[x0,X]×[0,T ] ≤ ≤ c0k0 x∫ x0 ||w(s, ε)||ds+ c0k1 X∫ x0 x∫ x0 ||w(s, ε)||dsdζ+ + c0X0 ε ||Φ(x, ε)||C[x0,X]×[0,T ] ≤ (c0k0 + c0k1X0) x∫ x0 ||w(s, ε)||ds+ + c0X0 ε ||Φ(x, ε)||C[x0,X]×[0,T ] where X0 = X − x0, ||K(x, t, s)||C[x0,X]×[0,T ] = k0, ||∂K(x,t,s) ∂x ||C[x0,X]×[0,T ] = k1. We got the inequality ||w(x, t, ε)|| ≤ c0X0 ε ||Φ(x, t, ε)||C[x0,X]×[0,T ] + (c0k0 + c0k1X0) x∫ x0 ||w(s, ε)||ds. Applying the Gronwall-Bellman lemma to this inequality [35], we have ∥w(x, t, ε)∥ ≤ c0X0 ε ∥Φ(x, t, ε)∥C[x0,X]×[0,T ]e (c0k0+c0k1X0) x∫ x0 ds = = c0X0 ε ∥Φ(x, t, ε)∥C[[x0,X]×[0,T ]e (c0k0+c0k1X0)(x−x0) ≤ ≤ ν0 ε ∥Φ(x, t, ε)∥C[x0,X]×[0,T ] ⇒ ⇒ ∥z(x, t, ε)∥C[x0,X]×[0,T ] ≤ ν0 ε ∥Φ(x, t, ε)∥C[x0,X]×[0,T ] where ν0 = c0X0 · max t∈[x0,X]×[0,T ] e(c0k0+c0k1X0)(x−x0). The Theorem 4 is proved. M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 14 of 20 Theorem 5. Let conditions (i)–(iii) be satisfied. Then for any ε ∈ (0, ε0], where ε0 > 0 is small enough, the problem (1.1) has a unique solution y(x, t, ε) ∈ C1 [x0, X] × [0, T ]; in this case, the estimate ∥y(x, t, ε)− uεN (x, t)∥C[x0,X]×[0,T ] ≤ CNε N+1(N = 0, 1, 2 . . .) holds true, where the constant CN > 0 does not depend on ε ∈ (0, ε0]. Proof. By the Lemma 1, the partial sum uεN (x, t) satisfies the problem (4.3), so the remainder rN (x, t, ε) ≡ y(x, t, ε)− uεN (x, t) satisfies the following problem: ε ∂rN (x, t, ε) ∂x = a(x)rN (x, t, ε) + x∫ x0 K(x, t, s)rN (s, t, ε)ds+ +εN+1RN (x, t, ε), rN (x0, t, ε) = 0 where Φ(x, t, ε) = −εN+1 x∫ x0 RN (s, t, ε)ds. By Theorem 4, we have the estimate ∥rN (x, t, ε)∥C[x0,X]×[0,T ] ≤ εN R̄N for all N = 0, 1, 2, . . . and all ε ∈ (0, εN ], which means that the partial sum uε,N+1(x, t) = = uεN (x, t) + εN+1uN+1(x, t, ψ(t) ε ) satisfies the inequality ||y(x, t, ε)− uε,N+1(x, t)||C[x0,X]×[0,T ] ≡ ≡ ||(y(x, t, ε)− y(x, t))− εN+1uN+1(x, t, ψ(x) ε )||C[x0,X]×[0,T ] ≤ C̄N+1ε N+1. Using the inequality ||a− b|| ≥ |||a|| − ||b|||, valid for any numbers a and b, we will have ||y(x, t, ε)− uεN (x, t)||C[x0,X]×[0,T ] ≤ ( C̄N + ∥∥∥∥uN+1 ( x, t, ψ(x) ε )∥∥∥∥ C[x0,X]×[0,T ] ) εN+1 whence we derive the estimate ||y(x, t, ε)− uεN (x, t)||C[x0,X]×[0,T ] ≤ CNε N+1 where the constant CN > 0 does not depend on ε ∈ (0, εN ]. 5. Construction of the solution of the first iteration problem in the space U Using Theorem 2, we will try to find a solution to the first iterative problem (3.1k). Since the right-hand side h1(t) + h2(x, t)e τ2 of the equation (3.10), satisfies the condition (3.3), this equation has (according to (3.6)) a solution in the space U in the form u0 (t, τ) = u (0) 0 (x, t) + α (0) 1 (x, t) eτ1 + [λ2(x)− λ1(x)] −1 h2(x, t)e τ2 (5.1) M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 15 of 20 where α(0) 1 (x, t) ∈ C∞[x0, X] × [t0, T ] are arbitrary function, u(0)0 (x, t) is the solution of the integral equation −a (x)u0 (x, t)− x∫ x0 K (x, t, s)u0 (s, t) ds = h1 (x, t) . Subordinating (5.1) to the initial condition u0 (x0, t, 0) = y0(t), we obtain values u (0) 0 (x0, t) + α (0) 1 (x0, t) + [λ2(x0)− λ1(x0)] −1 h2(x0, t) = y0(t) ⇔ ⇔ α (0) 1 (x0, t) = y0(t) + λ−1 1 (x0)h1 (x0, t)− [λ2(x0)− λ1(x0)] −1 h2(x0, t). (5.2) For a complete calculation of the function α (0) 1 (x, t), we pass to the next iterative problem (3.11). Substituting the solution (5.1) of the equation (3.10) into it, we arrive at the following equation: Lu1 (t, τ) = − ∂ ∂x ( u (0) 0 (x, t) ) − ∂ ∂x ( α (0) 1 (x, t) ) eτ1− − ∂ ∂x ( [λ2(x)− λ1(x)] −1 h2(x, t) ) eτ2+ (5.3) + 2∑ j=1  ( K (x, t, x)α (0) j (x, t) ) λj (x) eτi − ( K (x, t, x0)α (0) j (x0, t) ) λj (x0)  . Subordinating the right-hand side of this equation to the solvability conditions (3.3), we obtain the equation of ordinary differential equations − ∂ ∂x ( α (0) 1 (x, t) ) + K (x, t, x) λ1 (x) α (0) 1 (x, t) = 0. Adding the initial condition (5.2) to them, we find α (0) 1 (x, t) = α (0) 1 (x0, t) e x∫ x0 K(θ,t,θ) λ1(θ) dθ and therefore, the solution (5.1) of the problem will be found uniquely in the space U. In this case, the leading term of the asymptotics has the following form: yε0 (x, t) = u (0) 0 (x, t) + α (0) 1 (x0, t) e x∫ x0 K(θ,t,θ) λ1(θ) dθ+ 1 ε x∫ x0 λ1(θ)dθ + + [λ2(x)− λ1(x)] −1 h2(x, t)e 1 ε x∫ x0 λ2(θ)dθ (5.30) where u(0)0 (x, t) is the solution of the integral equation −a (t)u0 (x, t)− x∫ x0 K (x, t, s)u0 (s, t) ds = h1 (x, t) . M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 16 of 20 Example 1. Let us consider the following problem: ε ∂y(x, t, ε) ∂x = −y + x∫ x0 xtsy (s, t, ε) ds+ 2xt+ xte i(x+x3) ε , y (x0, t, ε) = y0(t), (5.4) where a(x) = −1, K (x, t, s) = xts, h1(x, t) = 2xt, h2(x, t) = xt. Let’s try to construct its main term of the asymptotic solution. In this equation we has the following spectrum: {λ1 (t) , λ2 (t)} = { −1, i ( 1 + 3x2 )} . Regularizing problem (5.4) using the functions τ1 = 1 ε x∫ x0 λ1 (θ) dθ, ⇔ τ1 = −1 ε (x− x0), τ2 = i ε x∫ x0 ( 1 + 3θ2 ) dθ = i ε [ (x− x0 + x3 − x0 3) ] we get the following extended problem: Lεu (x, t, τ, ε) ≡ ε∂u∂x + λ1 ∂u ∂τ1 + λ2 ∂u ∂τ2 − λ1(s)ỹ − J̃u = = h1(x, t) + h2(x, t)e τ2σ, u (x0, t, τ, ε) |τ=(0,0) = y0(t) (5.5) where τ = (τ1, τ2) , σ = exp−i(x0+x0 3) ε , J̃ is extension of the integral operator J on series of the form u (x, t, τ, ε) = ∞∑ k=0 εkuk (x, t, τ) (5.6) with coefficients uk (x, t, τ) from the space U of vector functions u (x, t, τ) = u0 (x, t) + 2∑ j=1 uj (x, t) e τj , uk (x, t) ∈ C∞ ([x0, X]× [0, T ] ,C) , k = 0, 2. This extension has the form J̃u(x, t, τ, ε) = ∞∑ r=0 εr r∑ s=0 Rr−sus(x, t, τ) where the operators Rν : U → U are calculated by the formulas : R0u(x, t, τ) = x∫ x0 xtsu0(s, t)ds, R1u(x, t, τ) = 2∑ j=1 [ x2t·uj(x,t) λj(x) eτj − xtx0·uj(x0,t) λj(x0) ] , ....... M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 17 of 20 (operators Rν at ν ≥ 2 we do not write out, because we do not need them when constructing the leading term of the asymptotics). Defining the solution of the problem (5.5) in the form of series (5.6), we obtain the following iterative problems: Lu0 ≡ λ1 ∂u0 ∂τ1 + λ2 ∂u0 ∂τ2 − λ1u0 −Ru0 = = h1 (x, t) + h2 (x, t) e τ2σ, u0 (x0, t, τ) |τ=0 = y0(t); (5.7) Lu1 = −∂u0 ∂x +R1y0, u1 (x0, t, 0) = 0; (5.8) ... The solution of the first iterative problem (5.7) will be as follows: u0 (x, t, τ) = u0 (x, t) + α1 (x, t) e τ1 + (λ2(x)− λ1) −1 h2(x, t)e τ2 (5.9) where α1 (x, t) ∈ C∞[x0, X]× [t0, T ] are arbitrary function, u0 (x, t) is the solution of the integral equation −u0 (x, t)− x x∫ x0 tsu0 (s, t) ds = 2xt. (5.10) Subordinate (5.8) to the initial condition u0 (x, t, τ) |x=x0,τ=0 = y0(t). Taking into account the form of function (5.10) and h2 (x, t) , we obtain the equation α1 (x0, t) = y0(t). For a complete calculation of the functions α1 (x, t), we pass to the next iterative problem (5.8). Taking into account that under the conditions of solvability (3.2) of problem (5.8) only exponentials eτ1 and eτ2 are involved we keep in its right-hand side only terms depending on these exponentials: −∂ (α1(x, t)) ∂x eτ1 − x2t · α1 (x, t) e τ1 = 0. We will have α̇1 (x, t) + x2t · α1 (x, t) = 0 or α̇1 (x, t) = −x2t · α1 (x, t) . Adding to these equations the initial conditions α1 (x0, t) = y0(t), found earlier, we uniquely find the functions α1 (x, t) = y0(t)e− x3 3 t and hence, we will uniquely construct solution (5.9) of the first iterative problem (5.7). Making a narrowing in it at τ1 = −x−x0 ε , τ2 = i ε [ (x− x0 + x3 − x0 3) ] , we obtain the leading term of the asymptotic solution of the problem (23): yε0 (x, t) = y0 (x, t) + u0(t)e− x3 3 te− x−x0 ε + [ i(1 + 3x2)− 1 ] xte i ε [(x−x0+x 3−x03)] (5.11) where u0 (x, t) is the solution of the integral equation (5.10). It is seen from (5.11) that, at the exact solution y (x, t, ε) of the problem (5.4) does not tend to the solution y0 (x, t) of the integral equation (5.10) at ε→ +0, but performs quick oscillations near it. M. Begaidarov, D. Bibulova, B. Kalimbetov / Eur. J. Pure Appl. Math, 18 (4) (2025), 6488 18 of 20 6. Conclusion From the expression (5.30) for yε0(t), it can be seen that the construction of the leading term of the asymptotics of the solution to problem (1.1) is significantly influenced by both the rapidly oscillating inhomogeneity and the kernel of the integral operator. Note that the application of other asymptotic methods (for example, the method of boundary functions [36–38]) to problems of type (1) with rapidly oscillating inhomo- geneities is problematic, since many of them rely heavily on the fact that all points of the spectrum (including the spectral value of the inhomogeneity) lie in the open half-plane Re λ < 0. References [1] S. A. Lomov. 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