1 Introduction and Main Results

The paper concerns the inverse spectral theory for integro-differential operators. Inverse spectral problems consist in reconstruction of operators, by using their spectral characteristics. The most complete theory of such problems was developed for differential operators (see the monographs [1,2,3,4]). However, integro-differential operators are often more adequate for description of various processes in physics, biology, economics and engineering [5]. Inverse spectral theory for intergo-differential operators has not been sufficiently developed yet. It consists of fragmentary results, not forming a general picture (see [6,7,8,9,10,11,12,13] and references therein).

In this paper, we solve an inverse problem for a pencil of integro-differential operators with a kernel, depending on the spectral parameter. We note that investigation of inverse problems for differential pencils causes principal difficulties, comparing with usual differential operators (see e.g. [14]). Inverse problems for integro-differential equations with coefficients, depending on the spectral parameter, as far as we know, have not been studied before.

We investigate the boundary value problem L for the first-order integro-differential equation

$$\begin{aligned} i y' (x) + \int _0^x M(x - t, \lambda ) y(t) \, dt = \lambda y(x), \quad 0< x < \pi , \end{aligned}$$
(1)

with the boundary condition \(y(0) = y(\pi )\), where the convolution kernel depends linearly on the spectral parameter: \(M(x, \lambda ) = M_0(x) + \lambda M_1(x)\).

Define the class of functions

$$\begin{aligned} L_{2,\pi } := \{ f :(\pi - x) f(x) \in L_2(0, \pi ) \}. \end{aligned}$$

We assume that the functions \(M_0\) and \(M_1\) are real-valued, the function \(M_1\) is absolutely continuous on \([0, \pi )\), \(M_0\) and \(M_1'\) belong to \(L_{2, \pi }\), and \(M_1(0) = 0\).

The main results of the paper are Theorems 1 and 2, providing necessary and sufficient conditions on the spectrum of the problem L.

Theorem 1

The boundary value problem L has a countable set of complex eigenvalues, which can be numbered as \(\{ \lambda _n \}_{n \in \mathbb Z}\), counting with their multiplicities, so that the following asymptotic relation holds

$$\begin{aligned} \lambda _n = 2 n + \varkappa _n, \quad \{ \varkappa _n \} \in l_2. \end{aligned}$$
(2)

Theorem 2

For arbitrary complex numbers \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\), satisfying the asymptotic relation (2), there exists a unique boundary value problem L in the form described above, such that \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\) is the spectrum of L.

Moreover, we develop a constructive procedure for solving the following inverse problem.

Inverse Problem 1

Given the spectrum \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\) of L, construct the functions \(M_0\) and \(M_1\).

Note that the case, when \(M_1 = 0\) and \(M_0\) is complex-valued, has been studied in [15]. In that case, the spectrum \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\) is also sufficient for recovering L, and the theorem similar to Theorem 2 is valid.

In order to solve Inverse Problem 1 and to prove Theorem 2, we develop an approach of [7, 12]. Our method is based on the reduction of the inverse problem to the system of nonlinear integral Eq. (15), called the main equations. In Sect. 2, we derive the system (15) and prove its unique solvability (Lemma 2). In Sect. 3, we prove the main results and obtain Algorithm 1 for solution of the inverse problem.

2 Main Equations of the Inverse Problem

Denote by \(e(x, \lambda )\) the solution of Eq. (1), satisfying the initial condition \(e(0, \lambda ) = 1\). Obviously, the eigenvalues of L coincide with the zeros of the entire characteristic function \(\Delta (\lambda ) := e(\pi , \lambda ) - e(0, \lambda )\). Introduce the following notations for convolutions

$$\begin{aligned} (f * g)(x)= & {} \int _0^x f(x - t) g(t) \, dt, \\ f^{*1}= & {} f, \quad f^{*n} = f^{*(n - 1)} * f, \, n \ge 1, \quad f^{*0} * g = g * f^{*0} = g. \end{aligned}$$

The solution \(e(x, \lambda )\) admits the representation

$$\begin{aligned} e(x, \lambda ) = \exp (-i \lambda x) + \int _0^x P(x, t, \lambda ) \exp (-i \lambda (x - t)) \, dt, \end{aligned}$$
(3)

where

$$\begin{aligned} P(x, t, \lambda ) = \sum _{\nu = 1}^{\infty } i^{\nu } \frac{(x - t)^{\nu }}{\nu !} M^{*\nu }(t, \lambda ). \end{aligned}$$
(4)

The relations (3) and (4) can be obtained similarly to [7], where Eq. (1) with \(M(x, \lambda ) \equiv M(x)\) has been considered.

Formal calculations show that

$$\begin{aligned} \Delta (\lambda ) = \exp (-i \lambda \pi ) - 1 + \int _0^{\pi } v(t, \lambda ) \exp (-i \lambda (\pi - t)) \, dt, \end{aligned}$$
(5)

where

$$\begin{aligned} v(t, \lambda )&= \sum _{\nu = 1}^{\infty } i^{\nu } \frac{(\pi - t)^{\nu }}{\nu !} (M_0 + \lambda M_1)^{*\nu }(t) \nonumber \\&= \sum _{\nu = 1}^{\infty }i^{\nu } \frac{(\pi - t)^{\nu }}{\nu !} \sum _{n = 0}^{\nu } \frac{\nu !}{n! (\nu - n)!} \lambda ^n (M_0^{*(\nu - n)} * M_1^{*n})(t) = \sum _{n = 0}^{\infty } \lambda ^n F_n(t), \end{aligned}$$
(6)
$$\begin{aligned} F_n(t)&= \sum _{s = \max \{1 - n, 0\}}^{\infty } \frac{i^{n + s} (\pi - t)^{n + s}}{n! s!} (M_0^{*s} * M_1^{*n})(t), \quad n \ge 0. \end{aligned}$$
(7)

Below we use the symbol C for various constants, independent of t, \(\lambda \), etc. Denote the functions

$$\begin{aligned} g_n(x) := \frac{x^n}{n!}, \, n \ge 0, \quad f * g_{-1} = g_{-1} * f = f, \quad {{\tilde{M}}}_1 := M_1'. \end{aligned}$$

Lemma 1

For \(n \ge 0\), the function \(F_n\) belongs to \(W_2^n[0, \pi ]\), and the following estimate holds

$$\begin{aligned} \Vert F_n^{(n)} \Vert _{L_2(0, \pi )} \le \frac{C^n}{[n/2]!}. \end{aligned}$$

Moreover, \(F_n^{(k)}(0) = F_n^{(k)}(\pi ) = 0\), \(0 \le k < n\).

Proof

Rewrite the relation (7) in the form

$$\begin{aligned} F_n(t)= & {} \sum _{s = \max \{1 - n, 0\}}^{\infty } F_{ns}(t), \nonumber \\ F_{ns}(t):= & {} \frac{i^{n + s} (\pi - t)^{n + s}}{n! s!} (M_0^{*s} * M_1^{*n})(t), \quad n \ge 0.\nonumber \end{aligned}$$
(8)

Note that

$$\begin{aligned} M_1 = {{\tilde{M}}}_1 * g_0, \quad M_1^{*n} = {{\tilde{M}}}_1^{*n} * g_{n - 1}, \quad (f * g_n)' = f * g_{n - 1}, \, n \ge 0. \end{aligned}$$

Using the latter formulas, we obtain

$$\begin{aligned} F_{ns}^{(k)}(t)= & {} \frac{i^{n + s} (n + s)!}{n! s!} \sum _{j = 0}^k (-1)^j g_{n + s - j}(\pi - t) (M_0^{*s} * {{\tilde{M}}}_1^{*n} * g_{n - k + j - 1})(t), \nonumber \\ 0\le & {} k \le n, \quad s \ge 0, \quad n + s \ge 1. \end{aligned}$$
(9)

Since the functions \(M_0\) and \({{\tilde{M}}}_1\) belong to \(L_{2,\pi }\), one can easily show (see e.g. [12]), that for \(n + s \ge 2\), \(\nu = \overline{0, n + s}\), the function \(g_{\nu }(\pi - t) (M_0^{*s} * {{\tilde{M}}}^{*n} * g_{n - 1 - \nu })(t)\) is absolutely continuous on \([0, \pi ]\), and

$$\begin{aligned} \left| g_{\nu }(\pi - t) (M_0^{*s} * {{\tilde{M}}}_1^{*n} * g_{n + s - 1 - \nu })(t) \right| \le \frac{C^{n + s}}{[(n + s)/2]!}, \quad t \in [0, \pi ], \end{aligned}$$

where [x] is an integer part of x. Consequently, the relation (9) for \(0 \le k < n\) and \(n + s \ge 2\) yields that the functions \(F_{ns}^{(k)}\) are absolutely continuous on \([0, \pi ]\), and \(F_{ns}^{(k)}(0) = F_{ns}^{(k)}(\pi ) = 0\). For \(n + s \ge 2\) the functions \(F_{ns}^{(n)}\) are also absolutely continuous on \([0, \pi ]\) and satisfy the estimate

$$\begin{aligned} \left| F_{ns}^{(n)}(t)\right| \le \frac{C^{n + s}}{[(n + s)/2]!}, \quad t \in [0, \pi ]. \end{aligned}$$

It remains to consider the two cases, when \(n + s = 1\):

$$\begin{aligned} F_{01}(t)&= i (\pi - t) M_0(t) \in L_2(0, \pi ), \\ F_{10}(t)&= i (\pi - t) ({{\tilde{M}}}_1 * g_0)(t) \in W_2^1[0, \pi ], \quad F_{10}(0) = F_{10}(\pi ) = 0. \end{aligned}$$

Thus, we have for all \(n, s \ge 0\), \(n + s \ge 1\), that \(F_{ns}^{(n)} \in L_2(0, \pi )\) and

$$\begin{aligned} \left\| F_{ns}^{(n)} \right\| _{L_2(0, \pi )} \le \frac{C^{n + s}}{[(n + s)/2]!}. \end{aligned}$$

Note that

$$\begin{aligned} \sum _{s = 0}^{\infty } \frac{C^{n + s}}{[(n + s)/2]!} \le C_1 \cdot \frac{C^n}{[n/2]!}, \end{aligned}$$

where \(C_1\) is a constant. Consequently, the series

$$\begin{aligned} \sum _{s = \max \{ 0, 1 - n \}}^{\infty } F_{ns}^{(n)}, \quad n \ge 0, \end{aligned}$$

converge in \(L_2(0, \pi )\). Taking (8) into account, we arrive at the assertion of the Lemma. \(\square \)

In view of Lemma 1, integration by parts yields

$$\begin{aligned} \lambda ^n \int _0^{\pi } F_n(t) \exp (- i \lambda (\pi - t)) \, dt = i^n \int _0^{\pi } F_n^{(n)}(t) \exp (- i \lambda (\pi - t)) \,dt. \end{aligned}$$
(10)

The relations (5), (6) and (10) imply

$$\begin{aligned} \Delta (\lambda ) = \exp (- i \lambda \pi ) - 1 + \int _0^{\pi } w(t) \exp (- i \lambda (\pi - t))\,dt, \end{aligned}$$
(11)

where

$$\begin{aligned} w(t) = \sum _{n = 0}^{\infty } i^n F_n^{(n)}(t). \end{aligned}$$
(12)

Lemma 1 implies \(w \in L_2(0, \pi )\).

Define the following functions for \(n \ge 1\), \(j = \overline{0, n}\):

$$\begin{aligned} \left. \begin{array}{l} Q_{nj}[M] := M_0^{*(n - j)} * {{\tilde{M}}}_1^{*j}, \quad \varphi _{nj}(x) := \dfrac{i^{n + j}}{j! (n - j)!} (\pi - x)^{n - 1}, \\ \Phi _{nj}(x, t) := \dfrac{i^{n + j} n!}{j! (n - j)!} \sum \limits _{s = 1}^j g_{n-s}(\pi - x) g_{s - 1}(x - t). \end{array} \right\} \end{aligned}$$
(13)

Substituting (7) into (12), we derive the relation

$$\begin{aligned} w(x) = (\pi - x) \sum _{n = 1}^{\infty } \sum _{j = 0}^n \left( \varphi _{nj}(x) Q_{nj}[M](x) + \int _0^x \Phi _{nj}(x, t) Q_{nj}[M](t) \, dt \right) .\nonumber \\ \end{aligned}$$
(14)

Considering the real part and the imaginary part of (14) separately, we arrive at the system of two nonlinear integral equations with respect to real functions \(M_0\) and \({{\tilde{M}}}_1\):

$$\begin{aligned} f_{\nu }(x) = \sum _{n = 1}^{\infty } \sum _{j = 0}^n \left( \psi _{\nu nj}(x) Q_{nj}[M](x) + \int _0^x \Psi _{\nu nj}(x, t) Q_{nj}[M](t) \, dt \right) , \quad \nu = 0, 1,\nonumber \\ \end{aligned}$$
(15)

where

$$\begin{aligned} \left. \begin{array}{l} \psi _{0nj}(x) = -i \text{ Im }\, \varphi _{nj}(x), \quad \psi _{1nj}(x) = -\text{ Re } \, \varphi _{nj}(x), \\ \Psi _{0nj}(x, t) = -i \text{ Im }\, \Phi _{nj}(x, t), \quad \Psi _{1nj}(x, t) = -\text{ Re } \, \Phi _{nj}(x, t), \\ f_0(x) = -i \text{ Im } \, w(x) / (\pi - x), \quad f_1(x) = -\text{ Re } \, w(x) / (\pi - x) \end{array} \right\} \end{aligned}$$
(16)

The following Lemma claims the unique solvability of (15), and plays a crucial role in investigation of Inverse Problem 1.

Lemma 2

The system of main Eq. (15) with the coefficients, defined by (16), has the unique solution \((M_0, \tilde{M}_1)\), \(M_0 \in L_{2, \pi }\), \({{\tilde{M}}}_1 \in L_{2, \pi }\), for any \(w \in L_2(0, \pi )\).

The proof of Lemma 2 is based on Proposition 1, which is a special case of [16, Theorem 3].

Proposition 1

Let \(\psi _{\nu n j}(x)\) and \(\Psi _{\nu n j}(x, t)\), \(\nu = 0, 1\), \(n \in {\mathbb {N}}\), \(j = \overline{0, n}\), be arbitrary functions, square integrable on (0, b) and \({\mathcal {S}} := \{ (x, t) :0< t< x < b \}\), respectively, and satisfying the estimates

$$\begin{aligned} \Vert \psi _{\nu n j} \Vert _{L_2(0, b)} \le A^n, \quad \Vert \Psi _{\nu n j} \Vert _{L_2({\mathcal {S}})} \le A^n, \quad \nu = 0, 1, \, n \in {\mathbb {N}}, \, j = \overline{0, n}, \end{aligned}$$
(17)

for some fixed \(A > 0\) independent of \(\nu \), n and j. Assume that \(\psi _{\nu 1 j}(x) = \delta _{\nu j}\), \(\nu , j = 0, 1\), where \(\delta _{\nu j}\) is the Kronecker delta. Then for every functions \(f_{\nu } \in L_2(0, b)\), \(\nu = 0, 1\), the system (15) has the unique solution \((M_0, {{\tilde{M}}}_1)\), \(M_0 \in L_2(0, b)\), \({{\tilde{M}}}_1 \in L_2(0, b)\).

Proof of Lemma 2

Let w be an arbitrary function from \(L_2(0, \pi )\). Obviously, the functions \(\psi _{\nu n j}\), \(\Psi _{\nu n j}\) and \(f_{\nu }\), \(\nu = 0, 1\), \(n \in {\mathbb {N}}\), \(j = \overline{0, n}\), defined by (16), satisfy the conditions of Proposition 1 for every \(b \in (0, \pi )\). Hence (15) has the unique solution \((M_0, {{\tilde{M}}}_1)\), \(M_0 \in L_2(0, b)\), \({{\tilde{M}}}_1 \in L_2(0, b)\) for every \(b \in (0, \pi )\). It remains to prove that \(M_0 \in L_{2, \pi }\) and \({{\tilde{M}}}_1 \in L_{2, \pi }\).

We represent the functions in the form \(M_0(x) = M_{01}(x) + M_{02}(x)\), \({{\tilde{M}}}_1(x) = M_{11}(x) + M_{12}(x)\), so that \(M_{\nu 1}(x) = 0\) for \(x \in (\pi /2, \pi )\) and \(M_{\nu 2}(x) = 0\) for \(x \in (0, \pi /2)\), \(\nu = 0, 1\). Note that \(M_{\nu _1 2} * M_{\nu _2 2} \equiv 0\) on \((0, \pi )\), \(\nu _k = 0, 1\), \(k = 1, 2\). Consequently, we have

$$\begin{aligned} Q_{nj}[M]= & {} Q_{nj}[M_{(1)}] + (n - j) M_{02} * Q_{n-1, j}[M_{(1)}] \nonumber \\&+ j M_{12} * Q_{n - 1, j - 1}[M_{(1)}], \quad n \ge 2, \quad j = \overline{0, n}, \end{aligned}$$
(18)

where

$$\begin{aligned} Q_{nj}[M_{(1)}] := M_{01}^{*(n - j)} * M_{11}^{*j}, \quad n \ge 1, \, j = \overline{0, n}. \end{aligned}$$

Denote

$$\begin{aligned} b_{0nj} := n - j, \quad b_{1nj} := j, \quad n \ge 2, \, j = \overline{0, n}. \end{aligned}$$

Substituting (18) into (15), we obtain the following relation for \(x > \pi /2\):

$$\begin{aligned} f_{\nu }(x)= & {} M_{\nu 2}(x) + \sum _{j = 0}^1 \int _0^x \Psi _{\nu 1 j}(x, t) M_{j2}(t) \, dt + \sum _{n = 2}^{\infty } \sum _{j = 0}^n \psi _{\nu n j}(x) Q_{nj}[M_{(1)}](x) \\&+ \sum _{n = 1}^{\infty } \sum _{j = 0}^n \int _0^x \Psi _{\nu n j}(x, t) Q_{nj}[M_{(1)}](t) \, dt + \sum _{n = 2}^{\infty } \sum _{j = 0}^n \sum _{\xi = 0}^1 b_{\xi n j} \\&\biggl ( \psi _{\nu n j}(x) (M_{\xi 2} * Q_{n - 1, j - \xi }[M_{(1)}])(x) \\&+ \int _0^x \Psi _{\nu n j} (x, t) (M_{\xi 2} * Q_{n-1, j - \xi }[M_{(1)}])(t) \, dt \biggr ), \quad \nu = 0, 1. \end{aligned}$$

Multiplying this relation by \((\pi - x)\), we arrive at the system of linear Volterra integral equations

$$\begin{aligned} \mu _{\nu }(x) = z_{\nu }(x) + \sum _{\xi = 0}^1 \int _{\pi /2}^x K_{\nu \xi }(x, t) z_{\xi }(t) \, dt, \quad \pi /2< x < \pi , \quad \nu = 0, 1, \end{aligned}$$
(19)

where

$$\begin{aligned} z_{\nu }(x) =\,&(\pi - x)M_{\nu 2}(x), \\ \mu _{\nu }(x) =\,&(\pi - x) \biggl ( f_{\nu }(x) - \sum _{n = 2}^{\infty } \sum _{j = 0}^n \psi _{\nu n j}(x) Q_{nj}[M_{(1)}](x) \\&- \sum _{n = 1}^{\infty } \sum _{j = 0}^n \int _0^x \Psi _{\nu n j}(x, t) Q_{nj}[M_{(1)}](t) \, dt \biggr ), \\ K_{\nu \xi }(x, t) =\,&\frac{\pi - x}{\pi - t} \biggl ( \Psi _{\nu 1 \xi }(x, t) + \sum _{n = 2}^{\infty } \sum _{j = 0}^n b_{\xi n j} \biggl ( \psi _{\nu n j}(x) Q_{n-1, j -\xi }[M_{(1)}](x - t) \\&+ \int _0^{x - t} \Psi _{\nu n j}(x, t + s) Q_{n - 1, j - \xi }[M_{(1)}](s) \, ds \biggr )\biggr ), \quad \nu , \xi = 0, 1. \end{aligned}$$

Note that \(f_{\nu } \in L_2(0, \pi )\), \(\Psi _{\nu 1 \xi } \in L_2({\mathcal {T}})\), \(\nu , \xi = 0, 1\), \({\mathcal {T}} := \{ (x, t) :\pi /2< t< x < \pi \}\). Using the estimates

$$\begin{aligned} \left| \frac{\pi - x}{\pi - t}\right|< 1, \quad \pi /2< t< x < \pi , \qquad |Q_{nj}[M_{(1)}](x)| \le \frac{C^n}{[n/2]!}, \quad n \ge 2, \end{aligned}$$

together with (17), we conclude that \(\mu _{\nu } \in L_2(\pi /2, \pi )\), \(K_{\nu \xi } \in L_2({\mathcal {T}})\), \(\nu , \xi = 0, 1\). Consequently, the Volterra integral Eq. (19) has the unique solution \((z_0, z_1)\), \(z_{\nu } \in L_2(0, \pi )\), \(\nu = 0, 1\), so we arrive at the assertion of the Lemma. \(\square \)

3 Proofs of the Main Results

In this section, we prove Theorems 1 and 2, and also provide an algorithm for solving Inverse Problem 1.

Proof of Theorem 1

Using (11), we obtain the following relation for the characteristic function:

$$\begin{aligned} \Delta (\lambda ) = \exp (-i \lambda \pi /2) \left( -2i \sin \tfrac{\lambda \pi }{2} + \int _{-\pi /2}^{\pi /2} w\bigl ( s + \tfrac{\pi }{2} \bigr ) \exp (i \lambda s) \, ds\right) . \end{aligned}$$
(20)

Applying to (20) the standard technique (see [4, Theorem 1.1.3]), based on Rouché‘s theorem, we derive the asymptotic relations (2) for the zeros of \(\Delta (\lambda )\). \(\square \)

Relying on (2) and (20), one can prove the following Proposition similarly to [11, Lemmas 1 and 2].

Proposition 2

The characteristic function is uniquely determined by its zeros by the formula

$$\begin{aligned} \Delta (\lambda ) = -i \pi \exp (-i \lambda \pi /2) (\lambda - \lambda _0) \prod _{\begin{array}{c} n = -\infty \\ n \ne 0 \end{array}}^{\infty } \frac{\lambda _n - \lambda }{2 n} \exp \left( \frac{\lambda }{2n}\right) . \end{aligned}$$
(21)

For arbitrary complex numbers \(\{\lambda _n\}_{n \in {\mathbb {Z}}}\) of the form (2), the function \(\Delta (\lambda )\), determined by (21), has the form (20) with a certain function \(w \in L_2(0, \pi )\).

Proof of Theorem 2

Consider an arbitrary sequence of complex numbers \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\), satisfying the asymptotic relations (2). Let \(\Delta (\lambda )\) be the functions, constructed by (21). By Proposition 2, \(\Delta (\lambda )\) admits the representation (20) with some function \(w \in L_2(0, \pi )\). Define the functions \(f_{\nu }\), \(\psi _{\nu n j}\) and \(\Psi _{\nu n j}\) for \(\nu = 0, 1\), \(n \in {\mathbb {N}}\), \(j = \overline{0, n}\), by (16). Then, by Lemma 2, the main Eq. (15) have the unique solution \((M_0, {{\tilde{M}}}_1)\), \(M_0 \in L_{2, \pi }\), \(\tilde{M}_1 \in L_{2, \pi }\). Define \(M_1(x) := \int _0^x {{\tilde{M}}}_1(t) \, dt\), and conisder the boundary value problem L with the kernel \(M(x, \lambda ) = M_0(x) + \lambda M_1(x)\), constructed by the found functions. By necessity, the characteristic function of L has the form (20) with the function w, satisfying the relation (14), equivalent to the system of the main Eq. (15). Thus, the characteristic function of L coincides with the function \(\Delta (\lambda )\), constructed by the given numbers \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\). Hence the spectrum of L coincides with \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\). \(\square \)

The proof of Theorem 2 leads to the following algorithm for solving Inverse Problem 1.

Algorithm 1

Let the complex numbers \(\{ \lambda _n \}_{n \in {\mathbb {Z}}}\) be given.

  1. 1.

    Construct the function \(\Delta (\lambda )\) as an infinite product by (21).

  2. 2.

    Find the function w(t), inverting the Fourier transform (20) by the formula

    $$\begin{aligned} w(t) = \frac{1}{\pi } \sum _{n = -\infty }^{\infty } \Delta (2 n) \exp (- 2 i n t). \end{aligned}$$
  3. 3.

    Construct the functions \(\varphi _{nj}(x)\), \(\Phi _{nj}(x, t)\), \(n \in {\mathbb {N}}\), \(j = \overline{0, n}\), using (13), and then \(f_{\nu }(x)\), \(\psi _{\nu n j}(x)\), \(\Psi _{\nu n j}(x, t)\), \(\nu = 0, 1\), \(n \in {\mathbb {N}}\), \(j = \overline{0, n}\), using (16).

  4. 4.

    Find the functions \(M_0(x)\) and \({{\tilde{M}}}_1(x)\) as the solution of the main Eq. (15), put \(M_1(x) := \int _0^x {{\tilde{M}}}_1(t) \, dt\).