Direct and inverse problems for nonlocal heat equation with boundary conditions of periodic type

A mathematical model of the process of heat diﬀusion in a closed metal wire is considered. This wire is wrapped around a thin sheet of insulation material. We assume that the insulation is slightly permeable. Because of this, the temperature at the point of the wire on one side of the insulation inﬂuences the diﬀusion process in the wire on the other side of the insulation. Thus, the standard heat equation will change and an extra term with involution will be added. When modeling of this process there arises an initial-boundary value problem for a one-dimensional heat equation with involution and with a boundary condition of periodic type with respect to a spatial variable. We prove the well-posedness of the formulated problem in the class of strong generalized solutions. The use of the method of separation of variables leads to a spectral problem for an ordinary diﬀerential operator with involution at the highest derivative. All eigenfunctions of the problem are constructed. In the case when all eigenvalues of the problem are simple, the system of eigenfunctions does not form an unconditional basis. A criterion when this spectral problem can have an inﬁnite number of multiple eigenvalues is proved. Corresponding root subspaces consist of one eigenfunction and one associated function. We prove that the system of root functions forms an unconditional basis and can be used for constructing a solution of the heat conduction problem by the method of separation of variables. We also consider an inverse problem. This is the problem on restoring (simultaneously with solving) of an unknown stationary source of external inﬂuence with respect to an additionally known ﬁnal state. The existence of a unique solution of this inverse problem and its stability with respect to initial and ﬁnal data are proved.


Introduction and statement of the problem
an extra term with involution will be added. When modeling this process there arises an initial-boundary value problem for a one-dimensional heat equation with involution and with a boundary condition of periodic type with respect to a spatial variable.
In the past few years, many studies have been devoted to problems for differential operators with involutional argument deviation. Problems with involution in the lowest terms of the equation and problems containing the argument deviation under the highest derivatives of the equation have been considered. Unlike many previous studies, in our paper we use nonlocal boundary conditions with respect to a spatial variable.
Let us consider a problem of modeling the thermal diffusion process. The related problem was described in the work of Cabada and Tojo [1]. They investigated a case describing a concrete situation in physics. Consider a closed metal wire (length 2) which is wrapped around a thin sheet of insulation material, as shown in Fig. 1.
Let x = 0 denote the position that is the lowest of the wire. The wire goes around the insulation up to the left to the point -1 and to the right to the point 1. The wire is closed, so the points -1 and 1 coincide.
The layer of insulation is assumed to be slightly permeable. Because of this, the temperature from one side of the insulation influences the diffusion process on the other side. Therefore, the classic heat equation changes with the addition of an additional term ε ∂ 2 ∂x 2 (-x, t) to ∂ 2 ∂x 2 (x, t) (where |ε| < 1). Here, (x, t) is the temperature at the point x of the wire at time t.
We will consider the process of heat diffusion that is described by a heat equation. Thus, this process is described by the nonlocal heat equation in the domain = {(x, t) : -1 < x < 1, 0 < t < T}. Here, F(x, t) is the influence of an external source; t = 0 is an initial time point and t = T is a final one.
The initial temperature distribution of (x, t) in the wire is considered known: As the wire is closed, it follows that the temperature at the ends of wire is the same: (-1, t) = (1, t), t ∈ [0, T]. (1.3) Consider the problem when an additional external thermal effect occurs at the junction of the ends of the wire. We consider the process in which the temperature flux at one end at each time t is proportional to the rate of change of the average temperature over the entire wire. We describe such an external influence with the help of the formula: (1. 4) here, γ = 0 is a proportionality coefficient. Hence, the process under study is reduced to the following problem: Find a solution (x, t) of the nonlocal heat equation (1.1) to the initial condition (1.2), the periodic boundary condition (1.3), and condition (1.4).
Note that in [2], instead of condition (1.4), an inverse problem of recovering the heatconduction process from nonlocal data was considered. In that work, when using condition (1.5), the arising spectral problem has only a simple spectrum. In contrast, in the case under consideration when using condition (1.4), the spectrum of the problem can be double. Note that the result of the work [2] was developed in [3] for the case of a nonlocal heat equation with a time-fractional derivative: Here, the derivative D α t that is defined as is a Caputo derivative for a differentiable function that is built on the Riemann-Liouville fractional integral

Reduction of the problem
As is easily seen, condition (1.4) is nonlocal. In this condition, there is the integral along inner lines of the domain. We transform this condition, using the idea of Samarskii. In consideration of equation (1.1) from (1.4), we obtain Hence, Then, we have where the notations a = 1 γ (1-ε) -1 and γ 1 = 1 (1-ε) are used.
Remark 2.1 As will be shown below (in Sect. 6), if a = 1, then for F ≡ 0 the problem with the zero initial conditions (1.2) and the boundary conditions (1.3), (2.1) has an infinite number of linearly independent solutions. That is, for a = 1 the problem is not Noetherian.
Therefore, throughout what follows we consider a = 1.
Let us set γ 2 = γ 1 2(1-a) and Then, for the new function u(x, t), we obtain the following problem:

2)
satisfying one initial condition

3)
and the boundary condition

4)
where ϕ(x) and f (x, t) are given sufficiently smooth functions; a = 1 is a real number; and ε is a nonzero real number such that |ε| < 1.
To formulate this problem we have used the following notations:

Summary of related papers
Note that initial-boundary value problems for the nonlocal heat equation (2.4) were repeatedly investigated earlier. In [4] the authors considered inverse problems on recovering the right-hand side of the nonlocal heat equation with the Dirichlet boundary conditions u(-1, t) = u(1, t) = 0 and the Neumann boundary In [5] inverse problems on recovering the right-hand side of the nonlocal heat equation with a time-fractional derivative were considered. The authors used Dirichlet boundary conditions, Neumann boundary conditions, the periodic boundary conditions u(- (1, t), and the antiperiodic boundary conditions u(-1, t) = -u(1, t), u x (-1, t) = -u x (1, t) as boundary conditions with respect to a spatial variable. In [6] an inverse problem with the Dirichlet boundary conditions u(-1, t) = u(1, t) = 0 for the time fractional evolution equations with an involution perturbation was considered.
References [4] and [6] contain a rather extensive bibliography on problems for differential operators with involution. A huge number of articles and a sizeable number of books either from the theoretical point of view or for practical considerations were devoted to differential evolution equations with time delays. Note that first Carleman (equations with shift (involution)) and then Przewoerska-Rolewicz, Aftabizadeh, Andreev, Burlutskayaa, Khromov, Gupta, Viner, Kirane, and Torebek paid great attention to differential equations with operations with respect to a spatial variable; the recent works of Kaliev, Sadybekov, Sarsenbi, Ashyralyev, Khromov, Kritskov, and Shkalikov were devoted to spectral problems and inverse problems for equations with involutions. In contrast to the previous works we consider problems with more complex boundary conditions. In this case, we obtain more complex spectral properties (the presence of double eigenvalues and associated vectors).
Also, note the recent work [7] in which the operator of the form L = JP + Q was studied, where J is an involution operator in the space L 2 (a, b). In this work P and Q are ordinary differential operators of order n and m generated by s-pieces of boundary conditions (where s = max{n, m}) on a bounded interval [a, b]. The authors announced theorems on the unconditional basis property and completeness of the root functions of the operator L depending on the type of boundary conditions.
We also note a few more works ( ) that are close to the topic of our investigation.
As is known, spectral problems for equation (4.1) were first considered in [25,26]. The authors considered cases of the Dirichlet and Neumann boundary conditions (a = -1). They singled out cases of the boundary conditions when the system of root vectors forms a Riesz basis in L 2 . Here, we consider a case a = -1. Until now, no one has investigated this case.
Spectral problems related to our topic were considered in the works [27][28][29][30][31][32]. In [27] a problem with the nonlocal conditions for equation (4.1) was studied. The authors proved that if r = √ (1ε)/(1 + ε) is irrational, then the system of eigenfunctions is complete and minimal in L 2 (-1, 1), but is not an unconditional basis. For rational r, it was proved that the system of root functions of the problem (for special choices of the associated functions) is the unconditional basis in L 2 (-1, 1). A similar result was proved in [28] for the case of the space L p (-1, 1).
In [31] a problem for equation (4.1) with the nonlocal boundary conditions was investigated for the case of the space L 2 (-1, 1) and in [32] for the space L p (-1, 1). In these papers it was also shown that the multiplicity of eigenvalues depends on the rationality or irrationality of the number r.
Since for equation (4.1) the spectral theory of boundary value problems is not yet fully formed, then each separate case of boundary conditions must be considered separately. The spectral problems with the nonlocal conditions (4.2) have not been previously considered. In this connection, we note the works  in which close problems related with spectral properties of nonlocal problems were considered.

General solution of equation (4.1)
To construct a general solution of equation (4.1), consider the Cauchy problem with data at the interior point with arbitrary constants A and B. Here, f (x) ∈ C[-1, 1]. By direct calculation it is easy to show that this problem (5.1) to (5.2) is equivalent to the integral equation Let us show that the integral equation (5.3) has a unique solution. For this, we introduce a new function where μ > 0 is a positive parameter that we will choose below.
Then, for Y (x) we obtain the integral equation where it is indicated Let I λ denote the integral operator in the left-hand side of (5.4). Estimating its norm in L 2 (-1, 1), we have Hence, it is easy to see that for any λ we always can choose a positive number μ > 0 such that the operator norm will be less than one: I λ ≤ δ < 1. Therefore, with this choice of μ, equation (5.4) has the unique solution Y (x) ∈ L 2 (-1, 1).
That is why equation (5.3) has the unique solution y(x) ∈ L 2 (-1, 1). It is easily seen that Further, since y(x) ∈ C[-1, 1], then it is easy to see that Therefore, from equation (5.3) we obtain that y( Thus, the following lemma is proved.

Lemma 5.1 For any values of the parameter λ, of the constants A and B and for any function f
As follows from this lemma, the general solution of equation (4.1) is two-parameter. As fundamental solutions we choose two functions c(x, λ) ∈ C 2 [-1, 1] and s(x, λ) ∈ C 2 [-1, 1] that are solutions of equation (4.1) and satisfy the Cauchy conditions: The existence of such solutions is ensured by Lemma 5.1. By direct calculation it is easy to obtain these solutions explicitly: It is also easy to verify that the chosen solutions have the following symmetry properties: Thus, the general solution of equation (4.1) has the form: with arbitrary constants C 1 and C 2 .

Eigenvalues of problem (4.1) to (4.2)
First, it is easy to see that λ = 0 is an eigenvalue of problem (4.1) to (4.2). The corresponding eigenfunction has the form: Consider a case λ = 0. Satisfying the general solution (5.6) of equation (4.1) to the boundary conditions (4.2), we obtain the linear system Its determinant will be the characteristic determinant of the spectral problem (4.1) to (4.2): Therefore, taking into account the symmetry conditions (5.5), we calculate First, from (6.2) we obtain that for a = 1 each number λ is the eigenvalue of problem (4.1) to (4.2), regardless of the value of ε. In this case, system (6.1) has the form Since |s (1, λ)| + |s(1, λ)| > 0, then it follows that C 2 = 0.
Thus, the following lemma is proved.
Now consider the case when a = 1. Then, from (6.2) we obtain c (1, λ)s(1, λ) = 0. Therefore, taking into account the explicit form of fundamental solutions, we have Thus, problem (4.1) to (4.2) has two series of the eigenvalues Proof Indeed, suppose that any two eigenvalues from different series coincide: This is equivalent to the equality That is, the coincidence of eigenvalues is possible if and only if for some k 0 , n 0 ∈ N r = n 0 /k 0 holds. That is, only if the value r is rational.

Spectral problem for irrational numbers r
Let r be an irrational number. Then, by virtue of Lemma 6.2, all eigenvalues of problem (4.1) to (4.2) are simple and are given by the formulas (6.4). By direct calculation from (6.1) we obtain that where k = 0, 1, 2, . . . and n = 1, 2, . . . , correspond to these eigenvalues.
Thus, the function f (x) turns out to be simultaneously even and odd almost everywhere on the interval (-1, 1). Consequently, f (x) = 0 holds almost everywhere on the interval (-1, 1). This proves the completeness of the system of functions (7.1) in L 2 (-1, 1).
Since the system under consideration (7.1) is a system of eigenfunctions of a linear operator, then it has a biorthogonal system consisting of eigenfunctions of an adjoint operator. We will not dwell here on a specific form of this system and the adjoint operator. However, from the existence of the biorthogonal system follows the minimality of the system of functions (7.1) in L 2 (-1, 1). Thus, the Lemma is proved. Now let us show that despite the fact that the system of functions (7.1) is complete and minimal in L 2 (-1, 1), it does not form an unconditional basis. For this, we use the necessary condition for the basis property from [36].
where ·, · is the inner products in H.
By virtue of this lemma, for the unconditional basis property in L 2 (-1, 1) of the system of functions (7.1), it is necessary to satisfy the strict inequality lim sup j→∞ y (1) n j y (2) n j < 1 (7.3) for all possible infinitely increasing subsequences k j and n j .
Calculating the norms of the eigenfunctions, we obtain Therefore, for n → ∞.
Calculate the inner products in L 2 (-1, 1): According to Dirichlet's approximation theorem (see, example, [37], Th. 1A, p. 34), for any irrational number α there exists an infinite set of irreducible fractions p q (where p and q are integers) such that Choosing here α = 1 r , we obtain that there exist infinite subsequences of the natural numbers k j and n j such that For these subsequences we will have Therefore, there exists the limit From this we have that the limit exists lim j→∞ y (1) k j , y (2) n j = |1 + a|r. From (7.4) it is easily seen that the limit exists lim j→∞ y (2) n j = |1 + a|r.
Finally, substituting everything obtained in (7.5), we obtain n j y (2) n j = 1 (7.5) for our chosen (according to the Dirichlet's approximation theorem) infinitely increasing subsequences k j and n j . That is, the necessary condition of the unconditional basis property (3.3) is not satisfied. Thus, the following lemma is proved.

Spectral problem for rational numbers r
Now consider the case when r is a rational number. Then, there exist natural numbers n 0 and k 0 such that r = n 0 k 0 . In this case, as follows from (6.4), problem (4.1) to (4.2) has an infinite number of double eigenvalues As mentioned above, the spectral problems for equation (4.1) with periodic boundary conditions (a = -1) were considered in [25,26]. For periodic problems it was shown that root subspaces, consisting of two eigenfunctions, correspond to the double eigenvalues. Here, we consider the case a = -1.
For a = -1, one eigenfunction and one associated function correspond to the double eigenvalues (8.1).
By direct calculation it is easily shown that for the cases when n k = n 0 k 0 , problem (4.1) to (4.2) has the eigenfunctions y (1) k (x) = cos(kπx), y (2) n (x) = (1 + a)r cos(nπ) cos nπx r + (a -1) sin nπ r sin(nπx), where k = 0, 1, 2, . . . and n = 1, 2, . . . , except for the cases when k = k 0 j, n = n 0 j for some j. For those cases when n k = n 0 k 0 (that is, when k = k 0 j, n = n 0 j for some j), problem (4.1) to (4.2) has the eigenfunctions y (1) k 0 j (x) and the corresponding associated functions y n 0 j,1 (x): x sin(k 0 jπx) + 1a 1 + a 1 r (-1) (n 0 +k 0 )j sin(n 0 jπx) . It is well known that the associated functions are not defined uniquely. Functions of the form for any constants C j are also the associated functions of problem (4.1) to (4.2) corresponding to the eigenvalues λ (1) k 0 j and the eigenfunctions y (1) k 0 j (x). "The problem of choosing associated functions" is also well known. This problem is related to the fact that with one choice of the constants C j the system can form an unconditional basis, and with other choices of these constants the system does not form an unconditional basis. To avoid this problem, we fix such a choice of associated functions by formula (8.3). The proof is similar to the proof of Lemma 7.1. Consider an arbitrary function f (x) orthogonal to the system of functions (8.2) to (8.3). Since it is orthogonal to all functions y (1) k (x), k = 0, 1, 2, . . . , then, as in the proof of Lemma 4, we have that f (x) + f (-x) = 0 (that is, this function is even) holds almost everywhere on the interval (-1, 1).
Further, from the orthogonality of f (x) to all functions y (2) n (x) from (8.2) we obtain that for all n = 1, 2, . . . , except the cases when n = n 0 j for some j. It follows from the oddness of f (x) that it is orthogonal to the functions x sin(k 0 jπx). Therefore, from the orthogonality of f (x) to all functions y k 0 j,1 (x) from (8.3) we obtain that (8.5) holds and for the cases when n = n 0 j for some j.
Thus, the function f (x) turns out to be simultaneously even and odd almost everywhere on the interval (-1, 1). Consequently, f (x) = 0 holds almost everywhere on the interval (-1, 1). This proves the completeness of the system of functions (8.2) to (8.3) in L 2 (-1, 1).
Since the system under consideration (8.2) to (8.3) is the system of eigen-and associated functions of a linear operator, then it has a biorthogonal system consisting of eigen-and associated functions of an adjoint operator. We will not dwell here on a specific form of this system and the adjoint operator. However, from the existence of the biorthogonal system follows the minimality of the system of functions (8.2) to (8.3) in L 2 (-1, 1). The Lemma is proved.
Now, let us prove that system (8.2) to (8.3) forms an unconditional basis in L 2 (-1, 1). For this we need a biorthogonal system. It is a system of eigen-and associated functions of the adjoint problem: Since the eigenvalues (6.4) of problem (4.1) to (4.2) are real, they are also the eigenvalues of the adjoint problem (8.6) to (8.7). The system of eigen-and associated functions of this problem can be constructed explicitly.
(8.10) When constructing this system of eigen-and associated functions of the adjoint problem, we have normalized the eigenfunctions so that the biorthogonality conditions hold for all k = 0, 1, 2, . . . and n = 1, 2, . . . , except for the cases when k = k 0 j, n = n 0 j for some j.
For the cases when n k = n 0 k 0 (that is, when k = k 0 j, n = n 0 j for some j), we have required the fulfilment of the biorthogonality conditions Here, by ·, · we denote the inner product in L 2 (-1, 1).
For what follows, we need to estimate the norms of the constructed eigen-and associated functions. By direct calculation we find ; y (1) n 0 j = 1; y n 0 j,1 2 = 1 (2k 0 jπ(1ε)) 2 Analyzing these explicit formulas, we see that only the asymptotic behavior of multipliers sin( nπ r ) and sin(rkπ) is not obvious. Let us show that these multipliers are strictly separated from zero.

Lemma 8.2
If r is a rational number: r = n 0 k 0 , then for all values of the indices n and k, when n = n 0 j and k = k 0 j, the inequalities hold The proof will be carried out by the method used in [30][31][32]. Since n = n 0 j, then the representation n = n 0 j + i holds for some j, i ∈ N, 1 ≤ i ≤ n 0 -1. Therefore, n r = k 0 j + k 0 i n 0 . Since n k = n 0 k 0 , then this number n r = k 0 j + k 0 i n 0 is not an integer. Consequently, we have: The second inequality from (8.11) is proved similarly. Since k = k 0 j, then the representation k = k 0 j + i holds for some j, i ∈ N, 1 ≤ i ≤ k 0 -1. Therefore, rk = n 0 j + n 0 i k 0 . Since n k = n 0 k 0 , this number rk = n 0 j + n 0 i k 0 is not an integer. Hence, we have:

Lemma 8.3
Let r be a rational number: r = n 0 k 0 . Then, each of the systems (8.2) to (8.3) and (8.8) to (8.10) (after the normalization in L 2 (-1, 1)) satisfies a Bessel-type inequality. Both systems form an unconditional basis in L 2 (-1, 1). Note that the system {ϕ j } has the Bessel property in a Hilbert space H, if there exists a constant B > 0 such that the Bessel-type inequality holds for all elements f ∈ H.
Proof By virtue of the above estimates of the eigen-and associated functions, in order to substantiate the Bessel property, we show the fulfilment of the Bessel property for the following three type of systems (j ∈ N): cos(jπx), sin(jπx); (8.12) x cos(jπx), x sin(jπx). (8.14) Since system (8.12) is orthonormal in L 2 (-1, 1), then it satisfies the Bessel-type inequality with constant B = 1. Since the multiplier x is bounded, the Bessel property of system (8.14) follows from the Bessel property of system (8.12). As a result, system (8.13) is the Bessel system by virtue of the following assertion proved in [30][31][32].  (-1, 1). The associated functions of problem (4.1) to (4.2) can be chosen in such a special way that this special system of eigen-and associated functions forms an unconditional basis in L 2 (-1, 1).

Construction of a formal solution of problem (2.2)-(2.4) by the method of separation of variables
As follows from Theorem 1, the system of root functions of the spectral problem (4.1) to (4.2) forms an unconditional basis if and only if the number r = √ (1ε)/(1 + ε) is rational. Therefore, using the method of separation of variables to solve problem (2.2)-(2.4) is possible only in the case when r is rational. Therefore, below we consider only such a case. 1) . (9.6) Here, by · 0 the norm in L 2 ( ) is denoted.

Main theorem on the well-posedness of the direct problem (2.2) to (2.4)
To complete the investigation of the problem, we need (similarly to the classical Fourier method) to justify the smoothness of the formal solution obtained, that is, we need to justify the convergence of all series met.
As usual, we denote by W 2,1 2 ( ) the Sobolev space with the norm Definition 10. 1 The function u(x, t) ∈ W 2,1 2 ( ) will be called a strong generalized solution of the problem (2.2) to (2.4), if there exists a sequence of the smooth functions u n (x, t) ∈ C 2,1 2 ( ) satisfying the boundary conditions (2.4) such that the sequences u n , Lu n , and u n (x, 0) converge to u, f , and ϕ(x) in norms of the spaces W 2,1 2 ( ), L 2 ( ), and W 2 2 (-1, 1), respectively.
From expansion (9.2), on the basis of the Bessel-type inequalities, it is easy to obtain the estimate From the explicit representation of the solutions of the problems (9.8) to (9.10) using the formulas (9.11) to (9.13), we calculate the estimates Substituting now (10.2) into (10.1), taking into account the inequalities (9.5) to (9.7), we obtain the estimate Thus, the obtained (in the form of a series) formal solution of problem (2.2) to (2.4) belongs to the class W 2,1 2 ( ). Obviously, for continuity on the interval [0, T] right-hand sides of equations (9.11) to (9.13) their solutions will be continuously differentiable on [0, T]. Therefore, by virtue of estimate (10.3), the existence of the sequence of the smooth functions u n (x, t) ∈ C 2,1 2 ( ) from the representation of the strong generalized solution is obvious. The uniqueness of the solution obtained is a consequence of the Riesz basis property of the system according to which the solution of the problem is expanded into the biorthogonal series (9.2).
Thus, the following main theorem on the well-posedness of the direct problem (2.2) to (2.4) is proved.

Formulation of inverse problem
Consider now an inverse problem, that is, the problem on recovering the right-hand side of a nonlocal differential equation with respect to initial-boundary conditions and an additional overdetermination condition.
In the domain = {(x, t) : -1 < x < 1, 0 < t < T} consider a problem on finding the righthand side of f (x) of the nonlocal heat equation Lu ≡ u t (x, t)u xx (x, t) + εu xx (-x, t) = f (x), (11.1) and its solution u(x, t) satisfying the initial condition where ϕ(x) and ψ(x) are given sufficiently smooth functions; a = 1 is real number; and ε is a nonzero real number such that |ε| < 1.
A fairly large number of works are devoted to the theory of inverse problems for evolutionary equations. This theory is quite well developed today. The principal difference between the formulation of the problem under consideration is that the differential equation is nonlocal. It contains a term with involutional argument deviation in the main part. Another difference is that we consider nonlocal boundary conditions with respect to a spatial variable.
In most cases, the inverse problems are ill-posed or conditionally well-posed. However, the case under consideration is (as we will show below) well-posed in a classical sense: a solution of the problem exists, is unique and stable with respect to the input data of the problem. Definition 11.1 A pair of functions {u(x, t), f (x)}, where u(x, t) ∈ W 2,1 2 ( ), f (x) ∈ L 2 (-1, 1) will be called a strong generalized solution of the inverse problem (11.1) to (11.4), if there exists a sequence of the smooth functions u n (x, t) ∈ C 2,1 2 ( ) satisfying the boundary conditions (2.4) such that the sequences u n , Lu n , u n (x, 0) and u n (x, T) converge to u(x, t), f (x), ϕ(x), and ψ(x) in norms of the spaces W 2,1 2 ( ), L 2 (-1, 1), W 2 2 (-1, 1), and W 2 2 (-1, 1), respectively.

Construction of a formal solution of the inverse problem (11.1) to (11.4)
As in the solution of the direct problem, we use here the method of separation of variables. A solution u(x, t) of the nonlocal heat equation (11.1) will be sought in the form of the biorthogonal series (9.2). We also converge the sought function f (x) and the given functions ϕ(x) and ψ(x) into the biorthogonal series with respect to the system of eigenand associated functions: n (x)f (2) n + j∈N jy n 0 j,1 (x)tjy (1) k 0 j (x) f (2) n 0 j , (12.1)