Theoretical Foundations of Incorporating Local Boundary Conditions Into Nonlocal Problems

Vol. XX (XXXX) No. X

THEORETICAL FOUNDATIONS OF INCORPORATING LOCAL BOUNDARY CONDITIONS INTO NONLOCAL PROBLEMS

Burak Aksoylu ∗ Department of Mathematics, TOBB University of Economics and Technology Sogutozu Caddesi No:43, Ankara, 06560, Turkey & Department of Mathematics, Wayne State University 656 W. Kirby, Detroit, MI 48202, USA. email: [email protected]

Horst Reinhard Beyer Department of Mathematics, TOBB University of Economics and Technology Sogutozu Caddesi No:43, Ankara, 06560, Turkey & Universidad Politécnicade Uruapan Carretera Carapan, 60120 Uruapan Michoacán,México& Theoretical Astrophysics, IAAT, Eberhard Karls University of Tübingen Tübingen72076, Germany. email: [email protected]

Fatih Celiker † Department of Mathematics, Wayne State University 656 W. Kirby, Detroit, MI 48202, USA. email: [email protected] (Received 2016)

We study nonlocal equations from the area of peridynamics on bounded domains. We present four main results. In our recent paper, we discover that, on R, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, as main result 1, we construct an abstract convolution operator on bounded domains which is a generalization of the standard convolution based on integrals. The abstract convolution operator is a function of the classical operator, deﬁned by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. As main result 2, we prove that the solution operator can be uniquely decomposed into a Hilbert-Schmidt operator and a multiple of the identity operator. As main result 3, we prove that Hilbert-Schmidt operators provide a smoothing of the input data in the sense a square integrable function is mapped into a function that is smooth up to boundary of the domain. As main result 4, for the homogeneous nonlocal wave equation, we prove that continuity is preserved by time evolution. Namely, the solution is discontinuous if and only if the initial data is discontinuous. As a consequence, discontinuities remain stationary.

Keywords: Nonlocal wave equation, nonlocal operator, peridynamics, boundary condition, Hilbert-Schmidt operator, operator theory.

Mathematics Subject Classiﬁcation (2000): 35L05, 74B99.

∗ Burak Aksoylu was supported in part by the National Science Foundation DMS 1016190 grant, European Commission Marie Curie Career Integration 293978 grant, and Scientiﬁc and Technological Research Council of Turkey (TUB¨ ITAK)˙ MFAG 115F473 grant. Research visit of Horst R. Beyer was supported in part by the TUB¨ ITAK˙ 2221 Fellowship for Visiting Scientist Program. Sabbatical visit of Fatih Celiker was supported in part by the TUB¨ ITAK˙ 2221 Fellowship for Scientist on Sabbatical Leave Program. † Fatih Celiker was supported in part by the National Science Foundation DMS 1115280 grant.

[1] B. Aksoylu, H.R. Beyer, and F. Celiker 2

1. Introduction There are indications that a development of nonlocal theories is necessary for description of certain natural phenomena. It is part of the folklore in physics that the point particle model, which is the root for locality in physics, is the cause of unphysical singular behavior in the description of phenomena. On the other hand, all fundamental theories of physics are local. There are many more alternatives for formulating nonlocal theories compared to local ones. Therefore, the task of formulating a viable nonlocal theory consistent with experiment seems much harder than that of a local one. In any case, for such formulation, an understanding of the capacities of nonlocal theories appears inevitable. Considering wave phenomena only partially successfully described by a classical wave equation, it seems reasonable to expect that a more successful model can be obtained by employing the functional calculus of self-adjoint operators, i.e., by replacing the classical governing operator A by a suitable function f(A). We call f the regulating function. By choosing different regulating functions, we can define governing operators tailored to the needs of the underlying application. Since classical boundary conditions (BC) is an integral part of the classical operator, these BC are automatically inherited by f(A). In this way, we vision to model wave phenomena by using appropriate f(A) and, as a consequence, need to study the effect of f(A) on the solutions. One advantage of our approach is that every symmetry that commutes with A also commutes with f(A). As a result, required invariance with respect to classical symmetries such as translation, rotation and so forth is preserved. The choice of regulating functions appropriate for the physical situation at hand is an object for future research. We are interested in studying instances of successful modeling by nonlocal theories of phenomena that cannot be captured by local theories. There are noteworthy developments in the area of nonlocal modeling. For instance, crack propagation [1] and viscoelastic damping [2] are modeled by peridynamics (PD) and fractional derivatives, respectively, both of which are nonlocal. PD is a nonlocal extension of continuum mechanics developed by Silling [1], is capable of quantitatively predicting the dynamics of propagating cracks, including bifurcation. Its effectiveness has been established in sophisticated applications such as Kalthoff-Winkler experiments of the fracture of a steel plate with notches [3,4], fracture and failure of composites, nanofiber networks, and polycrystal fracture [5–8]. Further applications are in the context of multiscale modeling, where PD has been shown to be an upscaling of molecular dynamics [9, 10] and has been demonstrated as a viable multiscale material model for length scales ranging from molecular dynamics to classical elasticity [11]. Also see the review and news articles [12–15] for a comprehensive discussion, and the recent book [16]. Disturbances in solids propagate in form of waves. The wave equation is the basic model for the description of the evolution of deformations. The driving application is PD, whose equation of motion corresponds exactly to the nonlocal wave equation under consideration. The same operator is also employed in nonlocal diffusion [12, 17, 18]. Similar classes of operators are used in numerous applications such as population models, particle systems, phase transition, coagulation, etc. Nonlocal operators have also been used in image processing [19]. In addition, we witness a major effort to meet the need for mathematical theory for PD applications and related nonlocal problems addressing, for instance, conditioning analysis, domain decomposition and variational theory [20–22], discretization [22–25], nonlinear PD [26, 27], convergence of solutions [28–32]. Since PD is a nonlocal theory, one might expect only the appearance of nonlocal BC, and, indeed, so far the concept of local BC does not apply to PD. Instead, external forces must be supplied through the loading force density b [1]. On the other hand, we demonstrate that the anticipation that local BC are incompatible with nonlocal operators is not quite correct. Our approach presents an alternative to nonlocal BC and we hope that the ability to enforce local BC provides a remedy for surface effects seen in PD; see [16, Chap. 4, 5, 7, and 12] and [33, 34]. We also hope that the applicability of PD can be extended to problems that require local BC such as contact, shear, and traction. There are four main results in this paper. Throughout the paper, we label the related results with the following identifiers. • Main Result 1: On a bounded domain Ω, utilizing the eigenbasis of the classical operator which allows us to incorporate local BC, we construct a convolution operator C∗ that possesses the Hilbert- Schmidt property.

• Main Result 2: The solution operator of the nonlocal wave equation utt(x, t) + (c − C∗)u(x, t) = 0 B. Aksoylu, H.R. Beyer, and F. Celiker 3

can be uniquely decomposed into a Hilbert-Schmidt operator and a multiple of the identity operator.

• Main Result 3: Hilbert-Schmidt operators provide a smoothing of the input data in the sense an L2(Ω) function is mapped into a function that is smooth up to boundary of Ω.

• Main Result 4: The solution to the homogeneous nonlocal wave equation is discontinuous if and only if the initial data is discontinuous. Hence, discontinuities remain stationary. The rest of the article is structured as follows. Several aspects of constructing convolutions are studied in Sec. 2.. In Sec. 2.1., we introduce the proposed convolution definition on Hilbert spaces. The concept of regulating function is central to our study. In Sec. 2.2., we demonstrate that the regulating function for the bounded domain is similar to that for the unbounded domain. Hence, our construction of convolutions is an appropriate generalization. In Sec. 2.3., we study properties of operators which comprise Main Result 1, i.e., the convolution operator enjoys the Hilbert-Schmidt property. In Sec. 2.4., we provide diagonalizations of the classical governing operator A and functions, f(A), of A. The governing nonlocal operator is a function of A. On the other hand, the representation of the solution of the initial value problem corresponding to f(A) contains bounded functions of f(A). Hence, in Sec. 2.5., we prove that bounded functions of f(A) are bounded functions of A. When we refer to operators that are (bounded) functions of functions of A, we mean solution operators as in Sec. 2.5.. In Sec. 2.6., we choose an instance of f(A) that corresponds to the PD governing operator in the unbounded domain case. We present important properties such as boundedness and self-adjointness of the PD governing operator, and provide the conditions that make it a function of A. We also present the form the solution operators take with this concrete choice of f(A). The main purpose of this paper is to provide a rigorous construction which gives the ability to satisfy local BC for nonlocal operators. In Sec. 3., we find that Hilbert-Schmidt operators play a crucial role in satisfying BC. By reflecting on the representation of the solution, in Sec. 3.1., in order to satisfy BC, we outline a strategy built on obtaining the Hilbert-Schmidt property. In Sec. 3.2., we seek the tools to obtain the Hilbert-Schmidt property. We reveal that the aforementioned strategy is based on a unique decomposition of the solution operator into a Hilbert-Schmidt operator and a multiple of the identity operator, referred to as Main Result 2. The main tool to obtain Main Result 2 turns out to be holomorphic functional calculus. We prove that Hilbert-Schmidt operators remain to be Hilbert-Schmidt under the action of holomorphic functions that vanish at zero. In Sec. 3.3., we rigorously show that BC are satisfied. For satisfying BC, since the key ingredient turns out to be the Hilbert-Schmidt property, we prove how this property leads to a uniform convergence argument which allows us to interchange limits, thereby, automatically satisfying BC. In addition, the Hilbert-Schmidt property leads to smoothing of the input in the sense an L2 function is mapped into a function that is continuous up to the boundary, which is what we refer to as Main Result 3. With additional decay conditions of the eigenvalues, we reach a uniform convergence argument also for derivatives which allows us to interchange limits. As a consequence, BC that involve derivatives are also automatically satisfied (also, Main Result 3). We close with Main Result 4, namely with the proof that continuity is preserved by time evolution for the homogeneous nonlocal wave equation. In other words, the solution is discontinuous if and only if initial data is discontinuous. This important result indicates that discontinuities of the solution remain stationary. We conclude in Sec. 4. by depicting the solutions of the nonlocal wave equation with discontinuous and continuous initial data. We also numerically verify Main Result 4 by observing that, for the homogeneous equation, discontinuities enter the solution only when the initial data is discontinuous. As a consequence, discontinuities remain stationary. In addition, we also observe that away from discontinuities, the solution is smooth, which numerically verifies Main Result 3. A comprehensive study regarding the application and implementation of the proposed construction in this paper is reported in our paper [35]. To improve the accessibility of the paper, we collect the treatment of inhomogeneous nonlocal wave equation in a separate section, which comprises the Appendix. B. Aksoylu, H.R. Beyer, and F. Celiker 4

2. Construction for the Bounded Domain PD equation of motion falls into the following class of generalized wave equations for u, x ∈ R

utt(x, t) + f(A)u(x, t) = 0, (1) with a micromodulus function C ∈ L1(R), the governing operator is deﬁned by Z Z f(A)u(x, t) := C(y)dy u(x, t) − C(x − y)u(y, t)dy. (2) R R All results in the paper are also valid for x ∈ Rn. Furthermore, our previous paper [36, Rem. 4] provides a basis for the treatment of u ∈ Rn. Carrying out the details is the subject of ongoing work. Our driving application is PD which requires the dimension of u to be equal to that of x; for the correspondence of (1)–(2) to the bond based linearized PD, see [37, Eq. 23] and [38, Eq. 3]. That is why, we restrict the presentation to u, x ∈ R throughout the paper. The case of u ∈ R and x ∈ Rn corresponds to nonlocal diﬀusion [12, 18] to which our results apply in a straightforward manner. In the unbounded domain case, in [36], we discovered that PD uses as governing operator a function f(A) of the classical (local) operator A. 1 Practical applications call for a bounded domain. The question is the generalization of convolutions to functions on bounded domains, while preserving the Banach algebra structure of convolutions for functions on L1(R). Indeed, such a generalization is known for periodic functions. We aim to accommodate general BC, not just periodic, and we show this in Sec. 2.1..

2.1. Convolutions on Hilbert Spaces In the unbounded domain case, in [36], we discovered that the governing nonlocal operator is a function of a multiple of the classical governing operator. Therefore, for the bounded domain case, it is natural to define the governing operator as a function of the corresponding classical operator. This opens a gateway to incorporate local BC to nonlocal theories. For simplicity, we choose the classical operator to be a multiple of the Laplace operator with appropriate BC. In the bounded domain case, the spectrum of the Laplace operator with classical BC such as periodic, antiperiodic, Neumann, Dirichlet, and Robin is purely discrete. Furthermore, we can explicitly calculate the eigenfunctions ek corresponding to each BC and the subscript signifies the BC used; BC ∈ {p, a, N, D} where p, a, N, and D stand for periodic, antiperiodic, Neumann, Dirichlet, respectively. These eigenfunctions form a Hilbert basis (complete orthonormal basis) through which the abstract convolution can be defined as follows:

X 2 C ∗BC u := hek|Ci hek|ui ek, (3) k where Z 1 ∗ hek|ui := ek(y)u(y)dy. −1 Let Ω denote the interval (−1, 1) which we refer to as the bounded domain throughout the paper. Inspired by the governing equation (1) on the unbounded domain, we deﬁne the nonlocal wave equation

utt(x, t) + ϕ(ABC)u(x, t) = 0, x ∈ Ω, t > 0, (4) where ϕ : σ(ABC) → R is a bounded function and ABC is the classical operator with spectrum σ(ABC). For 3 instance, the convolution in (3) deﬁnes the governing operator c − C∗BC where c is a suitable constant regulating function ϕ : σ(ABC) → R is deﬁned as

ϕ(ABC) := c − C ∗BC . (5)

1Throughout the paper, we refer to the Laplace operator as the classical operator. 2If there is a speciﬁed BC in the context, then the corresponding eigenfunctions are assumed to satisfy that BC. A pedantic P BC BC BC BC notation would be C ∗BC u := hek |Ci hek |ui ek . For brevity, we suppress the BC identiﬁer of ek , use ek instead. 3 R In the PD context, the choice of c in practice is Ω C(y)dy with C ≥ 0. B. Aksoylu, H.R. Beyer, and F. Celiker 5

Representing u in this Hilbert basis, X u = hek|ui ek, we arrive at X X (c − C∗BC)u = [c − hek|Ci] hek|ui ek = ϕ(λk) hek|ui ek,

where (λk, ek) denotes an eigenpair of the classical operator. Since the last relation regulates which function of the classical operator is used to deﬁne the nonlocal operator, we call ϕ(λk) the regulating function:

ϕ(λk) = c − hek|Ci . (6)

A comparative study of nonlocal operators obtained by diﬀerent regulating functions is reported in our paper [39].

2.2. Similarity Between Regulating Functions

We demonstrate the similarity between the regulating functions f(A) and ϕ(ABC) for the cases of periodic and Neumann BC, i.e., BC = p, N. By borrowing operator theory tools from our recent paper [36], we ﬁrst 1 reveal the regulating function f(A) by diagonalizing it employing Fourier transforms F1 : L (R) → C∞(R, C) 2 2 and F2 : L (R) → L (R) where C∞(R, C) denotes the space of complex valued continuous functions on R vanishing at inﬁnity. In addition, let Tλ2 denote the maximal multiplication operator by the spectral value of the classical operator λ2 where λ ∈ [0, ∞). First, we diagonalize A and f(A):

−1 A = F2 ◦ Tλ2 ◦ F2, −1 f(A) = F2 ◦ Tf(λ2) ◦ F2. (7) Then, we directly diagonalize f(A) by using the expression given in (2)

−1 f(A) = F2 ◦ TF1C(0)−F1C(λ) ◦ F2. (8) Therefore, a comparison of (7) and (8) yields

2 f(λ ) = F1C(0) − F1C(λ). (9)

From this point forward, we assume that C is an even function. Namely,

C(−y) = C(y). (10)

By explicitly rewriting the Fourier transform in (9) and using (10), we get Z Z f(λ2) = e−i0yC(y)dy − e−iλyC(y)dy ZR Z R = ei0yC(y)dy − eiλyC(y)dy. R R Hence, we conclude that 2 f(λ ) = he0 − eλ|Ci . (11) BC Note that for BC = p, N, we have e0 = 1. By recalling (6), we conclude

2 BC BC ϕ(λk) = he0 − ek |Ci . (12)

2 2 We see from (11) and (12) that ϕ(λk) and f(λ ) project the micromodulus function C on the diﬀerence between the lowest and k-th eigenmode, and λ-th spectral mode, respectively. In conclusion, we infer that the regulating function ϕ(ABC) obtained from the abstract convolution (3) has a similar structure to f(A). B. Aksoylu, H.R. Beyer, and F. Celiker 6

2.3. Properties of the Convolution Operator We aim to generalize convolutions on R to bounded domains by preserving the Banach algebra structure of convolutions for functions on L1(R). The following is a formal deﬁnition of convolutions on Hilbert spaces which leads to a Banach algebra structure. The conventional convolution in the governing equation is in the form of an integral as in (2). Typically, the kernel function C is taken to belong to L1(Ω). Here, we modify the function space to be L2(Ω) instead of L1(Ω), a departure from the conventional construction. The following abstract Hilbert space X is taken to be L2(Ω) when we deal with the classical operator. Hence, the concrete choice of X is L2(Ω).

Theorem 1 (Main Result 1: Convolutions on Hilbert Spaces) Let K ∈ {R, C}, (X, h | i) a non-trivial K-Hilbert space with induced norm k k and M ⊂ X a Hilbert basis. By X ξ ∗ η := he|ξi he|ηi e (13) e∈M(ξ)∩M(η)

for ξ, η ∈ X, where M(ξ) := {e ∈ M : he|ξi= 6 0},M(η) := {e ∈ M : he|ηi= 6 0},

and the sum over the empty set is deﬁned as 0X , there exists a bilinear, commutative and associative map

∗ : X2 → X (ξ, η) 7→ ξ ∗ η

such that kξ ∗ ηk 6 kξkkηk for all ξ, η ∈ X and, as a consequence, (X, +,., ∗, k k) is a commutative Banach algebra.

Proof. We note for ξ, η ∈ X that, as intersection of at most countable subsets of X, M(ξ) ∩ M(η) is at most countable. Furthermore, for every ﬁnite subset S of M(ξ) ∩ M(η)

X 2 X 2 2 X 2 X 2 | he|ξi he|ηi | = | he|ξi | | he|ηi | 6 | he|ξi | | he|ηi | e∈S e∈S e∈S e∈S X 2 X 2 2 2 6 | he|ξi | | he|ηi | = kξk kηk . e∈M(ξ) e∈M(η)

2 X 2 2 2 kξ ∗ ηk = | he|ξi he|ηi | 6 kξk kηk . e∈M(ξ)∩M(η)

That ∗ is bilinear, commutative, and associative is obvious. Throughout the paper, we denote the classical operator by A and a function of A by f(A). We want to establish results for (4) which involve ABC and ϕ(ABC). When we emphasize that BC have been enforced on a bounded domain, we use ABC and ϕ(ABC) instead of A and f(A); see (1) and (4). The results we establish hold for general a self-adjoint operator A with purely discrete spectrum σ(A), a general f ∈ B(σ(A), R), and the abstract space X. Most of the time, we present the material using A and f(A) instead of ABC and ϕ(ABC). We present properties of operators induced by convolutions on Hilbert spaces. We prove that such operators are Hilbert-Schmidt and are bounded functions of A.

Corollary 1 (Main Result 1: Convolution operator is Hilbert-Schmidt) Let K, (X, h | i), k k, M and ∗ be as in Theorem1. Then for every ξ ∈ X, ξ ∗ · ∈ L(X,X) and, in addition, Hilbert-Schmidt and therefore also compact. B. Aksoylu, H.R. Beyer, and F. Celiker 7

Proof. Let ξ ∈ X. First, it follows from Theorem1 that ξ ∗· ∈ L(X,X). Furthermore, from the deﬁnition of ∗, it follows for every e ∈ M that ξ ∗ e = he|ξi e. In particular, this implies that the set

M(A) := {e ∈ M : ξ ∗ e 6= 0}

2 2 is at most countable and that k he|ξi ek e∈M(A) = | he|ξi | e∈M(A) is summable. Hence ξ ∗ · is in addition Hilbert-Schmidt and therefore also compact.

Remark 1 In the context of L2 spaces, the Hilbert-Schmidt property leads to smoothing of input functions; see Theorem6.

In the case that the Hilbert basis is an eigenbasis of an operator A, we give a condition that the operators in Corollary1 are functions of A.

Corollary 2 Let K ∈ {R, C}, (X, h | i) a non-trivial K-Hilbert space, with induced norm k k, A a densely- deﬁned, linear and self-adjoint operator in X with a purely discrete spectrum σ(A), M ⊂ X the Hilbert basis consisting of the eigenfunctions of A and, ﬁnally, ∗ the convolution corresponding to M. Then, for every ξ ∈ X, satisfying he|ξi = he0|ξi (14) for every e, e0 ∈ M corresponding to the same eigenvalue of A, ξ ∗ · is a bounded function of A.

Proof. Let ξ ∈ X and for every e ∈ M, λ(e) ∈ R the corresponding eigenvalue of A. Then the spectrum σ(A) of A is given by σ(A) := {λ(e): e ∈ M}.

We deﬁne f ∈ B(σ(A), C), where B(σ(A), C) denotes complex valued bounded functions on σ(A), by f(λ(e)) := he|ξi

for every e ∈ M. Due to (14), this deﬁnition leads to a well-deﬁned f for every e, e0 ∈ M satisfying λ(e) = λ(e0). Also, we note that f is bounded since

|f(λ(e))| = | he|ξi | 6 kξk, for every e ∈ M. Furthermore, from the spectral theorem for densely-deﬁned, linear and self-adjoint Hilbert spaces and the deﬁnition of ∗, it follows for every e ∈ M that

f(A)e = f(λ(e))e = he|ξi e = ξ ∗ e

and hence that ξ ∗ · = f(A).

Remark 2 Note that simple rotations of the eigenbasis lead to diﬀerent convolutions. If M ⊂ X is a Hilbert basis and α : M → S1, where S1 ⊂ C denotes the unit circle, then

Mα := {α(e): e ∈ M}

is also a Hilbert basis, and for every ξ, η ∈ X,

X ∗ ξ ∗α η = (α(e)) he|ξi he|ηi e, e∈M(ξ)∩M(η)

where ∗α denotes the convolution that is associated to Mα. B. Aksoylu, H.R. Beyer, and F. Celiker 8

For our theoretical construction, by deﬁning u : R → X, we resort to the following abstract version of (1) and (4) u00(t) + f(A)u(t) = 0. (15) The solution to (15) is well-known [40, Thm. 2.2.1 and Cor. 2.2.2] and is given by the following 4 sin tpf(A) p 0 u(t) = cos t f(A) u(0) + u (0), t ∈ R. (16) pf(A)

These functions are called solution operators and will be denoted by g(f(A)). Most of the construction will revolve around solution operators. Both cos(tpf(A) ) and sin(tpf(A))/pf(A) are bounded operators. When we refer to operators that are (bounded) functions of functions of A, we mean solution operators; see Sec. 2.5.. To avoid repetition in the upcoming discussions, we make the following assumptions for (15).

Assumption 1 In the following, let (X, h | i) be a non-trivial complex Hilbert Space, A : D(A) → X a densely-defined, linear and self-adjoint operator with a purely discrete spectrum σ(A), i.e., such that the (non-empty, closed and real) σ(A) is discrete and contains only eigenvalues of finite multiplicity. Note that this implies that σ(A) is at most countable, there is an at most countable Hilbert basis M ⊂ X of X consisting of eigenfunctions of A and that (X, h | i) is separable. In the following, we consider the particular case that X is infinite dimensional and hence that M is countable. As a consequence,

M = {e1, e2,... },

∗ where e1, e2,... are pairwise orthogonal normalized eigenfunctions of A. In particular, for every k ∈ N , let λk be the eigenvalue corresponding to ek. Then

∗ σ(A) = {λk : k ∈ N }.

2.4. Diagonalization of A and Representation of Bounded Functions of A With the knowledge of the eigenbasis, the diagonalization of A is straightforward.

2 2 P 2 Theorem 2 Let h·|·i denote the inner product in L (Ω). Deﬁne U : L (Ω) → ∗ = l by 2 C C k∈N C C

∗ U(ξ) := (hek|ξi2)k∈N for every ξ ∈ X. Then U is a Hilbert space isomorphism. In particular,

−1 X U ◦ A ◦ U = λk idC. (17) ∗ k∈N Proof. For every ξ ∈ D(A)

−1 h X i ∗ ∗ ∗ U ◦ A ◦ U (hek|ξi2)k∈N = U ◦ Aξ = (λk hek|ξi2)k∈N = λk idC (hek|ξi2)k∈N . (18) ∗ k∈N The last step above follows from

∗ hek|Aξi2 = hAek|ξi2 = λk hek|ξi2 , k ∈ N . The identity (18) implies that X −1 λk idC ⊃ U ◦ A ◦ U . ∗ k∈N 4Solution operators, cos(tpf(A) ) and sin(tpf(A))/pf(A), denote the unique extensions to entire holomorphic functions. " # " √ # √ sin(t ) A pedantic notation would be cos t (f(A)) and √ (f(A)), respectively. For brevity, we prefer σ(f(A)) σ(f(A)) to use the former. B. Aksoylu, H.R. Beyer, and F. Celiker 9

−1 P 2 Since U ◦ A ◦ U and ∗ (λk id ) are both densely-deﬁned, linear, and self-adjoint operators in l , we k∈N C C arrive at (17). The Hilbert space isomorphism U in Theorem2 also diagonalizes bounded functions of A. In the solution of the initial value problem of the wave equation with governing operator A, only bounded functions appear; see (16). Namely, for f ∈ B(σ(A), C) and ξ ∈ X,

∞ X f(A) ξ = f(λk) hek|ξi2 ek. (19) k=1

2.5. Functions of Functions of A

The governing operator in the nonlocal equation (4) is a bounded function, ϕ(ABC), of the classical operator ABC. As a consequence, in the solution of the initial value problem, bounded functions, g(ϕ(ABC)), of bounded function, ϕ(ABC), appear. Since ABC has a purely discrete spectrum, it is easy to see that g(ϕ(ABC)) is a bounded function of ABC,(g ◦ ϕ)(ABC), as indicated in the following theorem.

Theorem 3 Let f ∈ B(σ(A), R). (i) Then f(A) is self-adjoint with pure point spectrum, σ(f(A)), given by

∗ σ(f(A)) = {f(λk): k ∈ N }. (20)

(ii) If g ∈ B(σ(f(A)), C), k ∈ N∗, and η ∈ X, then g(f(A)) = (g ◦ f)(A). (21)

Proof. Part (i): If f ∈ B(σ(A), R), we conclude from the spectral theorem for densely-deﬁned, linear and self-adjoint Hilbert spaces, that f(A) is, in particular, self-adjoint and from (19) that every member of the

∗ Hilbert basis (ek)k∈N is an eigenfunction of f(A). Hence f(A) has a pure point spectrum, and its spectrum σ(f(A)) is given by (20). Part (ii): If g ∈ B(σ(f(A)), C), k ∈ N∗, and η ∈ X, it follows from the spectral theorem for densely- deﬁned, linear, and self-adjoint Hilbert spaces that

g(f(A)) ek = g(f(λk))ek, and hence,

∞ ∞ X X g(f(A)) η = g(f(A)) hek|ηi2 ek = hek|ηi2 g(f(A)) ek k=1 k=1 ∞ ∞ X X = g(f(λk)) hek|ηi2 ek = (g ◦ f)(λk) hek|ηi2 ek k=1 k=1 = (g ◦ f)(A)η

from which (21) follows.

Remark 3 The following functions appear in the solution of the initial value problem for (15) with abstract governing operator f(A); see (16). In particular, for all t ∈ R, η ∈ X,

∞ p X p cos t f(A) η = cos t f(λk) hek|ηi2 ek, k=1 p ∞ p sin t f(A) X sin t f(λk) p η = p hek|ηi2 ek. f(A) k=1 f(λk) B. Aksoylu, H.R. Beyer, and F. Celiker 10

2.6. The Case of Nonlocal Governing Operators Involving Convolutions Since the PD governing operator (2) involves a convolution, we generalized the notion of convolution to (3) in order to be able to treat bounded domains. In this case, the governing operator takes the form (5). Again, we turn to our abstract framework and denote the convolution in X by ∗ according to Theorem1

∗ that is associated to the Hilbert basis (ek)k∈N . We present important properties such as boundedness and self-adjointness of the PD governing operator (5) and provide the connection to functions of A.

Theorem 4 Let C ∈ X and c ∈ R. Then, the following statements hold. (i) c − C∗ is a bounded operator. (ii) c − C∗ is self-adjoint if and only if ∗ hek|Ci ∈ R, ∀k ∈ N . (22) Furthermore, c − C∗ is self-adjoint positive if and only if

∗ hek|Ci 6 c, ∀k ∈ N . (23)

(iii) In addition, if ∗ hek|Ci = hel|Ci , ∀k, l ∈ N with λk = λl, (24) then c − C∗ = (c − f)(A),

where f ∈ B(σ(A), C) is deﬁned by f(λk) := hek|Ci .

Proof. Part (i): For C, ξ, η ∈ X and c ∈ R, c − C∗ deﬁnes a linear operator in X. Note that

−1 X U(c − C∗)U = (c − hek|Ci) idC. ∗ k∈N Therefore, c − C∗ is self-adjoint if and only if (22) holds. Furthermore, c − C∗ is self-adjoint positive if and only if (23) holds. Part (iii): This follows from Corollary2.

Remark 4 For periodic, antiperiodic, Dirichlet, and Neumann BC, the abstract convolution is indeed a function of A because the condition (24) is satisﬁed in these cases. We provide the proof of this in [35] for the aforementioned BC types.

Remark 5 If c − C∗ is self-adjoint and positive, then, for all t ∈ R, η ∈ X, it follows from (21) that the solution operators in (16) takes the form

Fig. 1: The counter example u(x) = χ[−1/2,1/2](x) + 1 for which partial sums of eigenfunction expansion satisfy homogeneous Dirichlet boundary conditions. However, u(x) does not. The missing ingredient is a uniform convergence argument which allows the interchange limits in order to satisfy boundary conditions.

3. Smoothing Property of Hilbert-Schmidt Operators and Boundary Conditions The main goal of this paper is to provide a rigorous construction which gives the ability to satisfy local BC for nonlocal operators. In order to identify the essential ingredient, we consider the function 2 D kπ u(x) = χ[−1/2,1/2](x)+1 on the interval Ω = (−1, 1). Since u(x) ∈ L (Ω) and e (x) ∗ = sin (x + 1) ∗ k k∈N 2 k∈N forms a complete basis for L2(Ω), we can write the Dirichlet eigenfunction expansion of u as follows

∞ X D D u = hek|ui2 ek. k=1 Although the partial sum N X D D hek|ui2 ek k=1 satisfies the Dirichlet BC and is also infinitely differentiable on Ω, u neither satisfies the BC nor is continuous; see Figure1. The missing ingredient here is a uniform convergence argument which allows us to interchange the limits limx→±1 and limN→∞ in order to satisfy BC. This opens the important question under which conditions the solution will satisfy the BC. We address this question in this section and find that Hilbert- Schmidt operators play a crucial role in satisfying the BC. As discussed in Sec. 2.6., the PD governing operator is of the form c − C∗ where c ∈ R. The operator C∗ possesses the Hilbert-Schmidt property depending on the decay of its eigenvalues. We find that the required decay should be stronger for BC involving derivatives than the ones without derivatives. In particular, Neumann BC require stronger conditions on the decay of the eigenvalues of the operator C∗ than that of provided by the Hilbert-Schmidt property; compare the conditions in Theorem6 and Theorem7, respectively. Indeed, we find that both of these decay properties are satisfied for Dirichlet and Neumann BC [35].

3.1. Strategy to Satisfy the Boundary Conditions Recalling (16), the solution is explicitly given in terms of the governing operator c − C∗ as follows √ √ sin t c − C∗ u(x, t) = cos t c − C∗ u(x, 0) + √ ut(x, 0). c − C∗

For brevity of discussion, let us denote either one of these solution operators as g(c − C∗). We follow a two-step strategy to show how BC are satisﬁed: B. Aksoylu, H.R. Beyer, and F. Celiker 12

1. Decompose the solution operator as follows:

g(c − C∗) = [g(c − C∗) − g(c)] + g(c), (25)

so that g(c − C∗) − g(c) becomes a Hilbert-Schmidt operator because C∗ is Hilbert-Schmidt. This leads to a uniform convergence argument which allows us to interchange limits; see Theorem6 and Theorem 7. We immediately see that g(c − C∗) − g(c) part enforces the BC.

2. For the remaining part g(c), we choose initial data u(x, 0) and ut(x, 0) that satisfy the BC. In order to show that g(c − C∗) − g(c) is Hilbert-Schmidt when C∗ is Hilbert-Schmidt, we can utilize power series expansions. The operator c commutes with any operator Z. Let us deﬁne h(Z) := g(c − Z). Since g(Z) is entire, so is h(Z). Furthermore, we have a power series representation of g(c − Z) in powers of Z as follows: ∞ X h(k)(0) g(c − Z) = h(Z) = Zk, k! k=0 where h(k)(Z) = (−1)kg(k)(c − Z). Hence,

∞ k (k) X (−1) g (c) k g(c − C∗) = (C∗) . k! k=0 Consequently, ∞ k (k) X (−1) g (c) k g(c − C∗) − g(c) = (C∗) k! k=1 is a series expression that contains strictly positive powers of C∗. We have shown in Corollary1 that abstract convolution operators are Hilbert-Schmidt. Since C is Hilbert-Schmidt due to the deﬁnition by abstract convolution, any power series in C∗ that contains strictly positive powers is also Hilbert-Schmidt. Consequently, g(c − C∗) − g(c) is Hilbert-Schmidt. For details, see Lemma2.

Remark 6 Since ϕ(ABC) = c−C∗ in (5) is a perturbation of a multiple of the identity operator by a compact operator, the steady state form of the governing equation (4) satisﬁes the Fredholm alternative.

3.2. Tools to Establish the Hilbert-Schmidt Property

We start with reminding√ that the√ holomorphic√ functional calculus from our paper [36] and the fact that the functions cos t and sin t / , appearing in (16) are entire functions.

Lemma 1 (Holomorphic functional calculus) Let (X, h | i) be a non-trivial complex Hilbert space, A ∈ L(X,X) self-adjoint and σ(A) ⊂ R the (non-empty, compact) spectrum of A. Furthermore, let R > kAk and g : UR(0) → C be holomorphic. Then, the sequence g(k)(0) Ak k! k∈N is absolutely summable in L(X,X) and

∞ X g(k)(0) (g| )(A) = Ak. σ(A) k! k=0 Proof. See [36]. Consider a holomorphic function that vanishes at zero. We will prove that Hilbert-Schmidt operators remain to be Hilbert-Schmidt under the action of such functions. More precisely, if A is Hilbert-Schmidt and g is holomorphic with g(0) = 0, then g(A) is also Hilbert-Schmidt. We note that Hilbert-Schmidt operators form a ∗-ideal in L(L2 (Ω),L2 (Ω)), the space of bounded linear operators in L2 (Ω). We denote this ideal by C C C I2. B. Aksoylu, H.R. Beyer, and F. Celiker 13

Lemma 2 (Holomorphic functional calculus for Hilbert-Schmidt operators) Let (X, h | i) be a non- trivial complex Hilbert space and I2 be the complex Hilbert space consisting of the Hilbert-Schmidt operators on X with induced norm k k2. Furthermore, let A ∈ I2 be self-adjoint, σ(A) ⊂ R the (non-empty, compact) spectrum of A. Finally, let R > kAk2 and g : UR(0) → C be holomorphic such that g(0) = 0. Then

(g|σ(A))(A) ∈ I2. (26)

Proof. Since R > kAk2 > kAk and g(0) = 0, an application of Lemma1 gives

∞ X g(k)(0) (g| )(A) = Ak. σ(A) k! k=1

Also, since g : UR(0) → C is holomorphic and kAk2 < R, the sequence

(k) g (0) k · kAk2 k! ∗ k∈N is absolutely summable. Using that I2 is a ∗-ideal in L(X,X) and that for all G, H ∈ I2

kG ◦ Hk2 6 kGk2 · kHk2,

it follows for every non-empty ﬁnite subset J ⊂ N∗,

(k) (k) ∞ (k) X g (0) k X |g (0)| k X |g (0)| k A kAk kAk . k! 6 k! 2 6 k! 2 k∈J 2 k∈J k=1 As a consequence, g(k)(0) Ak k! ∗ k∈N

is absolutely summable in (I2, k k2). Since I2 ,→ L(X,X) is continuous, we conclude that (26) holds.

Remark 7 The same statement holds true for I1 which denotes complex Banach space of the trace class operators on X and k k1 is the corresponding trace norm. The proof is virtually identical to the previous one. √ As a consequence of Lemma2, we observe that cos( t A) is a perturbation of the identity operator by √ sin(t A) a Hilbert-Schmidt operator. Likewise, √ is a perturbation of a multiple of the identity operator by a A Hilbert-Schmidt operator. The discussion so far involved the sum of two operators, a multiple of the identity and a convolution type operator, i.e., c−C∗. More generally, applying√ similar methods used for a proof in√ our recent paper [36, Thm. 4.3], the following theorem gives that cos(t A + B) is a perturbation of cos(t A) √ sin(t A+B) by a Hilbert-Schmidt operator if B is Hilbert-Schmidt for t ∈ R. Likewise, √ is a perturbation of √ A+B sin(t A) √ by a Hilbert-Schmidt operator if B is Hilbert-Schmidt for t ∈ . The proof of this fact is based on A R expansion of solution operators in terms of generalized hypergeometric functions given in [36, Thm. 4.3].

Theorem 5 (Main Result 2: Solution√ operator involves a Hilbert-Schmidt part) Let (X, h | i) be a non-trivial complex Hilbert space, the complex square-root function, with domain C \ ((−∞, 0] × {0}). A, B ∈ L(X,X) self-adjoint such that [A, B] = 0 and σ(A), σ(A + B) ⊂ R the (non-empty, compact) spectra of A and A + B, respectively. Let B ∈ I2, then the operators √ √ √ √ sin t A + B sin t A cos t A + B − cos t A , √ − √ A + B A

are elements of I2. B. Aksoylu, H.R. Beyer, and F. Celiker 14

Proof. Since I2 is a ∗-ideal in L(X,X), for G ∈ L(X,X), H ∈ I2, we have

kG ◦ Hk2 6 kGk · kHk2.

In the following, 0F1 denotes the generalized hypergeometric function, deﬁned as in [41]. In a ﬁrst step, we note for every k ∈ N, z ∈ C that

∞ l ∞ l ∞ l 1 X z X |z| X |z| 0F1 −; k + ; z = 2 (k + 1 ) · l! 6 (k + 1 ) · l! 6 ( 1 )l · l! l=0 2 l l=0 2 l l=0 2 ∞ X |2z|l = = e2|z|, l! l=0 ∞ l ∞ l ∞ l 3 X z X |z| X |z| 0F1 −; k + ; z = 2 (k + 3 ) · l! 6 (k + 3 ) · l! 6 ( 3 )l · l! l=0 2 l l=0 2 l l=0 2 ∞ X |2z/3|l = = e2|z|/3 , l! l=0

and hence, if in addition k > 0, for t ∈ R that

2k 2 2k k t 1 t k t2kAk/2 |t| k (−1) · . 0F1 −; k + ; − .idσ(A) (A) B 6 e kBk2 , (2k)! 2 4 2 (2k)! 2k+1 2 2k+1 k t 3 t k 3t2kAk/2 |t| k (−1) · . 0F1 −; k + ; − .idσ(A) (A) B 6 e kBk2 . (2k + 1)! 2 4 2 (2k + 1)! As a consequence, the sequences

2k 2 k t 1 t k (−1) · . 0F1 −; k + ; − .idσ(A) (A) B , (2k)! 2 4 ∗ k∈N 2k+1 2 k t 3 t k (−1) · . 0F1 −; k + ; − .idσ(A) (A) B (2k + 1)! 2 4 ∗ k∈N

are absolutely summable in I2. Since kC ∗ k2 > kC ∗ k for every C∗ ∈ I2, this implies that the operators √ 1 t2 cos t A + B − F −; ; − .id (A), 0 1 2 4 σ(A) √ sin t A + B 3 t2 √ − 0F1 −; ; − .idσ(A) (A) A + B 2 4 are elements of I2. The√ statement of the theorem follows√ from this with the help of [36, Lemma 4.4]. In particular, cos t c − C∗ is a perturbation of cos(t c) by a Hilbert-Schmidt operator if C∗ is Hilbert- √ √ √ √ Schmidt for t ∈ R. Likewise, sin t c − C∗ / c − C∗ is a perturbation of sin (t c) / c by a Hilbert-Schmidt operator if C∗ is Hilbert-Schmidt for t ∈ R. From the functional calculus for bounded, linear, self-adjoint operators on Hilbert spaces, it is easy to conclude that functions of c−C∗ are functions of C∗ in the following obvious way.

Corollary 3 Let C∗ ∈ I2, c > 0, A = c, and B = −C∗. Then, for every t ∈ R, the operators √ √ √ √ sin t c − C∗ sin (t c) cos t c − C∗ − cos t c , √ − √ c − C∗ c

are elements of I2, where we deﬁne sin 0/0 := 1.

Proof. The statement is an immediate consequence of Theorem5. B. Aksoylu, H.R. Beyer, and F. Celiker 15

3.3. Satisfying Boundary Conditions Now we are in a position to study BC for operators with a pure point spectrum. In Theorem6, we study BC that do not involve derivatives and in Theorem7, we also study the ones that involve derivatives. The key ingredient is the Hilbert-Schmidt property. This leads to a uniform convergence argument which allows us to interchange limits. Hence, BC are automatically satisﬁed. In addition, the Hilbert-Schmidt property leads to smoothing of the input, in the sense that an L2 function is mapped into a function that is continuous up to the boundary.

Theorem 6 (Main Result 3: Smoothing property of Hilbert-Schmidt operators) Let c, K > 0, n ∈ N∗, Ω ⊂ R be non-empty, bounded, and open, A be a densely-deﬁned, linear and self-adjoint operator in 2 L (Ω) with a pure point spectrum σ(A), i.e., for which there is a Hilbert basis ek ∗ of eigenfunctions. C k∈N ∗ In particular, for every k ∈ N , let λk be the eigenvalue corresponding to ek. Furthermore, let Ωb ⊃ Ω be ∗ bounded and open, and for every k ∈ N let ek be the restriction of some ebk ∈ C(Ωb, C) satisfying

kebkk∞ 6 K. (27) Finally, let f ∈ U s(σ(A), ) 1 and u ∈ L2 (Ω). C C C (i) f(A) is Hilbert-Schmidt if and only if

2 ∗ (|f(λk)| )k∈N is summable. (28)

(ii) If f(A) is Hilbert-Schmidt, then f(A)u has an extension to a continuous function on Ω, and for every limit point x of Ω ∞ X lim [f(A)u](y) = f(λk) hek|ui lim ek(y) . y→x0 y→x0 k=1 Proof. Part (i): From the spectral theorem for densely-deﬁned, linear and self-adjoint Hilbert spaces, it follows for every k ∈ ∗ and h ∈ L2 (Ω) that f(A) e = f(λ ) e . Hence, N C k k k ∞ ∞ ∞ X X X f(A) h = f(A) hek|hi2 ek = hek|hi2 f(A) ek = f(λk) hek|hi2 ek. k=1 k=1 k=1

As a result, f(A) has a pure point spectrum which is given by (20). In particular, since for every k ∈ N∗

2 2 2 kf(A)ekk2 = kf(λk)ekk2 = |f(λk)| , f(A) is Hilbert-Schmidt if and only if (28) holds. 0 ∗ 0 ∗ Part (ii): For m, m ∈ N satisfying N 6 m 6 m , where N ∈ N is suﬃciently large, we have

0 0 m m X X f(λk) hek|ui2 ek 6 K |f(λk)| | hek|ui2 | b ∞ k=m k=m 0 0 m 1/2 m 1/2 X 2 X 2 6 K |f(λk)| | hek|ui2 | k=m k=m 0 m 1/2 X 2 6 K kuk2 |f(λk)| . k=m As a consequence, N X f(λk) hek|ui2 ek b N∈ ∗ k=1 N

1U s(σ(A), ) denotes the space of bounded complex-valued functions on σ(A) that are strongly measurable in the sense that C C they are everywhere σ(A) limit of a sequence of step functions. B. Aksoylu, H.R. Beyer, and F. Celiker 16

is a Cauchy sequence in (BC(Ωb, C), k k∞), bounded continuous functions deﬁned on Ω,b and hence converges uniformly to an extension of f(A)u which is bounded continuous on Ω.b In particular, this implies, for every limit point x0 of Ω, that

N X lim [f(A)u](y) = lim lim f(λk) hek|ui ek(y) y→x0 y→x0 N→∞ b k=1 N X = lim lim f(λk) hek|ui ek(y) N→∞ y→x0 b k=1 ∞ X = f(λk) hek|ui lim ek(y) . y→x0 k=1 For the last step, see [42, Thm. 2.41].

Remark 8 An important consequence of Theorem6 (ii) is the following. Consider the solution to the wave equation (4) with Dirichlet BC and initial data u(·, 0), u0(·, 0) ∈ L2 (Ω). Since C lim u(x, 0) = 0 x→x0

holds pointwise for x in some neighborhood of boundary point x0,

lim u(x, t) = 0 x→x0 also holds for all t ∈ R. We are now in a position to study BC involving derivatives for operators with a pure point spectrum. With additional decay conditions of the eigenvalues of f(A), also for such BC, we reach a uniform convergence argument which allows us to interchange limits.

Theorem 7 (Main Result 3: Smoothing property of Hilbert-Schmidt operators with boundary conditions involving derivatives) In addition to the assumptions of Theorem6, we assume that f(A) is ∗ 0 Hilbert-Schmidt and for every k ∈ N that ek is diﬀerentiable with a derivative that has an extension ebk to a continuous function on Ωb. Finally, we assume that

0 2 ( | ke k f(λ )| ) ∗ is summable. (29) bk ∞ k k∈N 1 Then f(A)u ∈ C (Ω, C), and for every limit point x0 of Ω

∞ X lim [f(A)u](y) = f(λk) hek|ui lim ek(y) and y→x0 y→x0 k=1 ∞ 0 X 0 lim [f(A)u] (y) = f(λk) hek|ui lim ek(y) . y→x0 y→x0 k=1

0 ∗ 0 ∗ Proof. For m, m ∈ N satisfying N 6 m 6 m , where N ∈ N is suﬃciently large, we have

As a consequence, N X 0 f(λk) hek|ui2 ek b N∈ ∗ k=1 N

is a Cauchy sequence in (C(Ωb, C), k k∞) and hence, converges uniformly to a continuous function on Ω.b In particular, this implies that f(A)u is continuously diﬀerentiable and for every limit point x0 of Ω that

∞ 0 X 0 lim [f(A)u] (y) = f(λk) hek|ui lim ek(y) . y→x0 y→x0 k=1

For the last step, see [42, Thm. 2.42].

Remark 9 As in Remark8, an important consequence of Theorem7 is the following. Consider the solution to the wave equation (4) with Neumann BC and initial data u(·, 0), u0(·, 0) ∈ L2 (Ω). Since C lim u0(x, 0) = 0 x→x0 holds pointwise for x in some neighborhood of boundary point x0,

lim u0(x, t) = 0 x→x0 also holds for all t ∈ R. We want to investigate how much regularity one can gain in solutions if the convolution operator in (13) is used. We compare against the solution obtained from a governing equation whose convolution is in the form of an integral as in (2), but on a ﬁnite domain Ω. For a kernel function in C ∈ L1(Ω), it was reported in [12, Eq. (4.21b)] that the solution does not gain any regularity. Namely, for inhomogeneity b ∈ L2(Ω),

kukL2(Ω) 6 kregkbkL2(Ω). When the convolution in (13) is utilized in the governing equation (4), the solution gains significant regularity if a multiple of the initial solution is subtracted from it. This difference is continuous or even continuously differentiable depending on the decay of the eigenfunctions. Next we present our regularity result.

∗ Corollary 4 (Main Result 4: Discontinuities remain stationary) Let the eigenbasis (ek)k∈N satisfy the conditions of either Theorem6 or Theorem7. Then, the solution is uniquely decomposed into a multiple of the initial data and a smooth part as follows √ √ sin t c u(x, t) = cos t c u(x, 0) + u (x, t) + √ u (x, 0) + u (x, t), t ∈ , smth,1 c t smth,2 R where usmth,1(·, t) and usmth,2(·, t) are continuous or continuously diﬀerentiable functions, depending on whether (27) or (29) holds. Consequently, the solution to the homogeneous nonlocal wave equation is discontinuous if and only if the initial data is discontinuous. Hence, discontinuities remain stationary.

Proof. The statement is an implication of Main Result 2 and Main Result 3. More precisely, an implication of (25), Corollary3, Theorem6 part (ii), and Theorem7.

4. Conclusion This paper came about from the result in our recent paper [36] that the peridynamic governing operator is a bounded function of the classical governing operator on R. The peridynamic operator contains a convolution. In this paper, we generalized convolutions to bounded domains with the help of an eigenbasis obtained from the classical operator on the bounded domain. This opened us a gateway to incorporate local boundary conditions into nonlocal governing operators. B. Aksoylu, H.R. Beyer, and F. Celiker 18

Fig. 2: Approximate numerical solution to the nonlocal wave equation on [−1, 1] for time t ∈ [0, 20] with homogeneous Neumann (left) and Dirichlet (right) boundary conditions with vanishing initial velocity and discontinuous initial displacement. The size of nonlocality i.e., the size of the support of C(x) is 2−1. Note that the solution is discontinuous if and only if initial displacement is discontinuous. Hence, discontinuities are introduced only in the initial displacement. They remain stationary.

For the homogeneous nonlocal wave equation, we proved that continuity is preserved in time. Namely, for t ∈ R, we proved that the solution is discontinuous if and only if the initial data is discontinuous. This implies that the position of discontinuity is determined by the initial data and remains stationary. This is in contrast to the classical case, in which discontinuities propagate along characteristics. We depict the fact that discontinuities remain stationary in Figure2. Conversely, observe from Figure3 that when the initial solution is continuous, then it remains continuous in time. Observe also that away from discontinuities, the solution is smooth, an implication of Main Result 3. Details of how these numerical results are obtained, a comprehensive study regarding applications of the theory developed in this paper, and its numerical implementation are reported in [35]. We proved that the solution gains significant regularity if a multiple of the initial solution is subtracted from it. This difference is continuous or even continuously differentiable depending on the decay of the eigenbasis functions. The relation between the abstract convolution and the kernel function C(x) is addressed in [43]. Therein, we present modified governing operators that agree with the original PD operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. The main ingredients are antiperiodic and periodic extensions of kernel functions together with even and odd parts of functions. To the best of our knowledge, this is the first systematic approach of incorporating local BC into nonlocal problems governed by bounded operators. Regulating functions are the key for capturing the essence of the underlying physics. The creation of a map from regulating functions to physical situations is an exciting research endeavor for future research. We conclude that we added valuable tools to the arsenal of methods to treat nonlocal problems.

A Inhomogeneous Nonlocal Wave Equation This section is dedicated to the study of the inhomogeneous wave equation in our framework provided in Sec. 2.1.. B. Aksoylu, H.R. Beyer, and F. Celiker 19

Fig. 3: Approximate numerical solution to the nonlocal wave equation on [−1, 1] for time t ∈ [0, 100] with homogeneous Neumann (left) and Dirichlet (right) boundary conditions with vanishing initial velocity and continuous initial displacement. The size of nonlocality, i.e., the size of the support of C(x) is 2−3. Note that the solution is continuous if and only if initial displacement is continuous.

A1. Representation of the Solution For vanishing initial data, we present the solution of the inhomogeneous wave equation with abstract governing operator f(A) corresponding to continuous inhomogeneity.

Theorem 8 Let A : D(A) → X be positive and b : → L2 (Ω) be continuous. Furthermore, let f : σ(A) → R C R be bounded. Then, v : R → X, deﬁned by Z sin (t − τ)pf(A) v(t) := p b(τ) dτ, (30) It f(A) where R denotes weak integration in X, ( [0, t] if t > 0 It := , (31) [t, 0] if t < 0

is twice continuously diﬀerentiable, such that

00 v (t) + f(A) v(t) = b(t), t ∈ R, v(0) = v0(0) = 0.

Proof. See the proof in [36, Thm. 3]. The general inhomogeneous solution is given by the superposition of the general homogeneous solution ∗ with v in (32). We calculate the expansion coeﬃcients hek|v(t)i2 , k ∈ N with respect to Hilbert basis corresponding to A: t p Z sin (t − τ) f(λk) hek|v(t)i2 = p hek|b(τ)i dτ. 0 f(λk) Hence, the explicit expression of v in (30) is the following

∞ Z t p X h sin (t − τ) f(λk) i v(t) = p hek|b(τ)i dτ ek. (32) k=1 0 f(λk) B. Aksoylu, H.R. Beyer, and F. Celiker 20

Reﬂecting on Remark5, for the PD governing operator c − C∗, the expression of v in (32) takes the form

∞ Z t p X h sin (t − τ) c − hek|Ci i v(t) = p hek|b(τ)i dτ ek. k=1 0 c − hek|Ci

A2. Supporting Estimates Used in Satisfying Boundary Conditions for the Inhomogeneous Equation In order to treat the inhomogeneous equation, we need supporting estimates regarding the decay of eigenvalues of the solution operator. The following estimate provides the foundation for such an estimate.

Lemma 3 Let c > 0 and λ 6 min{c, 1}. Then for every t ∈ R √ √ t2 |t| | cos(t c − λ ) − cos(t c )| + √ |λ|, 6 2c c √ √ 2 sin(t c − λ ) sin(t c ) t |t| √ − √ 6 + √ |λ|. c − λ c 6c 2 c Proof. For λ < c, after some algebraic manipulations, we have √ √ √ √ √ √ √ √ cos(t c − λ ) − cos(t c ) = {cos(t [ c − λ − c ]) − 1} cos(t c ) − sin(t [ c − λ − c ]) sin(t c ) t √ √ √ √ √ √ = −2 sin2 [ c − λ − c ] cos(t c ) − sin(t [ c − λ − c ]) sin(t c ) 2 and hence,

√ √ t2 √ √ √ √ | cos(t c − λ ) − cos(t c )| [ c − λ − c ]2 + |t| | c − λ − c | 6 2 2 t h −λ i2 −λ = √ √ + |t| √ √ 2 c − λ + c c − λ + c t2λ2 |t| |λ| + √ . 6 2c c

Furthermore, assuming x, y > 0, after some algebraic manipulations, we have

sin(x) sin(y) Z 1 − = {[cos((x − y)γ) − 1] cos(yγ) − sin((x − y)γ) sin(yγ)} dγ x y 0 Z 1 h x − y i = − 2 sin2 γ cos(yγ) − sin((x − y)γ) sin(yγ) dγ, 0 2 and hence Z 1 sin(x) sin(y) 2x − y − 6 −2 sin γ cos(yγ) − sin((x − y)γ) sin(yγ) dγ x y 0 2 2 Z 1 Z 1 2 |x − y| 2 |x − y| |x − y| 6 γ dγ + |x − y| γ dγ = + . 2 0 0 6 2 We arrive at √ √ √ √ 2 √ 2 √ sin(t c − λ ) sin(t c ) t | c − λ − c| |t| | c − λ − c| √ − √ 6 + c − λ c 6 2 2 t h −λ i2 |t| −λ = √ √ + √ √ 6 c − λ + c 2 c − λ + c t2λ2 |t| |λ| + √ . 6 6c 2 c B. Aksoylu, H.R. Beyer, and F. Celiker 21

This completes the proof. The following result is related to decay of eigenvalues for the Hilbert-Schmidt part of the solution operator. It is used in proving the smoothing property of Hilbert-Schmidt operators with BC not involving derivatives and the ones with derivatives; see Theorem 10 and Theorem 11, respectively. √ Theorem 9 Let (X, h | i) be a non-trivial complex Hilbert space, the complex square-root function, with domain C \ ((−∞, 0] × {0}), c > 0 and t ∈ R. Furthermore, let C∗ ∈ L(X,X) be Hilbert-Schmidt, ek k∈ ∗ N a corresponding basis of eigenfunctions, and λk, ek be the eigenpair. (i) Then, lim λk = 0. (33) k→∞

∗ ∗ (ii) If N ∈ N is such that λk 6 min{c, 1} for every k ∈ N satisfying k > N, then √ √ t2 |t| cos t c − C∗ − cos(t c ) (λ ) + √ |λ |, k 6 2c c k √ √ sin t c − C∗ sin(t c ) t2 |t| √ − √ (λk) 6 + √ |λk|. c − C∗ c 6c 2 c

Proof. Part (i): First, we note that C ∗ ek = λk ek. Then, for every ξ ∈ X,

∞ ∞ ∞ X X X C ∗ ξ = C ∗ hek|ξi2 ek = hek|ξi2 C ∗ ek = λk hek|ξi2 ek. k=1 k=1 k=1

In particular, since for every k ∈ N∗

2 2 2 kC ∗ ekk2 = kλkekk2 = |λk|

2 and C∗ is Hilbert-Schmidt, |λk| ∗ is summable. As a consequence, the result (33) holds. k∈N The statement of Part (ii) is a direct consequence of Lemma3.

A3. Satisfying Boundary Conditions for the Inhomogeneous Equation Let the assumptions of Theorem6 hold. Then, for inhomogeneous equations, Hilbert-Schmidt property leads to a smoothing property in the following sense.

Theorem 10 (Main Result 3: Smoothing property of Hilbert-Schmidt operators for the inhomogeneous equation) Let b : → L2 (Ω) be continuous. If f is real-valued such that f(A) is Hilbert- R C Schmidt and f(A) 6 c, then the following statements hold. (i) Let v : R → X be deﬁned by p Z sin (t − τ) c − f(A) v(t) := p b(τ) dτ, It c − f(A) R where denotes weak integration in X and It is deﬁned in (31). Then, ∞ p X h Z sin (t − τ) c − f(λk) i v(t) = p hek|b(τ)i dτ ek. (34) k=1 It c − f(λk)

(ii) Let v : → L2 (Ω) be deﬁned by c R C √ Z sin (t − τ) c vc(t) := √ b(τ) dτ. It c B. Aksoylu, H.R. Beyer, and F. Celiker 22

Then, for every t ∈ R, v(t) − vc(t) has an extension to a continuous function on Ω, and for every limit point x0 of Ω

∞ X h Z i lim [v(t) − vc(t)](y) = Sk(t − τ) hek|b(τ)i dτ lim ek(y) y→x0 y→x0 k=1 It where p √ sin (t − τ) c − f(λk) sin (t − τ) c Sk(t − τ) := p − √ . c − f(λk) c

Proof. Part (i): First, for all t ∈ R and every k ∈ N∗, it follows from the spectral theorem for densely- deﬁned, linear and self-adjoint in Hilbert spaces that p Z sin (t − τ) c − f(A) hek|v(t)i = hek| p b(τ)i dτ It c − f(A) p Z sin (t − τ) c − f(A) = h p ek|b(τ)i dτ It c − f(A) p Z sin (t − τ) c − f(λk) = p hek|b(τ)i dτ, It c − f(λk)

and hence (34) holds. ∗ Part (ii): If k ∈ N is such that f(λk) 6 min{c, 1}, then by using Theorem9 Part (ii), we obtain h(t − τ)2 |t − τ|i S (t − τ) + √ |f(λ )| k 6 6c 2 c k

and hence Z Z h(t − τ)2 |t − τ|i

Sk(t − τ) hek|b(τ)i dτ 6 + √ |f(λk)| |hek|b(τ)i| dτ It It 6c 2 c Z 6 kt,c |f(λk)| |hek|b(τ)i| dτ, It

t2 |t| where k := + √ . For m, m0 ∈ ∗ satisfying N m m0, where N ∈ ∗ is suﬃciently large, we have t,c 6c 2 c N 6 6 N

m0 m0 X Z Z h X i Sk(t − τ) hek|b(τ)i dτ ek 6 K kt,c |f(λk)| |hek|b(τ)i|dτ b ∞ k=m It It k=m 0 0 m 1/2 Z m 1/2 h X 2i h X 2i 6 K kt,c |f(λk)| |hek|b(τ)i| dτ k=m It k=m 0 m 1/2 Z h X 2i 6 K kt,c |f(λk)| kb(τ)k2 dτ. k=m It As a consequence, N X Z Sk(t − τ) hek|b(τ)i dτ ek b N∈ ∗ k=1 It N B. Aksoylu, H.R. Beyer, and F. Celiker 23

is a Cauchy sequence in (BC(Ωb, C), k k∞) and hence converges uniformly to an extension of v(t) − vc(t) which is bounded continuous on Ω.b In particular, this implies that for every limit point x0 of Ω that N X Z lim [v(t) − vc(t)](y) = lim lim Sk(t − τ) hek|b(τ)i dτ ek(y) y→x0 y→x0 N→∞ k=1 It N X Z = lim lim Sk(t − τ) hek|b(τ)i dτ ek(y) N→∞ y→x0 k=1 It ∞ X Z = Sk(t − τ) hek|b(τ)i dτ lim ek(y) . y→x0 k=1 It For the last step, see [42, Thm 2.41]. Under the assumptions in Theorem7, analogous to the homogeneous case, the Hilbert-Schmidt property leads to smoothing of the solution of the inhomogeneous equation. Theorem 11 (Main Result 3: Smoothing property of Hilbert-Schmidt operators for the inhomogeneous equation with boundary conditions involving derivatives) If f is real-valued such that 1 f(A) 6 c and v, vc are deﬁned as in Theorem 10, then for every t ∈ R, v(t) − vc(t) ∈ C (Ω, C) and for every limit point x0 of Ω ∞ X Z lim [v(t) − vc(t)](y) = Sk(t − τ) hek|b(τ)i dτ lim ek(y) and y→x0 y→x0 k=1 It ∞ Z 0 X 0 lim [v(t) − vc(t)] (y) = Sk(t − τ) hek|b(τ)i dτ lim ek(y) . y→x0 y→x0 k=1 It 0 ∗ 0 ∗ Proof. For t ∈ R, m, m ∈ N satisfying N 6 m 6 m , where N ∈ N is suﬃciently large, then by using Theorem9 Part (ii), we obtain

is a Cauchy sequence in (C(Ωb, C), k k∞) and hence converges uniformly to a continuous function on Ω.b In particular, this implies that v(t) − vc(t) is continuously diﬀerentiable and for every limit point x0 of Ω that N Z 0 X 0 lim [v(t) − vc(t)] (y) = lim lim Sk(t − τ) hek|b(τ)i dτ ek(y) y→x0 y→x0 N→∞ b k=1 It N Z X 0 = lim lim Sk(t − τ) hek|b(τ)i dτ ek(y) N→∞ y→x0 b k=1 It ∞ Z X 0 = Sk(t − τ) hek|b(τ)i dτ lim ek(y) . y→x0 b k=1 It For the last step, see [42, Thm 2.41]. B. Aksoylu, H.R. Beyer, and F. Celiker 24

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