Nonlocal Operators with Local Boundary Conditions: An Overview
Abstract
We present novel governing operators in arbitrary dimension for nonlocal diffusion in homogeneous media. The operators are inspired by the theory of peridynamics (PD). They agree with the original PD operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of kernel functions together with even and odd parts of bivariate functions. We present different types of BC in 2D which include pure and mixed combinations of Neumann and Dirichlet BC. Our construction is systematic and easy to follow. We provide numerical experiments that validate our theoretical findings. When our novel operators are extended to vectorvalued functions, they will allow the extension of PD to applications that require local BC. Furthermore, we hope that the ability to enforce local BC provides a remedy for surface effects seen in PD.
We recently proved that the nonlocal diffusion operator is a function of the classical operator. This observation opened a gateway to incorporate local BC to nonlocal problems on bounded domains. The main tool we use to define the novel governing operators is functional calculus, in which we replace the classical governing operator by a suitable function of it. We present how to apply functional calculus to general nonlocal problems in a methodical way.
Keywords
Nonlocal wave equation Nonlocal operator Peridynamics Boundary condition Integral operatorIntroduction
When PD is considered, the dimension of u must be equal to that of x. In that case, the governing operator in (1) restricted to 1D corresponds to the bondbased linearized PD; see Silling et al. (2003, Eq. 23) and Weckner and Abeyaratne (2005, Eq. 3). For the discussion of PD, it is implied that \(u,x \in {\mathbb {R}}\). The case of \(u \in {\mathbb {R}}\) and \(x \in {\mathbb {R}}^d\) corresponds to nonlocal diffusion (Du et al., 2012; Seleson et al., 2013).
Our approach to nonlocal problems is fundamentally different because we exclusively want to use local BC. In Beyer et al. (2016), one of our major results was the finding that the governing operator of PD equation in \({\mathbb {R}}\) and nonlocal diffusion in \({\mathbb {R}}^d\) are functions of the Laplace operator. This result opened the path to the introduction of local boundary conditions into PD theory. Since PD is a nonlocal theory, one might expect only the appearance of nonlocal BC while employing \({\mathcal {L}}_{\text{orig}}\) as the governing operator. In the original PD formulation, the concept of local BC does not apply to PD. Instead, external forces must be supplied through the loading force density (Silling, 2000, p.201). On the other hand, we demonstrate that the anticipation that local BC are incompatible with nonlocal operators is not quite correct. Our novel operators present an alternative to nonlocal BC, and we hope that the ability to enforce local BC will provide a remedy for surface effects seen in PD; see Madenci and Oterkus (2014, Chap. 4, 5, 7, and 12) and Kilic (2008), Mitchell et al. (2015). Furthermore, our approach will provide us the capability to solve important elasticity problems that require local BC such as contact, shear, and traction.
We studied various aspects of local BC in nonlocal problems (Aksoylu et al., 2017a,b, 2016, 2017, Submitted; Beyer et al., 2016). Building on Beyer et al. (2016), we generalized the results in \({\mathbb {R}}\) to bounded domains (Aksoylu et al., 2017a,b), a critical feature for all practical applications. In Aksoylu et al. (2017b), we laid the theoretical foundations, and in Aksoylu et al. (2017a), we applied the foundations to prominent BC such as Dirichlet and Neumann, as well as presented numerical implementation of the corresponding wave propagation. In Aksoylu et al. (2017), we constructed the first 1D operators that agree with the original bondbased PD operator in the bulk of the domain and simultaneously enforce local Neumann and Dirichlet BC which we denote by \({\mathcal {M}}_{\mathtt {N}}\) and \({\mathcal {M}}_{\mathtt {D}}\), respectively. We carried out numerical experiments by utilizing \({\mathcal {M}}_{\mathtt {N}}\) and \({\mathcal {M}}_{\mathtt {D}}\) as governing operators in Aksoylu et al. (2017a). In Aksoylu et al. (2016), we studied other related governing operators. In Aksoylu and Kaya (2018), we study the condition numbers of the novel governing operators. Therein, we prove that the modifications made to the operator \({\mathcal {L}}_{\text{orig}}\) to obtain the novel operators are minor as far as the condition numbers are concerned.
Our approach is not limited to PD; the abstractness of the theoretical methods used allows generalization to other nonlocal theories. Our approach presents a unique way of combining the powers of abstract operator theory with numerical computing (Aksoylu et al., 2017a). Similar classes of operators are used in numerous applications such as nonlocal diffusion (AndreuVaillo et al., 2010; Du et al., 2012; Seleson et al., 2013), image processing (Gilboa and Osher, 2008), population models, particle systems, phase transition, and coagulation. See the review and news articles Du et al. (2012, 2014), and Silling and Lehoucq (2010) for a comprehensive discussion and the book (Madenci and Oterkus, 2014). In addition, see the studies dedicated to conditioning analysis, domain decomposition and variational theory (Aksoylu and Kaya, 2018; Aksoylu and Mengesha, 2010; Aksoylu and Parks, 2011; Aksoylu and Unlu, 2014), discretization (Aksoylu and Unlu, 2014; Emmrich and Weckner, 2007; Tian and Du, 2013), and kernel functions (Mengesha and Du, 2013; Seleson and Parks, 2011).
The rest of the paper is structured as follows. In section “The Novel Operators in 2D”, first we prove that the operators \({\mathcal {L}}_{\text{orig}}\) and \({\mathcal {L}}\) agree in the bulk in 2D. We define the novel operators using orthogonal projections on bivariate functions for which we utilize the periodic, antiperiodic, and mixed extensions of the kernel function C(x). We give the main theorem in 2D. In section “Operators in 1D”, first we prove that the novel operators are selfadjoint. In 1D, we give the main theorem which states they all agree with \({\mathcal {L}}_{\text{orig}}\) in the bulk and simultaneously enforce the corresponding local BC. In section “The Construction of 2D Operators”, we exploit the properties of the operators in 1D to construct the novel operators in 2D. We transfer the agreement in the bulk property established for univariate functions to bivariate ones and eventually prove that the novel operators agree with \({\mathcal {L}}_{\text{orig}}\) in the bulk in 2D. In section “Verifying the Boundary Conditions”, we make use of the Leibniz rule , the Fubini theorem, and the Lebesgue dominated convergence theorem to prove that the novel operators enforce the local BC stated in the main theorem. In section “Operators in Higher Dimensions”, we present the operators in 3D which can be easily extended to arbitrary dimension. In section “Numerical Experiments”, we report the numerical experiments. In section “The Treatment of General Nonlocal Problems Using Functional Calculus”, we present the treatment of general nonlocal problems using functional calculus. We conclude in section “Conclusion”.
The Novel Operators in 2D
Lemma 1
Proof
Building on our 1D construction in Aksoylu et al. (2017), in higher dimensions, we discovered the operators that enforce local pure and mixed Neumann and Dirichlet BC. We present the main theorem in 2D with the following 4 types of BC.
Theorem 1 (Main Theorem in 2D)
Proof
The proofs of agreement in the bulk and the verification of BC are given in sections “The Construction of 2D Operators” and “Verifying the Boundary Conditions”, respectively . □
Remark 1
Although we assume a separable kernel function as in (11), note that we do not impose a separability assumption on the solution u(x, y).
Operators in 1D
Lemma 2
Proof
The value of each extension of the function C
Interval  \({\widehat {C}}_{\mathtt {p}}(x)\)  \({\widehat {C}}_{\mathtt {a}}(x)\)  \({\widehat {C}}_{{\mathtt {p}}{\mathtt {a}}}(x)\)  \({\widehat {C}}_{{\mathtt {a}}{\mathtt {p}}}(x)\) 

x ∈ [−2, −2 + δ)  μ(x + 2)  − μ(x + 2)  μ(x + 2)  − μ(x + 2) 
x ∈ [−2 + δ, −δ]  0  0  0  0 
x ∈ (−δ, δ)  μ(x)  μ(x)  μ(x)  μ(x) 
x ∈ [δ, 2 − δ]  0  0  0  0 
x ∈ (2 − δ, 2]  μ(x − 2)  − μ(x − 2)  − μ(x − 2)  μ(x − 2) 
We present a commutativity property that allows us to identify the kernel functions associated with the operators \({\mathcal {M}}_{\mathtt {N}}\) and \({\mathcal {M}}_{\mathtt {D}}\).
Lemma 3
Proof
Remark 2
The above commutativity property plays an important role in determining the spectrum of the operators \({\mathcal {M}}_{\mathtt {N}}\) and \({\mathcal {M}}_{\mathtt {D}}\); see Aksoylu and Kaya (2018). It also helps in identifying the associated kernel functions; see (22) and (23). Note that the above commutativity property does not hold for the operators \({\mathcal {C}}_{{\mathtt {p}}{\mathtt {a}}}\) and \({\mathcal {C}}_{{\mathtt {a}}{\mathtt {p}}}\). Identification of the associated kernel functions can be done by direct manipulation.
Theorem 2 (Main Theorem in 1D)
We define the operators \({\mathcal {M}}_{\mathtt {N}}\), \({\mathcal {M}}_{\mathtt {D}}\), \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\), and \({\mathcal {M}}_{{\mathtt {D}}{\mathtt { N}}}\) as bounded, linear operators. More precisely, \({\mathcal {M}}_{\mathtt {D}},~ {\mathcal {M}}_{\mathtt {N}}, ~{\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}, ~{\mathcal {M}}_{{\mathtt {D}}{\mathtt {N}}} \in L(X,X)\) where X = L ^{2}(Ω) ∩ C ^{1}(∂Ω). For \({\mathcal {M}}_{\mathtt {D}}\), the choice of X can be relaxed as L ^{2}(Ω) ∩ C ^{0}(∂Ω). This choice is implied when we study \({\mathcal {M}}_{\mathtt {D}}\). The assumptions for the operators \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\) and \({\mathcal {M}}_{{\mathtt {D}}{\mathtt {N}}}\) are also implied in a similar way.
Imposing Neumann (also periodic and antiperiodic) BC requires differentiation. For technical details regarding differentiation under the integral sign, see the discussion on the Leibniz rule in Aksoylu et al. (2017) whose proof relies on the Lebesgue dominated convergence theorem . In addition, the limit in the definition of the Dirichlet BC can be interchanged with the integral sign, again by the Lebesgue dominated convergence theorem.
Remark 3
When we assume homogeneous Neumann and Dirichlet BC on u, then the operators \({\mathcal {M}}_{\mathtt {N}}\) and \({\mathcal {M}}_{\mathtt {D}}\) enforce homogeneous Dirichlet and Neumann BC, respectively. More precisely, for u(±1) = 0 and u′(±1) = 0, we obtain \(\displaystyle\frac {d}{dx}{\mathcal {M}}_{\mathtt {N}} u(\pm 1) = 0\) and \({\mathcal {M}}_{\mathtt {D}} u(\pm 1) = 0\), respectively. The same line of argument applies to the operators \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\) and \({\mathcal {M}}_{{\mathtt {D}}{\mathtt {N}}}\).
Remark 4
The boundedness of \({\mathcal {M}}_{\mathtt {N}}\), \({\mathcal {M}}_{\mathtt {D}}\), \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\), and \({\mathcal {M}}_{{\mathtt {D}}{\mathtt { N}}}\) follow from the choices of (5) and (24). In addition, all of them fall into the class of integral operators ; hence, their selfadjointness follows from the fact that the corresponding kernels are symmetric (due to evenness of C), i.e., K _{ BC }(x, x′) = K _{ BC }(x′, x) and BC ∈{N, D, ND, DN}. The cases of BC ∈{ND, DN} are more involved than the rest. One useful identity is \({\widehat {C}}_{{\mathtt {a}}{\mathtt {p}}}({x'}x) = {\widehat {C}}_{{\mathtt {p}}{\mathtt {a}}}({x'}+x)\).
In the upcoming proofs, we want to report a minor caveat. We use \({\widehat {C}}_{\mathtt {a}}({x'}+1) = {\widehat {C}}_{\mathtt {a}}({x'}1)\) which holds for x′≠ 0. For x′ = 0, i.e., \({\widehat {C}}_{\mathtt {a}}({x'}+1) = C(1) \neq  C(1) = {\widehat {C}}_{\mathtt {a}}({x'}1)\). Since x′ = 0 is only a point, it does not change the value of the integral. We choose not to point it out each time we run into this case.
Proof (Proof of Theorem 2)
 The operator \({\mathcal {M}}_{\mathtt {N}}\): First we remove the points at which the partial derivative of K _{ N }(x, x′) does not exist from the set of integration. Note that such points form a set of measure zero and, hence, do not affect the value of the integral. We differentiate both sides of (21). In Aksoylu et al. (2017), we had proved that the differentiation in the definition of the Neumann BC can interchange with the integral. We can differentiate the integrand K _{ N }(x, x′) piecewise and obtainwhere$$\displaystyle \begin{aligned} \frac{d}{dx} \big[ \big( {\mathcal{M}}_{\mathtt{N}}  c \big) u \big] (x) =  \int_\varOmega \frac{\partial K_{\mathtt{N}}}{\partial x}(x,{x'}) u({x'})d{x'}, \end{aligned} $$(29)We check the boundary values by plugging x = ±1 in (29).$$\displaystyle \begin{aligned} \frac{\partial K_{\mathtt{N}}}{\partial x}(x,{x'}) = \frac{1}{2} \big\{ \big[ {\widehat{C}}_{\mathtt{p}}^{\prime}({x'}x) + {\widehat{C}}_{\mathtt{p}}^{\prime}({x'}+x)\big] + \big[{\widehat{C}}_{\mathtt{a}}^{\prime}({x'}x)  {\widehat{C}}_{\mathtt{a}}^{\prime}({x'}+x) \big] \big\}. \end{aligned}$$The functions \({\widehat {C}}_{\mathtt {p}}^{\prime }\) and \({\widehat {C}}_{\mathtt {a}}^{\prime }\) are 2periodic and 2antiperiodic because they are the derivatives of 2periodic and 2antiperiodic functions, respectively. Hence,$$\displaystyle \begin{aligned} \frac{d}{dx} \big[ \big( {\mathcal{M}}_{\mathtt{N}}  c \big) u \big] (\pm 1) =  \int_\varOmega \frac{\partial K_{\mathtt{N}}}{\partial x}(\pm 1,{x'}) u({x'})d{x'}. \end{aligned} $$(30)Hence, the integrand in (30) vanishes, i.e.,$$\displaystyle \begin{aligned} {\widehat{C}}_{\mathtt{p}}^{\prime} ({x'} \mp 1) = {\widehat{C}}_{\mathtt{p}}^{\prime} ({x'} \pm 1) \quad\text{and}\quad {\widehat{C}}_{\mathtt{a}}^{\prime} ({x'} \mp 1) =  {\widehat{C}}_{\mathtt{a}}^{\prime} ({x'} \pm 1). \end{aligned}$$Therefore, we arrive at$$\displaystyle \begin{aligned} \frac{\partial K_{\mathtt{N}}}{\partial x}(\pm 1,{x'}) = 0. \end{aligned}$$When we assume that u satisfies homogeneous Neumann BC, i.e., u′(±1) = 0, we conclude that the operator \({\mathcal {M}}_{\mathtt {N}}\) enforces homogeneous Neumann BC as well.$$\displaystyle \begin{aligned} \frac{d}{dx} {\mathcal{M}}_{\mathtt{N}} u(\pm 1) = c u'(\pm 1). \end{aligned}$$
 The operator \({\mathcal {M}}_{\mathtt {D}}\): By the Lebesgue dominated convergence theorem , the limit in the definition of the Dirichlet BC can be interchanged with the integral. Now, we check the boundary values by plugging x = ±1 in (23).Since \({\widehat {C}}_{\mathtt {p}}\) and \({\widehat {C}}_{\mathtt {a}}\) are 2periodic and 2antiperiodic, respectively, we have$$\displaystyle \begin{aligned} \big( {\mathcal{M}}_{\mathtt{D}}  c \big) u(\pm 1) =  \int_\varOmega K_{\mathtt{D}}(\pm 1, {x'}) u({x'})d{x'}. \end{aligned} $$(31)Hence, the integrand in (31) vanishes, i.e., K _{ D }(±1, x′) = 0. Therefore, we arrive at$$\displaystyle \begin{aligned} {\widehat{C}}_{\mathtt{p}} ({x'} \mp 1) = {\widehat{C}}_{\mathtt{p}} ({x'} \pm 1) \quad\text{and}\quad {\widehat{C}}_{\mathtt{a}} ({x'} \mp 1) =  {\widehat{C}}_{\mathtt{a}} ({x'} \pm 1). \end{aligned}$$When we assume that u satisfies homogeneous Dirichlet BC, i.e., u(±1) = 0, we conclude that the operator \({\mathcal {M}}_{\mathtt {D}}\) enforces homogeneous Dirichlet BC as well.$$\displaystyle \begin{aligned} {\mathcal{M}}_{\mathtt{D}} u(\pm 1) = c u(\pm 1). \end{aligned}$$
 The operator \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\): First we prove that \({\mathcal {C}}_{{\mathtt {a}}{\mathtt {p}}} P_e u(+1)=0\). We use a change of variable in the second piece. Then, we split the integrals into two parts as follows:For x′∈ [−1, 0], we have x′− 1 ∈ [−2, −1]. By using the definition of \({\widehat {C}}_{{\mathtt {a}}{\mathtt {p}}}\) and the evenness of C, we obtain$$\displaystyle \begin{aligned} \begin{aligned} {\mathcal{C}}_{{\mathtt{a}}{\mathtt{p}}} P_e u(+1) & = \frac{1}{2} \int_{1}^0 \big[ {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) + {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) \big] P_e u({x'}) d{x'} \\ & \quad + \frac{1}{2} \int_0^1 \big[ {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) + {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) \big] P_eu({x'}) d{x'}. \end{aligned} \end{aligned} $$(32)For x′∈ [0, 1], we have x′− 1 ∈ [−1, 0]. By using the definition of \({\widehat {C}}_{{\mathtt {a}}{\mathtt {p}}}\) and the evenness of C, we obtain$$\displaystyle \begin{aligned} {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) = {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}+1) = C({x'}+1) = C({x'}1) =  {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1). \end{aligned} $$(33)Combining (33) and (34) with (32), we conclude that \({\mathcal {C}}_{{\mathtt {a}}{\mathtt {p}}} P_e u(+1)=0\). Similarly, we can conclude that \({\mathcal {C}}_{{\mathtt {p}}{\mathtt {a}}} P_o u(+1)=0\). Consequently, we arrive at$$\displaystyle \begin{aligned} {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1) = C({x'}1) = C({x'}+1) = {\widehat{C}}_{{\mathtt{a}}{\mathtt{p}}}({x'}1). \end{aligned} $$(34)$$\displaystyle \begin{aligned} {\mathcal{C}}_{{\mathtt{N}}{\mathtt{D}}} u(+1) = 0. \end{aligned}$$We prove that \(\displaystyle\frac {d}{dx} {\mathcal {C}}_{{\mathtt {p}}{\mathtt {a}}} P_o u(1)=0\). We use a change of variable in the second piece. Then, we split the integrals into two parts as follows:For x′∈ [−1, 0], we have x′ + 1 ∈ [0, 1]. By using the definition of \({\widehat {C}}_{{\mathtt {p}}{\mathtt {a}}}\) and the oddness of C′, we obtain$$\displaystyle \begin{aligned} \begin{aligned} \frac{d}{dx} {\mathcal{C}}_{{\mathtt{p}}{\mathtt{a}}} P_o u(1) & =  \frac{1}{2} \int_{1}^0 \big[ {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1)  {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1) \big] P_o u({x'}) d{x'} \\ & \quad  \frac{1}{2} \int_0^1 \big[ {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1)  {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1) \big] P_ou({x'}) d{x'}. \end{aligned} \end{aligned} $$(35)For x′∈ [0, 1], we have x′ + 1 ∈ [1, 2]. By using the definition of \({\widehat {C}}_{{\mathtt {p}}{\mathtt {a}}}\) and the oddness of C′, we obtain$$\displaystyle \begin{aligned} {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1) = C'({x'}+1) = C'({x'}1) = {\widehat{C}}^{\prime}_{{\mathtt{p}}{\mathtt{a}}}({x'}1) = {\widehat{C}}^{\prime}_{{\mathtt{p}}{\mathtt{a}}}({x'}+1). \end{aligned} $$(36)Combining (36) and (37) with (35), we conclude that \(\frac {d}{dx} {\mathcal {C}}_{{\mathtt {p}}{\mathtt {a}}} P_o u(1)=0\). Similarly, we can conclude that \(\frac {d}{dx} {\mathcal {C}}_{{\mathtt {a}}{\mathtt {p}}} P_e u(1)=0\). Consequently, we arrive at$$\displaystyle \begin{aligned} {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}+1) =  {\widehat{C}}_{{\mathtt{p}}{\mathtt{a}}}^{\prime}({x'}1) =  C'({x'}1) = C'({x'}+1) = {\widehat{C}}^{\prime}_{{\mathtt{p}}{\mathtt{a}}}({x'}+1).\end{aligned} $$(37)$$\displaystyle \begin{aligned} \frac{d}{dx} {\mathcal{C}}_{{\mathtt{N}}{\mathtt{D}}} u(1) = 0. \end{aligned}$$

The operator \({\mathcal {M}}_{{\mathtt {D}}{\mathtt {N}}}\): The proof is similar to the case of \({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}}\).
Remark 5
The Construction of 2D Operators
Remark 6
Verifying the Boundary Conditions
The operators \(\big ({\mathcal {M}}_{\mathtt {N}}  c \big )\), \(\big ({\mathcal {M}}_{\mathtt {D}} c \big )\), \(\big ({\mathcal {M}}_{{\mathtt {N}}{\mathtt {D}}, \, {\mathtt {N}}{\mathtt {D}}}  c \big )\), and \(\big ({\mathcal {M}}_{{\mathtt {N}}, \, {\mathtt {D}}{\mathtt {N}}}  c \big )\) given in (51), (52), (53), and (54), respectively, are the product of two 1D operators. As we mentioned, the limit in the definition of the BC can be interchanged with the integral sign due to the Lebesgue dominated convergence theorem and the Leibniz rule . Then, using the change in the order of integration as in (42) and (38), we can prove that the pure and mixed Neumann and Dirichlet BC are enforced.
Operators in Higher Dimensions
Numerical Experiments
The Treatment of General Nonlocal Problems Using Functional Calculus
Our main tool that allows us to incorporate local BC into nonlocal operators is functional calculus. More precisely, the novel governing operators are obtained by employing the functional calculus of selfadjoint operators, i.e., by replacing the classical governing operator A by a suitable function of A, f(A). We call f the regulating function. Since classical BC is an integral part of the classical operator, these BC are automatically inherited by f(A). 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.
 FC1.

Apply limit to the horizon parameter, i.e., δ → 0, to identify a local counterpart A of NL.
 FC2.

Apply the Fourier transform to “diagonalize” NL and A to obtain the corresponding spectra.
 FC3.

Read off the regulating function f by comparing the spectra of NL and A. Spectra on \({\mathbb {R}}^d\) are continuous.
 FC4.

Restrict A to Ω with a prescribed BC. Denote the new operator by A _{ BC }. Spectrum of A _{ BC }, σ(A _{ BC }) is now discrete. Find the eigenfunctions of A _{ BC }.
 FC5.

Define a generalized convolution as in (58) by using eigenfunctions of A _{ BC }.
 FC6.
 Rewrite (recycle) the regulating function with discrete spectrum.
 FC7.
 Construct f _{ BC }(A _{ BC }) using the spectral theorem. Namely, for \(u = \sum _k \langle {e_k^{\mathtt {BC}}u}\rangle e_k^{\mathtt {BC}}\), we have$$\displaystyle \begin{aligned} f_{\mathtt{BC}}(A_{\mathtt{BC}}) u = \sum_k f_{\mathtt{BC}}(\lambda_k^{\mathtt{BC}}) \langle e_k^{\mathtt{BC}}u\rangle_k^{\mathtt{BC}}. \end{aligned} $$(64)
 FC8.

Find a computationally feasible expression of f _{ BC }(A _{ BC }) such as an integral representation.
Now, we show how we use the FC steps to construct the governing operators \({\mathcal {M}}_{\mathtt {BC}}, ~{\mathtt {BC}} \in \{{\mathtt {p}}, {\mathtt {a}} \}\) in 1D. Namely, we want to verify \(f_{\mathtt {BC}}(A_{\mathtt {BC}})u = {\mathcal {M}}_{\mathtt {BC}} u\).
Remark 7
Fractional diffusion and fractional PDEs also fall into the class of nonlocal problems; see some of the recent developments (AndreuVaillo et al., 2010; Caffarelli et al., 2007; Di Nezza et al., 2012; Nochetto et al., 2015). There is a fundamental difference between these operators and ours: our governing operators are bounded. Note that the regulating function in (63) is bounded and that is why the application of the spectral theorem in (64) is valid. Since our ultimate goal is to capture discontinuities or cracks, we are mainly interested in bounded governing operators. Fractional operators become unbounded for such discontinuities, and hence, we exclude them from our discussion.
Conclusion
We presented novel governing operators in arbitrary dimension for nonlocal diffusion. The operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local BC. We presented methodically how to verify the BC by using a change in the order of integration. We presented different types of BC in 2D which include pure and mixed combinations of Neumann and Dirichlet BC. We presented numerical experiments for the nonlocal wave equation . We verified that the novel operators enforce local BC for all time. We also observed that the property we proved for 1D, namely, discontinuities remain stationary, also holds for 2D.
Our ongoing work aims to extend the novel operators to vectorvalued problems which will allow the extension of PD to applications that require local BC. Furthermore, we hope that our novel approach potentially will avoid altogether the surface effects seen in PD.
Footnotes
 1.
We do not explicitly denote the dimension of the domain Ω. The dimension is implied by the number of iterated integrals present in the operator. The domain Ω is equal to [−1, 1], [−1, 1] × [−1, 1], and [−1, 1] × [−1, 1] × [−1, 1] in 1D, 2D, and 3D, respectively.
Notes
Acknowledgements
Burak Aksoylu was supported in part by the European Commission Marie Curie Career Integration 293978 grant, and Scientific and Technological Research Council of Turkey (TÜB\(\dot {\mathrm {I}}\)TAK) MFAG 115F473 grant.
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