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On Nonlocal Problems with Inhomogeneous Local Boundary Conditions

Abstract

This work aims to provide a comprehensive treatment on how to enforce inhomogeneous local boundary conditions (BC) in nonlocal problems in 1D. In prior work, we have presented novel governing operators with homogeneous BC. Here, we extend the construction to inhomogeneous BC. The construction of the operators is inspired by peridynamics. The operators agree with the original peridynamic operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. We explain methodically how to construct forcing functions to enforce local BC and their relationship to initial values. We present exact solutions with both homogeneous and inhomogeneous BC and utilize the resulting error to verify numerical experiments. We explain the critical role of the Hilbert-Schmidt property in enforcing local BC rigorously. For the Neumann BC, we prescribe an interpolation strategy to find the appropriate value of the forcing function from its derivative. We also present numerical experiments with unknown solution and report the computed displacement and strain fields.

Introduction

We consider the following nonlocal wave equations with inhomogeneous local Dirichlet and local Neumann boundary conditions (BC), respectively:

$$ \begin{array}{@{}rcl@{}} u_{tt}^{\texttt{D}}(x,t) + \mathcal{M}_{\texttt{D}} u^{\texttt{D}}(x,t) & =& b^{\texttt{D}}(x,t), \quad (x,t) \in {\Omega} \times (0,T), \end{array} $$
(1.1a)
$$ \begin{array}{@{}rcl@{}} u^{\texttt{D}}(\pm 1,t) & = &\alpha_{\pm}^{\texttt{D}}(t), \end{array} $$
(1.1b)
$$ \begin{array}{@{}rcl@{}} u^{\texttt{D}}(x,0) & = &\phi_{\texttt{D}}(x), \end{array} $$
(1.1c)
$$ \begin{array}{@{}rcl@{}} u_{t}^{\texttt{D}}(x,0) & =& \psi_{\texttt{D}}(x), \end{array} $$
(1.1d)
$$ \begin{array}{@{}rcl@{}} u_{tt}^{\texttt{N}}(x,t) + \mathcal{M}_{\texttt{N}} u^{\texttt{N}}(x,t) & = &b^{\texttt{N}}(x,t), \quad (x,t) \in {\Omega} \times (0,T), \end{array} $$
(1.2a)
$$ \begin{array}{@{}rcl@{}} u_{x}^{\texttt{N}}(\pm 1,t) & = &\alpha_{\pm}^{\texttt{N}}(t), \end{array} $$
(1.2b)
$$ \begin{array}{@{}rcl@{}} u^{\texttt{N}}(x,0) & =& \phi_{\texttt{N}}(x), \end{array} $$
(1.2c)
$$ \begin{array}{@{}rcl@{}} u_{t}^{\texttt{N}}(x,0) & =& \psi_{\texttt{N}}(x), \end{array} $$
(1.2d)

on the domain Ω := (− 1, 1) for some T > 0 where the variable uBC represents the displacement. The problems Eqs. 1.1a1.1d and 1.2a1.2d fall into the class of initial boundary value problems. We have studied the above nonlocal wave equations with homogeneous local BC in prior work [1, 2, 7]. We extended the treatment to inhomogeneous BC in a preliminary study [8]. The primary purpose of this study is to present a comprehensive treatment.

The main theoretical contributions in this study are:

  1. (1)

    We present exact solutions and utilize the resulting error to verify numerical experiments.

  2. (2)

    We explain the critical role of Hilbert-Schmidt property in satisfying BC rigorously.

  3. (3)

    We provide the relationships between forcing function, boundary condition, and initial values used to enforce local BC.

  4. (4)

    For the Neumann BC, we prescribe an interpolation strategy to find the appropriate value of the forcing function from its derivative.

To the authors’ knowledge, our operators are the first nonlocal operators that can enforce local displacement and strain BC. When extended to vector valued problems, they will help apply peridynamics to problems that require local BC. The operators are inspired by the theory of peridynamics, a nonlocal formulation of continuum mechanics developed by Silling [18]. They agree with the original peridynamic operator in the bulk of the domain and simultaneously enforce local BC.

We studied various aspects of local BC in nonlocal problems [1,2,3,4,5,6,7, 9, 11]. Building on [11], we generalized the results in \(\mathbb {R}\) to a bounded domain [1, 2], a critical feature for all practical applications. In [2], we laid the theoretical foundations and in [1], we applied the foundations to numerically solve wave propagation problems using local BC. In [4], we constructed the first 1D operators that agree with the original bond-based peridynamic operator in the bulk of the domain and simultaneously enforce local Neumann or Dirichlet BC which we denote by \({\mathcal{M}}_{\texttt {N}}\) and \({\mathcal{M}}_{\texttt {D}}\), respectively. We carried out numerical experiments by utilizing \({\mathcal{M}}_{\texttt {N}}\) and \({\mathcal{M}}_{\texttt {D}}\) as governing operators in [1]. We extended the operators to higher dimensions in [7]. In [6], we methodically apply functional calculus to general nonlocal problems. In [9], we study the conditioning of nonlocal operators together with error analysis. In [5], we proved that the collocation methods presented in this work have optimal order convergence with respect to polynomial degree. Also in [5], we established the asymptotic compatibility of the collocation methods under consideration.

Our approach is not limited to peridynamics; 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 [1]. Nonlocal modeling is an emerging field. See the relevant review and news articles [13,14,15, 19] for a comprehensive discussion, and the book [17].

The rest of the paper is structured as follows. In Section 2, we outline the key steps to construct the nonlocal operators using functional calculus. In Section 3, we present the main construction for boundary value problems by providing the relationships between forcing function, BC, and initial values. In Section 4, we explain how the Hilbert-Schmidt property gives rise to uniform convergence, which in turn is used to satisfy BC rigorously. The series solutions from the classical theory cannot guarantee such rigor, and hence, qualify only as formal solutions. In Section 5, we present exact solutions with homogeneous BC. In Section 6, we present exact solutions with inhomogeneous BC using the method of shifting the data. In Section 7, we set the stage for numerical experiments by choosing kernel functions. We introduce the appropriate scaling so that the discretized nonlocal operator captures the discretized Laplace operator when δ = h. The discretized nonlocal operator enjoys the zero row sum property which can be spoiled due round-off for small δ. We carefully explain how to avoid it. Enforcing the Neumann BC involves taking the spatial derivative of the forcing function. However, the forcing function itself is present in the governing equation. In Section 8, we prescribe an interpolation strategy to find the appropriate value of the forcing function from its derivative. In Section 9, we present the implementation and the numerical experiments. Finally, we conclude in Section 10.

The Convolution and the Governing Operators

In this section, we explain the key steps in the construction of the governing operator \({\mathcal{M}}_{\texttt {BC}}\). We observe that the peridynamic governing operator contains a convolution operator. First, we construct the convolution operators \(\mathcal {C}_{\texttt {a}}\) and \(\mathcal {C}_{\texttt {p}}\) with antiperiodic and periodic BC, respectively, using the eigenfunctions

$$ e_{k}^{\texttt{a}}(x) := \frac{1}{\sqrt{2}} e^{i \pi (k+\frac12)x}, \quad k \in \mathbb{N}, \quad \text{and} \quad e_{k}^{\texttt{p}}(x) := \frac{1}{\sqrt{2}} e^{i \pi k x}, \quad k \in \mathbb{N}, $$

of the classical operator Aa and Ap in which the BC information is already encoded. For a given kernel function CL2(Ω), the convolution operator, for uL2(Ω), is defined as

$$ \mathcal{C}_{\texttt{BC}} u(x) := \sqrt{2} \sum\limits_{k \in \mathbb{N}} \langle{e_{k}^{\texttt{BC}}|C}\rangle \langle{e_{k}^{\texttt{BC}}|u}\rangle e_{k}^{\texttt{BC}}(x), \quad \texttt{BC} \in \{\texttt{a}, \texttt{p}\}, $$

where 〈⋅|⋅〉 denotes the L2(Ω) inner product. We define \(\mathbb {N}_{\texttt {D}} := \mathbb {N} \setminus \{0\}\) and \(\mathbb {N}_{\texttt {N}} := \mathbb {N}\). The operators \(\mathcal {C}_{\texttt {BC}}\) turn out to be bounded functions of the classical operator ABC, thereby maintaining the connection to ABC.

In this study, we consider only the operators \({\mathcal{M}}_{\texttt {D}}\) and \({\mathcal{M}}_{\texttt {N}}\) where D and N denote the Dirichlet and Neumann BC. Hence, in the rest of the discussion, we set BC ∈{D,N}. The operator \({\mathcal{M}}_{\texttt {BC}}\) is constructed using functional calculus on the classical self-adjoint operator ABC. We are in search of a suitable regulating function \(f_{\texttt {BC}}:\sigma (A_{\texttt {BC}}) \rightarrow \mathbb {R}\) that would connect the nonlocal operator \({\mathcal{M}}_{\texttt {BC}}\) to ABC, i.e., \({\mathcal{M}}_{\texttt {BC}} = f_{\texttt {BC}}(A_{\texttt {BC}})\). This regulating function should be bounded so that the end product \({\mathcal{M}}_{\texttt {BC}}\) is a bounded operator. Eventually, we end up with the nonlocal governing operator \({\mathcal{M}}_{\texttt {BC}}\) that is densely defined in L2(Ω) with a domain that encodes the prescribed BC, bounded, and self-adjoint. Therefore, the operator \({\mathcal{M}}_{\texttt {BC}}\) has a unique bounded extension to L2(Ω). Consequently, we find that a construction involving densely defined operators provides a suitable framework for treating local BC in the nonlocal wave equation.

In this work, the choice of fBC is inspired by the theory of peridynamics. In prior work, we discovered that the peridynamic governing operator for the case \({\Omega } = \mathbb {R}\) is a function of the classical operator [11]. We reuse that regulating function for the case of Ω = (− 1,1). Our choice of regulating functions is

$$ f_{\texttt{BC}}: \sigma(A_{\texttt{BC}}) \rightarrow \mathbb{R}, \quad f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}}) = \langle{1|C}\rangle - \sqrt{2} \left\{\begin{array}{ll} \langle{e_{k/2}^{\texttt{p}}|C}\rangle & \text{ if } k \in \mathbb{N}_{\texttt{BC}} \text{ is even}, \\ \langle{e_{(k-1)/2}^{\texttt{a}}|C}\rangle & \text{ if } k \in \mathbb{N}_{\texttt{BC}} \text{ is odd}. \end{array}\right. $$

Utilizing the convolution operators \(\mathcal {C}_{\texttt {a}}\) and \(\mathcal {C}_{\texttt {p}}\) obtained by functional calculus on Aa and Ap, respectively, defining c := 〈1|C〉, we proved in [1, 4] that

$$ \begin{array}{@{}rcl@{}} f_{\texttt{D}} (A_{\texttt{D}}) u^{\texttt{D}} &=& \left( c - \mathcal{C}_{\texttt{a}} P_{e} - \mathcal{C}_{\texttt{p}} P_{o} \right) u^{\texttt{D}} = \mathcal{M}_{\texttt{D}} u^{\texttt{D}}, \\ f_{\texttt{N}} (A_{\texttt{N}}) u^{\texttt{N}} &=& \left( c - \mathcal{C}_{\texttt{p}} P_{e} - \mathcal{C}_{\texttt{a}} P_{o} \right) u^{\texttt{N}} = \mathcal{M}_{\texttt{N}} u^{\texttt{N}}, \end{array} $$

where we denote the orthogonal projections that give the even and odd parts, respectively, by \(P_{e},P_{o}:~L^{2}({\Omega })~\rightarrow ~L^{2}({\Omega }), \) whose definitions are

$$ P_{e} u(x) := \frac{u(x) + u(-x)}{2}, \quad P_{o} u(x) := \frac{u(x) - u(-x)}{2}. $$

The crucial step in the construction of \({\mathcal{M}}_{\texttt {BC}}\) is the application of the spectral theorem for bounded operators. Namely, for \(u^{\texttt {BC}} = {\sum }_{k} \langle {e_{k}^{\texttt {BC}}|u^{\texttt {BC}}}\rangle e_{k}^{\texttt {BC}}\), we have

$$ \mathcal{M}_{\texttt{BC}} u^{\texttt{BC}} = f_{\texttt{BC}}(A_{\texttt{BC}}) u^{\texttt{BC}} = \sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}}) \langle{e_{k}^{\texttt{BC}}|u^{\texttt{BC}}}\rangle e_{k}^{\texttt{BC}}. $$
(2.1)

For an extended discussion on the treatment of general nonlocal problems using functional calculus, see [6].

An integral representation of the series (2.1) is more convenient for implementation. We gave such representations in [1] and the governing operators take the form

$$ \left( \mathcal{M}_{\texttt{BC}} - c \right) u^{\texttt{BC}}(x,t) = - {\int}_{\Omega} K_{\texttt{BC}}(x,x^{\prime})u^{\texttt{BC}}(x^{\prime},t) \mathrm{d}x^{\prime}, $$
(2.2)
$$ \begin{array}{@{}rcl@{}} K_{\texttt{D}}(x,x^{\prime}) & :=& \frac{1}{2} \left\{ \left[ \widehat{C}_{\texttt{a}}(x^{\prime}-x) + \widehat{C}_{\texttt{a}}(x^{\prime}+x) \right] + \left[ \widehat{C}_{\texttt{p}}(x^{\prime}-x) - \widehat{C}_{\texttt{p}}(x^{\prime}+x) \right] \right\}, \\ K_{\texttt{N}}(x,x^{\prime}) & := & \frac{1}{2} \left\{ \left[ \widehat{C}_{\texttt{p}}(x^{\prime}-x) + \widehat{C}_{\texttt{p}}(x^{\prime}+x) \right] + \left[ \widehat{C}_{\texttt{a}}(x^{\prime}-x) - \widehat{C}_{\texttt{a}}(x^{\prime}+x) \right] \right\}, \end{array} $$

where we denote the periodic and antiperiodic extensions of C(x) from (− 1,1) to (− 2,2), respectively, as follows

$$ \widehat{C}_{\texttt{p}}(x):= \left\{ \begin{array}{ll} C(x+2), & x \in (-2,-1), \\ C(x), & x \in (-1,1), \\ C(x-2), & x \in (1,2), \end{array} \right. \quad \widehat{C}_{\texttt{a}}(x):= \left\{ \begin{array}{llll} - & C(x+2), && x \in (-2,-1), \\ & C(x), && x \in (-1,1), \\ - & C(x-2), && x \in (1,2). \end{array} \right. $$

In addition, \(u_{x}^{\texttt {N}}\) represents strain.

Forcing Function, BC, and Initial Value Relationships

In order to find the suitable forcing function that enforces the prescribed BC, we need to identify the governing ordinary differential equation (ODE) on the boundary. For this identification, we assume that \(u^{\texttt {D}} \in \mathcal {C}^{2}(\overline {\Omega } \times [0,T])\), \(u^{\texttt {N}} \in \mathcal {C}^{3}(\overline {\Omega } \times [0,T])\), and \(b^{\texttt {D}} \in \mathcal {C}^{0}(\overline {\Omega } \times [0,T])\), \(b^{\texttt {N}} \in \mathcal {C}^{1}(\overline {\Omega } \times [0,T])\). Let us explain a crucial point in the choice of these subspaces and elaborate on the subspace of uD only. The case of Neumann BC easily follows. In fact, the solution uD(⋅, t) belongs to L2(Ω) for any t ∈ [0, T]. But, L2(Ω) ignores the values of u(⋅, t) on the boundary of Ω. In order to determine the boundary behavior of u(⋅, t), we visualize it as a limit of a sequence from \(\mathcal {C}^{2}(\overline {\Omega })\), a space that is aware of the function values on the boundary of Ω. We accomplish this by the density of \(\mathcal {C}^{2}(\overline {\Omega })\) in L2(Ω). Note that this density is in alignment with \({\mathcal{M}}_{\texttt {D}}\)’s property of being densely defined. So, when we write uD, we implicitly mean a sequence \(u_{n}^{\texttt {D}}(\cdot ,t) \in \mathcal {C}^{2}(\overline {\Omega })\) approaching uD(⋅, t), i.e., \(\lim _{n \to \infty } u_{n}^{\texttt {D}}(\cdot ,t) = u^{\texttt {D}}(\cdot ,t)\) in the L2(Ω)-norm. In this section, with a slight abuse of notation, we prefer to use uD(⋅, t) instead of \(u_{n}^{\texttt {D}}(\cdot ,t)\). The action of \({\mathcal{M}}_{\texttt {D}}\) on uD(⋅, t) is seen as

$$ \begin{array}{@{}rcl@{}} \mathcal{M}_{\texttt{D}} u^{\texttt{D}}(\cdot,t) & = & \mathcal{M}_{\texttt{D}} \lim_{n \to \infty} u_{n}^{\texttt{D}}(\cdot,t)\\ & = & \lim_{n \to \infty} \mathcal{M}_{\texttt{D}} u_{n}^{\texttt{D}}(\cdot,t) \quad \text{using the boundedness of } \mathcal{M}_{\texttt{D}}. \end{array} $$

For more rigorous discussion of the function spaces, see [8, Sec. 1].

We emphasize that our construction does not assume any smoothness on the initial displacement and initial velocity. We can treat

$$ u^{\texttt{BC}}(x,0), u_{t}^{\texttt{BC}}(x,0) \in L^{2}({\Omega}). $$
(3.1)

We have substantiated the validity of assumption (3.1) by choosing discontinuous initial displacement profiles for numerical experiments in 1D [1] as well as in 2D [7]. The construction only assumes the existence of the following limits

$$ \lim_{x \to \pm 1} u^{\texttt{D}}(x,t), \lim_{x \to \pm 1} u_{x}^{\texttt{N}}(x,t), \text{~and~} \lim_{x \to \pm 1} u_{t}^{\texttt{BC}}(x,t), $$

and they should be provided as boundary data to set up the problem.

On the boundary, denote the displacement, the strain and the forcing functions by

$$ \begin{array}{@{}rcl@{}} u_{\pm}^{\texttt{D}} (t) & := &\lim_{x \rightarrow \pm 1} u^{\texttt{D}}(x,t) \quad {\kern7.3pt}\text{and} \quad b_{\pm}^{\texttt{D}} (t) {\kern6.7pt}:= \lim_{x \rightarrow \pm 1} b^{\texttt{D}}(x,t),\\ u_{x,\pm}^{\texttt{N}} (t) & := & \lim_{x \rightarrow \pm 1} \frac{\partial u^{\texttt{N}}}{\partial x}(x,t) \quad \text{and} \quad b_{x,\pm}^{\texttt{N}} (t) := \lim_{x \rightarrow \pm 1} \frac{\partial b^{\texttt{N}}}{\partial x}(x,t). \end{array} $$

In order to investigate the behavior of the solution on the boundary, first we study the action of the governing operator \({\mathcal{M}}_{\texttt {BC}}\) on the boundary. By the Lebesgue Dominated Convergence Theorem and the design of the kernel functions \(K_{\texttt {BC}}(x,x^{\prime })\), we have

$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left( \mathcal{M}_{\texttt{D}} - c \right)u^{\texttt{D}}(x,t) & =& - \lim_{x \rightarrow \pm 1} {\int}_{\Omega} K_{\texttt{D}}(x,x^{\prime})u^{\texttt{D}}(x^{\prime},t) \mathrm{d}x^{\prime} \end{array} $$
(3.2)
$$ \begin{array}{@{}rcl@{}} & = & - {\int}_{\Omega} \lim_{x \rightarrow \pm 1} K_{\texttt{D}}(x,x^{\prime})u^{\texttt{D}}(x^{\prime},t) \mathrm{d}x^{\prime} = 0, \\ \lim_{x \rightarrow \pm 1} \frac{\partial}{\partial x} \left( \mathcal{M}_{\texttt{N}} - c \right)u^{\texttt{N}}(x,t) & =& - \lim_{x \rightarrow \pm 1} \frac{\partial}{\partial x} {\int}_{\Omega} K_{\texttt{N}}(x,x^{\prime})u^{\texttt{N}}(x^{\prime},t) \mathrm{d}x^{\prime}\\ & =& - {\int}_{\Omega} \lim_{x \rightarrow \pm 1} \frac{\partial K_{\texttt{N}}}{\partial x} (x,x^{\prime})u^{\texttt{N}}(x^{\prime},t) \mathrm{d}x^{\prime} = 0. \end{array} $$
(3.3)

The governing Eqs. 1.1a and 1.2a under the action of \(\lim _{x \rightarrow \pm 1}\) and \(\lim _{x \rightarrow \pm 1} \frac {\partial }{\partial x}\), respectively, reduce to the following ODE:

$$ \begin{array}{@{}rcl@{}} \frac{\mathrm{d}^{2} u_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + c u_{\pm}^{\texttt{D}} (t) & = & b_{\pm}^{\texttt{D}} (t), \quad {\kern6.5pt}t \in (0,T), \end{array} $$
(3.4)
$$ \begin{array}{@{}rcl@{}} \frac{\mathrm{d}^{2} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + c u_{x,\pm}^{\texttt{N}} (t) & = & b_{x,\pm}^{\texttt{N}} (t), \quad t \in (0,T). \end{array} $$
(3.5)

In order to obtain a unique solution to Eqs. 3.4 and 3.5, we need to prescribe the two initial values \(u_{\pm }^{\texttt {D}}(0)\) and \(\frac {\mathrm {d} u_{\pm }^{\texttt {D}}}{\mathrm {d} t}(0)\) and \(u_{x,\pm }^{\texttt {N}}(0)\) and \(\frac {\mathrm {d} u_{x,\pm }^{\texttt {N}}}{\mathrm {d} t}(0)\), respectively.

By taking \(\lim _{x \rightarrow \pm 1}\) in Eqs. 1.1c and 1.1d and \(\lim _{x \rightarrow \pm 1} \frac {\partial }{\partial x}\) in Eqs. 1.2c and 1.2d, we immediately identify the initial displacement and velocity for the Dirichlet problem and initial strain and strain rate for the Neumann problem as

$$ \begin{array}{@{}rcl@{}} u_{\pm}^{\texttt{D}}(0) & = & \phi_{\texttt{D}}(\pm 1) \quad \text{and} \quad \frac{\mathrm{d} u_{\pm}^{\texttt{D}}}{\mathrm{d} t}(0) {\kern6.5pt}= \psi_{\texttt{D}}(\pm 1), \end{array} $$
(3.6)
$$ \begin{array}{@{}rcl@{}} u_{x,\pm}^{\texttt{N}}(0) & = & \phi_{\texttt{N}}^{\prime}(\pm 1) \quad \text{and} \quad \frac{\mathrm{d} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t}(0) = \psi_{\texttt{N}}^{\prime}(\pm 1). \end{array} $$
(3.7)

Putting together Eqs. 3.4 and 3.6, we arrive at the initial value problem (IVP) on the boundary for the Dirichlet problem:

$$ \begin{array}{@{}rcl@{}} &&\frac{\mathrm{d}^{2} u_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + c u_{\pm}^{\texttt{D}} (t) = b_{\pm}^{\texttt{D}} (t), \quad t \in (0,T), \\ && u_{\pm}^{\texttt{D}} (0) = \phi_{\texttt{D}}(\pm 1) \quad \text{and} \quad \frac{\mathrm{d} u_{\pm}^{\texttt{D}}}{\mathrm{d} t}(0) = \psi(\pm 1). \end{array} $$
(3.8)

Similarly, putting Eqs. 3.5 and 3.7 together, we arrive at the IVP on the boundary for the Neumann problem:

$$ \begin{array}{@{}rcl@{}} &&\frac{\mathrm{d}^{2} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + c u_{x,\pm}^{\texttt{N}} (t) = b_{x,\pm}^{\texttt{N}} (t), \quad t \in (0,T), \\ && u_{x,\pm}^{\texttt{N}} (0) = \phi_{\texttt{N}}^{\prime}(\pm 1) \quad \text{and} \quad \frac{\mathrm{d} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t}(0) = \psi_{\texttt{N}}^{\prime}(\pm 1). \end{array} $$
(3.9)

On the other hand, the BC Eqs. 1.1b and 1.2b demand a solution from Eqs. 3.8 and 3.9 that are equal to \(\alpha _{\pm }^{\texttt {D}} (t)\) and \(\alpha _{\pm }^{\texttt {N}} (t)\), respectively. Hence, we identify the initial displacement and velocity, for the Dirichlet problem and initial strain and initial strain rate, for the Neumann problem, as well as the corresponding forcing functions. When the following choices are made,

$$ \begin{array}{@{}rcl@{}} \text{Dirichlet:} \quad b_{\pm}^{\texttt{D}} (t) &=& \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + c \alpha_{\pm}^{\texttt{D}}(t),\\ \phi_{\texttt{D}}(\pm 1) &=& \alpha_{\pm}^{\texttt{D}}(0),\\ \psi_{\texttt{D}}(\pm 1) &=& \frac{\mathrm{d} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t}(0), \end{array} $$
(3.10)
$$ \begin{array}{@{}rcl@{}} \text{Neumann:} \quad b_{x,\pm}^{\texttt{N}} (t) & = & \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + c \alpha_{\pm}^{\texttt{N}}(t),\\ \phi_{\texttt{N}}^{\prime}(\pm 1) &=& \alpha_{\pm}^{\texttt{N}}(0),\\ \psi_{\texttt{N}}^{\prime}(\pm 1) &=& \frac{\mathrm{d} \alpha_{\pm}^{\texttt{N}}}{\mathrm{d} t}(0), \end{array} $$
(3.11)

the IVP (3.8) for the Dirichlet problem takes the form

$$ \begin{array}{@{}rcl@{}} &&\frac{\mathrm{d}^{2} u_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + c u_{\pm}^{\texttt{D}} (t) = \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + c \alpha_{\pm}^{\texttt{D}}(t), \quad t \in (0,T), \\ && u_{\pm}^{\texttt{D}} (0) = \alpha_{\pm}^{\texttt{D}} (0) \quad \text{and} \quad \frac{\mathrm{d} u_{\pm}^{\texttt{D}}}{\mathrm{d} t} (0) = \frac{\mathrm{d} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t} (0). \end{array} $$

Similarly, the IVP (3.9) for the Neumann problem takes the form

$$ \begin{array}{@{}rcl@{}} && \frac{\mathrm{d}^{2} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + c u_{x,\pm}^{\texttt{N}} (t) = \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + c \alpha_{\pm}^{\texttt{N}}(t), \quad t \in (0,T), \\ &&\quad u_{x,\pm}^{\texttt{N}}(0) = \alpha_{\pm}^{\texttt{N}}(0) \quad\text{\!and} \quad \frac{\mathrm{d} u_{x,\pm}^{\texttt{N}}}{\mathrm{d} t}(0) = \frac{\mathrm{d} \alpha_{\pm}^{\texttt{N}}}{\mathrm{d} t}(0). \end{array} $$

Consequently, we guarantee that the solutions to Eqs. 3.8 and 3.9 are exactly \(\alpha _{\pm }^{\texttt {D}}(t)\) and \(\alpha _{\pm }^{\texttt {N}}(t)\), respectively. As seen above, the way to enforce inhomogeneous local BC is by the use of a forcing function on the boundary only, not in the interior of Ω. This is a major difference between enforcing local and nonlocal BC.

Remark 3.1

Since \(u^{\texttt {D}} \in \mathcal {C}^{2}(\overline {\Omega } \times [0,T])\), the choices Eqs. 3.102 and 3.103 correspond to the continuity of uD and \(u_{t}^{\texttt {D}}\), respectively, at the corner points (± 1,0). More precisely, they are implications for the following interchange of limits.

$$ \begin{array}{@{}rcl@{}} \phi_{\texttt{D}}(\pm 1) & =& \lim_{x \rightarrow \pm 1} \lim_{t \rightarrow 0} u^{\texttt{D}}(x,t) = \lim_{t \rightarrow 0} \lim_{x \rightarrow \pm 1} u^{\texttt{D}}(x,t) = \alpha_{\pm}^{\texttt{D}}(0),\\ \psi_{\texttt{D}}(\pm 1) & =& \lim_{x \rightarrow \pm 1} \lim_{t \rightarrow 0} u_{t}^{\texttt{D}}(x,t) = \lim_{t \rightarrow 0} \lim_{x \rightarrow \pm 1} u_{t}^{\texttt{D}}(x,t) = \frac{\mathrm{d} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t}(0). \end{array} $$

Similarly, since \(u^{\texttt {N}} \in \mathcal {C}^{3}(\overline {\Omega } \times [0,T])\), the choices Eqs. 3.112 and 3.113 correspond to the continuity of \(u_{x}^{\texttt {N}}\) and \(u_{xt}^{\texttt {D}}\), respectively, at the corner points (± 1,0).

$$ \begin{array}{@{}rcl@{}} \phi_{\texttt{N}}^{\prime}(\pm 1) & = & \lim_{x \rightarrow \pm 1} \lim_{t \rightarrow 0} \frac{\partial u^{\texttt{N}}}{\partial x} (x,t) = \lim_{t \rightarrow 0} \lim_{x \rightarrow \pm 1} \frac{\partial u^{\texttt{N}}}{\partial x} (x,t) = \alpha_{\pm}^{\texttt{N}}(0), \\ \psi_{\texttt{N}}^{\prime}(\pm 1) & =& \lim_{x \rightarrow \pm 1} \lim_{t \rightarrow 0} \frac{\partial u_{t}^{\texttt{N}}}{\partial x} (x,t) = \lim_{t \rightarrow 0} \lim_{x \rightarrow \pm 1} \frac{\partial u_{t}^{\texttt{N}}}{\partial x} (x,t) = \frac{\mathrm{d} \alpha_{\pm}^{\texttt{N}}}{\mathrm{d} t}(0). \end{array} $$

The Hilbert-Schmidt Property and the Governing Operator

Resorting to the integral representation of the operators, since \(K_{\texttt {BC}}(x,x^{\prime }) \in L^{2}({\Omega } \times {\Omega })\), we see that the operator \(\left ({\mathcal{M}}_{\texttt {BC}} - c \right )\) is Hilbert-Schmidt. The main tool to prove that the BC are satisfied is this property. An operator that possesses the Hilbert-Schmidt property “feels the boundary” of Ω. For the sake of clarity, we restrict the discussion to the case of the Dirichlet BC.

Unlike differential operators, integral operators can increase the regularity of the function on which they act. More precisely, given uD(⋅, t) ∈ L2(Ω), the function \(\left ({\mathcal{M}}_{\texttt {D}} - c \right ) u^{\texttt {D}}(x,t)\) has an extension to a continuous function on \(\overline {\Omega }\) for t ∈ [0, T]. Hence, the boundary value can be obtained by simply taking the limit as shown in Eq. 3.2. Furthermore, for the Neumann BC, the function \(({\mathcal{M}}_{\texttt {N}} - c)u^{\texttt {N}}(x,t)\) has an extension to a continuously differentiable function on \(\overline {\Omega }\) for t ∈ [0, T]. Hence, the boundary value can be obtained by simply taking the limit of the derivative as shown in Eq. 3.3. In conclusion, the Hilbert-Schmidt property is the mechanism that guarantees the required regularity to enforce the BC.

The operator \({\mathcal{M}}_{\texttt {D}}\) acts on uD(⋅, t) ∈ L2(Ω), but the space L2(Ω) altogether ignores the values of uD(⋅, t) on the boundary of Ω. Next, take a closer look at the boundary behavior of the governing (1.1a). To ensure that the limits of the left hand side of Eq. 1.1a exist, write them as

$$ \lim_{x \rightarrow \pm 1} u_{tt}^{\texttt{D}}(x,t) + \lim_{x \rightarrow \pm 1} (\mathcal{M}_{\texttt{D}} - c)u^{\texttt{D}}(x,t) + c \lim_{x \rightarrow \pm 1}u^{\texttt{D}}(x,t) = \lim_{x \rightarrow \pm 1} b^{\texttt{D}}(x,t). $$

Since Eqs. 1.1a1.1d is a second order initial value problem in time, it naturally assumes the existence of \(\lim _{x \rightarrow \pm 1} u_{tt}^{\texttt {D}}(x,t), ~t \in [0,T]\) and \(\lim _{x \rightarrow \pm 1} u^{\texttt {D}}(x,t)\) is provided as boundary data for all t ∈ [0, T]. One crucial question remains: Why do the limits

$$ \lim_{x \rightarrow \pm 1} (\mathcal{M}_{\texttt{D}} - c)u^{\texttt{D}}(x,t) $$
(4.1)

exist? The answer is due to a subtle point in our construction. As we mentioned above, the bounded operator \({\mathcal{M}}_{\texttt {D}}\) after subtracting c, i.e., \({\mathcal{M}}_{\texttt {D}} - c\), possesses the Hilbert-Schmidt property. Consequently, due to the aforementioned continuous extension, the limits in Eq. 4.1 exist and in fact are equal to 0. To see the latter, we expand uD in the Hilbert basis as

$$ u^{\texttt{D}}(x,t) = \sum\limits_{k=1}^{\infty} \langle{e_{k}^{\texttt{D}}|u^{\texttt{D}}}\rangle e_{k}^{\texttt{D}}(x). $$

By the spectral theorem for bounded operators, we can reproduce the same limit result in Eq. 3.2

$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left( \mathcal{M}_{\texttt{D}} - c \right) u^{\texttt{D}}(x,t) & = & \lim_{x \rightarrow \pm 1} \sum\limits_{k=1}^{\infty} \left( \lambda_{k}(\mathcal{M}_{\texttt{D}}) - c \right) \langle{e_{k}^{\texttt{D}}|u^{\texttt{D}}}\rangle e_{k}^{\texttt{D}}(x) \\ & = & \sum\limits_{k=1}^{\infty} \left( \lambda_{k}(\mathcal{M}_{\texttt{D}}) - c \right) \langle{e_{k}^{\texttt{D}}|u^{\texttt{D}}}\rangle \underbrace{ \lim_{x \rightarrow \pm 1} e_{k}^{\texttt{D}}(x)}_{=~0} \\ & = & 0 \end{array} $$
(4.2)

The interchange of \(\lim _{x \rightarrow \pm 1}\) with \({\sum }_{k=1}^{\infty }\) is justified by the uniform convergence due to the Hilbert-Schmidt property. For details, see [2].

Two important consequences follow. First, since \(\lim _{x \rightarrow \pm 1} u^{\texttt {D}}(x,t)\) is assumed to exist for all t ∈ [0, T], Eq. 4.1 implies that

$$ \lim_{x \rightarrow \pm 1} \mathcal{M}_{\texttt{D}} u^{\texttt{D}}(x,t) = c \lim_{x \rightarrow \pm 1} u^{\texttt{D}}(x,t). $$
(4.3)

Second, a compatibility condition arises. Since we proved that the limits of the left hand side of Eq. 1.1a exist, the governing (1.1a) becomes well-defined if we admit only a forcing function bD(⋅, t) ∈ L2(Ω) that satisfies the compatibility condition:

$$ \lim_{x \rightarrow \pm 1} b^{\texttt{D}}(x,t) = \lim_{x \rightarrow \pm 1} u_{tt}^{\texttt{D}}(x,t) + c \lim_{x \rightarrow \pm 1} u^{\texttt{D}}(x,t), \quad t \in [0,T]. $$

For the compatibility condition of the stationary problem, see [4, Sec. 5].

Uniform Convergence and the Classical Solutions

Our nonlocal problem requires continuity at corner points (± 1, 0) as pointed out in Remark 3.1. Thanks to uniform convergence guaranteed by the Hilbert-Schmidt property, the BC are automatically satisfied. The situation is different in the classical problem. There is no continuity requirement at corner points. This provides the freedom that initial conditions can disagree with the BC. However, the classical problem suffers from a major complication. The solutions do not guarantee that the BC are satisfied unless a uniform convergence of the series solution is in place. Even for initial value problems, uniform convergence of a series representation is a requirement for obtaining a solution to the classical wave equation [12, p. 29]. Typically, a series solution must satisfy the Weirstrass M-test to guarantee uniform convergence; see [16, Sec. 18.3.2]. Since this is not always the case, the series solutions qualify only as formal solutions [16, p. 980].

The Hilbert-Schmidt Property, Solution Operators, and Boundary Conditions

The explicit expression of the solution to Eqs. 1.1a1.1d is given as

$$ u^{\texttt{D}}(x,t) = \cos (t \sqrt{\mathcal{M}_{\texttt{D}}}) \phi_{\texttt{D}}(x) + \frac{\sin (t \sqrt{\mathcal{M}_{\texttt{D}}})}{\sqrt{\mathcal{M}_{\texttt{D}}}} \psi_{\texttt{D}}(x) + {{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{\mathcal{M}_{\texttt{D}}} \right)} {\sqrt{\mathcal{M}_{\texttt{D}}}} b^{\texttt{D}}(x,\tau) \mathrm{d}\tau. $$
(4.4)

See [2, Eq. (16) and Thm. 8] and [11, Thm. 1 and Thm. 3] for the expressions for bounded and unbounded domain, respectively. Giving a rigorous proof for the fact that BC are satisfied for all t ∈ [0, T] calls for establishing the Hilbert-Schmidt property of the solution operators. The solution representation in Eq. 4.4 suggests defining the following solution operators.

$$ \begin{array}{@{}rcl@{}} g_{\texttt{D},0}(\mathcal{M}_{\texttt{D}}) \phi_{\texttt{D}}(x) & := & \cos (t \sqrt{\mathcal{M}_{\texttt{D}}}) \phi_{\texttt{D}}(x) \\ g_{\texttt{D},1}(\mathcal{M}_{\texttt{D}}) \psi_{\texttt{D}}(x) & := & \frac{\sin (t \sqrt{\mathcal{M}_{\texttt{D}}})}{\sqrt{\mathcal{M}_{\texttt{D}}}} \psi_{\texttt{D}}(x)\\ g_{\texttt{D},2}(\mathcal{M}_{\texttt{D}}) b^{\texttt{D}}(x,t) & := & {{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{\mathcal{M}_{\texttt{D}}} \right)} {\sqrt{\mathcal{M}_{\texttt{D}}}} b^{\texttt{D}}(x,\tau) \mathrm{d}\tau. \end{array} $$

Note that all solution operators are bounded functions of \({\mathcal{M}}_{\texttt {D}}\) (see [2, Sec. 2.5]). We decompose the solution operators in the following way to extract a Hilbert-Schmidt term:

$$ g_{\texttt{D},i}(\mathcal{M}_{\texttt{D}}) = \left[ g_{\texttt{D},i}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},i}(c) \right] + g_{\texttt{D},i}(c), \quad i=0,1,2. $$
(4.5)

In [2, Sec. 3.1], we proved that the term \(\left [ g_{\texttt {D},i}({\mathcal{M}}_{\texttt {D}}) - g_{\texttt {D},i}(c) \right ]\) is a Hilbert-Schmidt operator. Hence, \(\left [ g_{\texttt {D},i}({\mathcal{M}}_{\texttt {D}}) - g_{\texttt {D},i}(c) \right ]v(x)\) has an extension to a continuous function on \(\overline {\Omega }\). As a result, the following limits all exist and are equal to zero

$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left[ g_{\texttt{D},0}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},0}(c) \right] \phi_{\texttt{D}}(x) & = & 0 \end{array} $$
(4.6)
$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left[ g_{\texttt{D},1}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},1}(c) \right] \psi_{\texttt{D}}(x) & = & 0 \end{array} $$
(4.7)
$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left[ g_{\texttt{D},2}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},2}(c) \right] b^{\texttt{D}}(x,t) & = & 0. \end{array} $$
(4.8)

To prove the latter for Eq. 4.6, we proceed similarly as for the result in Eq. 4.2. We apply the spectral theorem for bounded operators and obtain

$$ \begin{array}{@{}rcl@{}} \lim_{x \rightarrow \pm 1} \left[ g_{\texttt{D},0}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},0}(c) \right] \phi_{\texttt{D}}(x) & = & \lim_{x \rightarrow \pm 1} \sum\limits_{k=1}^{\infty} \left( g_{\texttt{D},0}(\lambda_{k}(\mathcal{M}_{\texttt{D}})) - g_{\texttt{D},0}(c) \right) \langle{e_{k}^{\texttt{D}}|u^{\texttt{D}}}\rangle e_{k}^{\texttt{D}}(x) \\ & = & \sum\limits_{k=1}^{\infty} \left( g_{\texttt{D},0}(\lambda_{k}(\mathcal{M}_{\texttt{D}})) - g_{\texttt{D},0}(c) \right) \langle{e_{k}^{\texttt{D}}|u^{\texttt{D}}}\rangle \underbrace{ \lim_{x \rightarrow \pm 1} e_{k}^{\texttt{D}}(x)}_{=~0} \\ & = & 0. \end{array} $$

Again, the interchange of \(\lim _{x \rightarrow \pm 1}\) with \({\sum }_{k =1}^{\infty }\) is justified by the uniform convergence due to the Hilbert-Schmidt property of the operator \(\left [ g_{\texttt {D},0}({\mathcal{M}}_{\texttt {D}}) - g_{\texttt {D},0}(c) \right ]\). Consequently, similar to Eq. 4.3, we arrive at

$$ \lim_{x \rightarrow \pm 1} g_{\texttt{D},0}(\mathcal{M}_{\texttt{D}}) \phi_{\texttt{D}}(x) = g_{\texttt{D},0}(c) \lim_{x \rightarrow \pm 1} \phi_{\texttt{D}}(x) = \cos(t \sqrt{c}) \phi_{\texttt{D}}(\pm 1). $$

When we write the solution expression in Eq. 4.4 in terms of the solution operators and, for each term, utilize the decomposition Eq. 4.5, we arrive at

$$ \begin{array}{@{}rcl@{}} u^{\texttt{D}}(x,t) &=& \cos (t \sqrt{c}) \phi_{\texttt{D}}(x) + \frac{\sin (t \sqrt{c})}{\sqrt{c}} \psi_{\texttt{D}}(x) + {\int}_{0}^{t} \frac{\sin \left( (t-\tau) \sqrt{c} \right)} {\sqrt{c}} b^{\texttt{D}}(x,\tau) \mathrm{d}\tau \\ && +\left[ g_{\texttt{D},0}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},0}(c) \right] \phi_{\texttt{D}}(x) + \left[ g_{\texttt{D},1}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},1}(c) \right] \psi_{\texttt{D}}(x)\\ &&+ \left[ g_{\texttt{D},2}(\mathcal{M}_{\texttt{D}}) - g_{\texttt{D},2}(c) \right] b^{\texttt{D}}(x,t). \end{array} $$

Taking \(\lim _{x \rightarrow \pm 1}\), using Eqs. 4.64.74.8, and making all the necessary choices in Eq. 3.10, the expression of the solution on the boundary takes the form

$$ u^{\texttt{D}}_{\pm}(t) = \cos (t \sqrt{c}) \alpha_{\pm}^{\texttt{D}}(0) + \frac{\sin (t \sqrt{c})}{\sqrt{c}} \frac{\mathrm{d} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t}(0)+ {{\int}_{0}^{t}} \frac{\sin \left( (t - \tau) \sqrt{c} \right)} {\sqrt{c}} \left( \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(\tau) + c \alpha_{\pm}^{\texttt{D}}(\tau) \right) \mathrm{d}\tau. $$
(4.9)

Applying integration by parts twice on Eq. 4.9 eventually leads to satisfying the BC:

$$ u^{\texttt{D}}_{\pm}(t) = \alpha_{\pm}^{\texttt{D}}(t), \quad t \in [0,T]. $$

Exact Solutions with Homogeneous BC

Thanks to functional calculus, it is possible to find exact solutions to Eqs. 1.1a1.1d and 1.2a1.2d. We can generalize the solution operators in Eq. 4.4 and the expressions for the solution to Eqs. 1.1a1.1d and 1.2a1.2d are given as [2, 11]

$$ \begin{array}{@{}rcl@{}} u^{\texttt{BC}}(x,t) & = &\cos \left( t \sqrt{\mathcal{M}_{\texttt{BC}}}\right) \phi_{\texttt{BC}}(x) + \frac{\sin \left( t \sqrt{\mathcal{M}_{\texttt{BC}}}\right)}{\sqrt{\mathcal{M}_{\texttt{BC}}}} \psi_{\texttt{BC}}(x) \\ && +{{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{\mathcal{M}_{\texttt{BC}}} \right)} {\sqrt{\mathcal{M}_{\texttt{BC}}}} b^{\texttt{BC}}(x,\tau) \mathrm{d}\tau. \end{array} $$
(5.1)

Using the Hilbert basis and the spectral theorem for bounded operators, expression Eq. 5.1 can be written in terms of the following series representation.

$$ \begin{array}{@{}rcl@{}} u^{\texttt{BC}}(x,t) & = & \sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} \cos \left( t \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}\right) \langle{e_{k}^{\texttt{BC}}|\phi_{\texttt{BC}}}\rangle e_{k}^{\texttt{BC}}(x)\\ &&+ \sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} \frac{\sin \left( t \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}\right)} {\sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}} \langle{e_{k}^{\texttt{BC}}|\psi_{\texttt{BC}}}\rangle e_{k}^{\texttt{BC}}(x)\\ &&+ \sum\limits_{k \in \mathbb{N}_{\texttt{BC}}}\left[ {{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})} \right)} {\sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}} \langle{e_{k}^{\texttt{BC}}|b^{\texttt{BC}}(\tau)}\rangle \mathrm{d}\tau \right] e_{k}^{\texttt{BC}}(x). \end{array} $$

The series can be collapsed by using the orthonormality of \(e_{k}^{\texttt {BC}}\). For instance, the choice of

$$ b^{\texttt{BC}}(x,t) \equiv 0, \quad \phi_{\texttt{BC}}(x) = e_{m}^{\texttt{BC}}(x), \quad \psi_{\texttt{BC}}(x) \equiv 0, $$
(5.2)

for some \(m \in \mathbb {N} \setminus \{0\}\), leads to

$$ u^{\texttt{BC}}(x,t) = \cos \left( t \sqrt{f_{\texttt{BC}}(\lambda_{m}^{\texttt{BC}})}\right) e_{m}^{\texttt{BC}}(x). $$

Classical Exact Solutions with Homogeneous BC

We also study the local analogs of the problems Eqs. 1.1a1.1d and 1.2a1.2d. We consider the classical wave equation with homogeneous Dirichlet and Neumann BC with the same choice given in Eq. 5.2

$$ \begin{array}{@{}rcl@{}} u_{tt}^{\texttt{BC}}(x,t) - \frac{4}{\pi^{2}} u_{xx}^{\texttt{BC}}(x,t) & = & 0, \quad (x,t) \in {\Omega} \times (0,T),\\ u^{\texttt{D}}(\pm 1,t) &=& 0 \quad \text{or} \quad u_{x}^{\texttt{N}}(\pm 1,t) = 0, \\ u^{\texttt{BC}}(x,0) & =& e_{m}^{\texttt{BC}}(x), \\ u_{t}^{\texttt{BC}}(x,0) &=& 0, \end{array} $$
(5.3)

for some \(m \in \mathbb {N} \setminus \{0\}\). It is possible to obtain a closed form solution using d’Alembert’s formula together with the method of images or reflections. After some algebra, we obtain

$$ u^{\texttt{BC}}(x,t) = \cos \left( t \sqrt{m^{2}} \right) e_{m}^{\texttt{BC}}(x). $$

Since the classical governing Eq. 5.3 contain the classical operators ABC, the regulating function is nothing but the identity function. Using the expression of the spectrum \(\sigma (A_{\texttt {BC}}) =\{k^{2}: k \in \mathbb {N}_{\texttt {BC}} \}\), we have

$$ f_{\texttt{BC}}^{\text{classi}} (\lambda_{k}^{\texttt{BC}}) = \lambda_{k}^{\texttt{BC}} = k^{2}, \quad k \in \mathbb{N}_{\texttt{BC}}. $$

Even though \(f_{\texttt {BC}}^{\text {classi}}: \sigma (A_{\texttt {BC}}) \rightarrow \mathbb {R}\) is not a bounded function, \({\cos \limits } \left (t \sqrt {f_{\texttt {BC}}^{\text {classi}}(\lambda _{k}^{\texttt {BC}})} \right )\) is a bounded function of \(\lambda _{k}^{\texttt {BC}}\). The solution expression obtained from the formula Eq. 5.1 still captures the expression obtained from d’Alembert’s formula due to the spectral theorem for bounded operators.

Exact Solutions with Inhomogeneous BC

We treat inhomogeneous BC by the method of shifting the data [20, p. 149]. A shift function GBC(x, t) is designed to satisfy the BC and is defined as

$$ G^{\texttt{D}}(\pm 1, t) = \alpha_{\pm}^{\texttt{D}}(t) \quad \text{and} \quad G_{x}^{\texttt{N}}(\pm 1, t) = \alpha_{\pm}^{\texttt{N}}(t) $$
(6.1)

GBC(x, t) can be any function that satisfies Eq. 6.1. A practical choice is

$$ \begin{array}{@{}rcl@{}} G^{\texttt{D}}(x,t) & =& \frac{1-x}{2} \alpha_{-}^{\texttt{D}}(t) + \frac{1+x}{2} \alpha_{+}^{\texttt{D}}(t), \end{array} $$
(6.2)
$$ \begin{array}{@{}rcl@{}} G^{\texttt{N}}(x,t) & = & -\frac{(1-x)^{2}}{4} \alpha_{-}^{\texttt{N}}(t) + \frac{(1+x)^{2}}{4} \alpha_{+}^{\texttt{N}}(t). \end{array} $$
(6.3)

The boundary data are assumed to have the following regularity

$$ \alpha_{\pm}^{\texttt{D}} \in \mathcal{C}^{2}([0,T]) \quad \text{and} \quad \alpha_{\pm}^{\texttt{N}} \in \mathcal{C}^{3}([0,T]). $$
(6.4)

As a result of Eq. 6.4, the shift function should have the following regularity.

$$ G^{\texttt{D}} \in \mathcal{C}^{2}([0,T],L^{2}({\Omega})) \quad \text{and} \quad G^{\texttt{N}} \in \mathcal{C}^{3}([0,T],L^{2}({\Omega})). $$

Here, for instance by \(G^{\texttt {D}} \in \mathcal {C}^{2}([0,T],L^{2}({\Omega }))\), we mainly mean a twice continuously differentiable function in the time variable and a square integrable function in the space variable. Eventually, an equivalent IVP with homogeneous BC is obtained by defining

$$ w^{\texttt{BC}}(x,t) := u^{\texttt{BC}}(x,t) - G^{\texttt{BC}}(x,t). $$
(6.5)

Combining Eqs. 1.1b and 1.2b with 6.1, we obtain the homogeneous BC, i.e., wD(± 1, t) = 0 and \(w_{x}^{\texttt {N}}(\pm 1,t)=0\). Substituting the expression for uBC(x, t) from Eqs. 6.5 into Eqs. 1.1a1.1d and 1.2a1.2d, we arrive at the equivalent problem with homogeneous BC:

$$ \begin{array}{@{}rcl@{}} w_{tt}^{\texttt{BC}}(x,t) + \mathcal{M}_{\texttt{BC}} w^{\texttt{BC}}(x,t) & =& b^{\texttt{BC},w}(x,t), \quad (x,t) \in {\Omega} \times (0,T), \\ w^{\texttt{D}}(\pm 1,t) &=& 0 \quad \text{or} \quad w_{x}^{\texttt{N}}(\pm 1,t) = 0, \\ w^{\texttt{BC}}(x,0) & =& \phi_{\texttt{BC}}^{w}(x), \\ w_{t}^{\texttt{BC}}(x,0) & =& \psi_{\texttt{BC}}^{w}(x), \end{array} $$

where we define

$$ \begin{array}{@{}rcl@{}} b^{\texttt{BC},w}(x,t) & := & b^{\texttt{BC}}(x,t) - G_{tt}^{\texttt{BC}}(x,t) - \mathcal{M}_{\texttt{BC}} G^{\texttt{BC}}(x,t) \\ \phi_{\texttt{BC}}^{w}(x) & := & \phi_{\texttt{BC}}(x) - G^{\texttt{BC}}(x,0) \\ \psi_{\texttt{BC}}^{w}(x) & := & \psi_{\texttt{BC}}(x) - G_{t}^{\texttt{BC}}(x,0). \end{array} $$

Then, the explicit expression for the solution uBC(x, t) from Eq. 5.1 takes the form

$$ \begin{array}{@{}rcl@{}} u^{\texttt{BC}}(x,t) &=& G^{\texttt{BC}}(x,t) + \cos (t \sqrt{\mathcal{M}_{\texttt{BC}}}) \left( \phi_{\texttt{BC}}(x) - G^{\texttt{BC}}(x,0) \right) \\ && +\frac{\sin (t \sqrt{\mathcal{M}_{\texttt{BC}}})}{\sqrt{\mathcal{M}_{\texttt{BC}}}} \left( \psi_{\texttt{BC}}(x) - G_{t}^{\texttt{BC}}(x,0) \right) \\ && + {{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{\mathcal{M}_{\texttt{BC}}} \right)} {\sqrt{\mathcal{M}_{\texttt{BC}}}} \left( b^{\texttt{BC}}(x,\tau) - G_{tt}^{\texttt{BC}}(x,\tau) - \mathcal{M}_{\texttt{BC}} G^{\texttt{BC}}(x,\tau) \right) \mathrm{d}\tau. \end{array} $$

The corresponding series representation takes the form

$$ \begin{array}{@{}rcl@{}} u^{\texttt{BC}}(x,t) &=& G^{\texttt{BC}}(x,t) + \sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} \cos \left( t \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}\right) \langle{e_{k}^{\texttt{BC}}|\phi_{\texttt{BC}} - G^{\texttt{BC}}(\cdot,0)}\rangle e_{k}^{\texttt{BC}}(x) \\ && +\sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} \frac{\sin \left( t \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}\right)} {\sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}} \langle{e_{k}^{\texttt{BC}}|\psi_{\texttt{BC}} - G_{t}^{\texttt{BC}}(\cdot,0)}\rangle e_{k}^{\texttt{BC}}(x) \\ && +\sum\limits_{k \in \mathbb{N}_{\texttt{BC}}} \left[ {{\int}_{0}^{t}} \frac{\sin \left( (t-\tau) \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})} \right)} {\sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}} \langle e_{k}^{\texttt{BC}}|b^{\texttt{BC}}(\cdot,\tau) - G_{tt}^{\texttt{BC}}(\cdot,\tau)\right. \\ &&\qquad\qquad \left. -\mathcal{M}_{\texttt{BC}} G^{\texttt{BC}}(\cdot,\tau) \rangle \mathrm{d}\tau \vphantom{\frac{\sin \left( (t-\tau) \sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})} \right)} {\sqrt{f_{\texttt{BC}}(\lambda_{k}^{\texttt{BC}})}}}\right] e_{k}^{\texttt{BC}}(x). \end{array} $$
(6.6)

To find an exact solution with inhomogeneous BC, we make the following choices for the series representation Eq. 6.6:

$$ \begin{array}{@{}rcl@{}} b^{\texttt{BC}}(x,t) & =& G_{tt}^{\texttt{BC}}(x,t) + \mathcal{M}_{\texttt{BC}} G^{\texttt{BC}}(x,t), \\ \phi_{\texttt{BC}}(x) & =& G^{\texttt{BC}}(x,0), \\ \psi_{\texttt{BC}}(x) & =& G_{t}^{\texttt{BC}}(x,0). \end{array} $$
(6.7)

With this choice, note that all the terms in Eq. 6.6 vanish except the first term. Eventually, we arrive at the exact solution

$$ u^{\texttt{BC}}(x,t) = G^{\texttt{BC}}(x,t). $$
(6.8)

One can easily construct other exact solutions by making different choices in Eq. 6.7.

The Choice of Kernel Functions, Scaling, and Discretization

A collocation method with piecewise linear nodal basis functions is employed to discretize the governing Eqs. 1.1a and 1.2a. A family of kernel functions with horizon δ is chosen as

$$ \begin{array}{@{}rcl@{}} C_{1}(x) & :=& \left\{ \begin{array}{ll} 1, & x \in (-\delta, \delta) \\ 0, & \text{otherwise}, \end{array} \right. \\ C_{2}(x) & :=& \left\{ \begin{array}{ll} 1 - \left|\frac{x}{\delta}\right|^{s}, &x \in (-\delta, \delta) \\ 0, &\text{otherwise}, \end{array} \right. \end{array} $$

with s > 0. Next, we want to elaborate on the choice of kernel functions. For univariate and bivariate kernel functions, see Figs. 1 and 2, respectively. First note that

$$ \lim_{s \to \infty} C_{2}(x) = C_{1}(x), $$
(7.1)

such that C2 is a family that approaches C1. The parameter s associated with the C2 family provides a way to monitor matrix properties as \(s \to \infty \).

Fig. 1
figure1

Plot of the univariate kernel function C(x) with δ = 2− 1

Fig. 2
figure2

Plot of the bivariate kernel functions \(K_{\texttt {D}}(x,x^{\prime })\) (left) and \(K_{\texttt {N}}(x,x^{\prime })\) (right) with δ = 2− 1

After suitable scaling, in \(\mathbb {R}\), it is well-known that the peridynamic governing operator converges to the Laplace operator as δ → 0 [10]. When collocation with piecewise linear nodal basis is used, the suitable scaling turns out to be

$$ \text{scaling} = \frac{2}{\delta^{3}}. $$
(7.2)

The scaling (7.2) is always inserted in the governing operators to capture the local operator. Likewise, the discretized nonlocal operator should capture the discretized Laplace operator in some sense. With the choice of C1 as a kernel function, when δ = h, except first and last rows, the discretized operator satisfies

$$ \frac{2}{\delta^{3}} \mathcal{M}_{\texttt{BC}}^{h} = \frac{1}{h^{2}} \text{tridiag}(-1,2,-1). $$

Similarly when δ = h, the kernel function C2 leads to

$$ \frac{2}{\delta^{3}} \mathcal{M}_{\texttt{BC}}^{h} = \frac{1}{h^{2}} \text{tridiag}(-\frac{s}{s+2}, \frac{2s}{s+2},-\frac{s}{s+2}). $$

Similar to convergence in Eq. 7.1, the discretization of \({\mathcal{M}}_{\texttt {BC}}\) with C2 converges to that with C1. More precisely,

$$ \lim_{s \to \infty} \text{tridiag}(-\frac{s}{s+2}, \frac{2s}{s+2},-\frac{s}{s+2}) = \text{tridiag}(-1,2,-1). $$

Hence, capturing the discretized Laplace operator in some sense corresponds to obtaining \(\frac {1}{h^{2}} \text {tridiag}(-1,2,-1)\) or attaining it as \(s \to \infty \) in our context.

The Zero Row Sum Property

Note that the constant function u(x, t) ≡ k is in the kernel of the \({\mathcal{M}}_{\texttt {N}}\) operator. Since \({\mathcal{M}}_{\texttt {N}} k \equiv 0\), the discretized operator \({\mathcal{M}}_{\texttt {N}}^{h}\) satisfies

$$ \mathcal{M}_{\texttt{N}}^{h} 1_{h} = 0_{h}. $$

In other words, \({\mathcal{M}}_{\texttt {N}}^{h}\) has zero row sum property for all of its rows. Since \({\mathcal{M}}_{\texttt {N}}\) and \({\mathcal{M}}_{\texttt {D}}\) agree in the bulk, the zero row sum property holds for \({\mathcal{M}}_{\texttt {D}}^{h}\) for all rows corresponding to the bulk. We pay a special attention to maintain the zero row sum property at machine precision. One crucial step is to incorporate the scaling (7.2) after \({\mathcal{M}}_{\texttt {BC}}^{h} u_{h}\) takes place. See how we reflect this to our time stepping iteration:

$$ \begin{array}{@{}rcl@{}} v^{n} & =& \mathcal{M}_{\texttt{BC}}^{h} {u_{h}^{n}}, \\ u_{h}^{n+1} & =& 2 {u_{h}^{n}} - u_{h}^{n-1} + \frac12 dt^{2} \left( {b_{h}^{n}} - \frac{2}{\delta^{3}} v^{n} \right), \quad n=1, {\ldots} . \end{array} $$

Otherwise, for small δ, the round-off errors spoil the zero row sum property which leads to distortions in the wave pattern.

Interpolation Strategy to Enforce Neumann Boundary Condition

The expression (3.9), which determines the suitable forcing function to enforce Neumann BC, involves \(b_{x}^{\texttt {N}}(\pm 1, t)\). On the other hand, the forcing function in the governing Eq. 1.2a involves bN(± 1, t). We will prescribe an interpolation strategy to find the appropriate value of bN(± 1, t) that uses the value of \(b_{x}^{\texttt {N}}(\pm 1, t)\). For sake of simplicity, let us consider only the BC on the left boundary point, i.e., of x = x1 = − 1, and the kernel function C1(x).

Using nodal collocation with Lagrange basis functions, the discrete solution takes the form

$$ u_{h}^{\texttt{N}}(x,t) = \sum\limits_{j=1}^{N} u_{h}^{\texttt{N}}(x_{j},t) \phi_{j}(x). $$

After discretization and incorporating the scaling (7.2), the governing (1.2a) becomes

$$ \frac{2}{\delta^{3}} c_{1} \sum\limits_{j=1}^{N} u_{h}^{\texttt{N}}(x_{j},t) \phi_{j}(x) - \sum\limits_{j=1}^{N} u_{h}^{\texttt{N}}(x_{j},t) {\int}_{\Omega} \frac{2}{\delta^{3}} K_{\texttt{N}}(x^{\prime}, x) \phi_{j}(x^{\prime}) \mathrm{d}x^{\prime} = b^{\texttt{N}}(x,t). $$
(8.1)

On substituting x = x1, incorporating \(c_{1} := {\int \limits }_{\Omega } C_{1}(x) \mathrm {d} x = 2\delta \), and collapsing the sum only in the first term, the expression (8.1) reduces to

$$ \frac{4}{\delta^{2}} u_{h}^{\texttt{N}}(x_{1},t) - \sum\limits_{j=1}^{N} u_{h}^{\texttt{N}}(x_{j},t) {\int}_{\Omega} \frac{2}{\delta^{3}} K_{\texttt{N}}(x^{\prime}, x_{1}) \phi_{j}(x^{\prime}) \mathrm{d}x^{\prime} = b^{\texttt{N}}(x_{1},t). $$
(8.2)

The kernel function \(K_{\texttt {N}}(x^{\prime }, x_{1})\) has the support of [x1, x1 + δ] and becomes identically the constant function 2. Hence,

$$ K_{\texttt{N}}(x^{\prime}, x_{1}) = 2 \chi_{[x_{1},x_{1}+\delta]}. $$

Note that only the supports of ϕ1 and ϕ2 intersect [x1, x1 + δ]. The Eq. 8.2 reduces to

$$ \begin{array}{@{}rcl@{}} &&\frac{4}{\delta^{2}} u_{h}^{\texttt{N}}(x_{1},t) - \frac{2}{\delta^{3}} u_{h}^{\texttt{N}}(x_{1},t) {\int}_{[x_{1},x_{1}+\delta]} 2 \phi_{1}(x^{\prime}) \mathrm{d}x^{\prime} - \frac{2}{\delta^{3}} u_{h}^{\texttt{N}}(x_{1} + \delta,t) {\int}_{[x_{1},x_{1}+\delta]} 2 \phi_{2}(x^{\prime}) \mathrm{d}x^{\prime}\\ &=& b^{\texttt{N}}(x_{1},t). \end{array} $$

After using

$$ {\int}_{[x_{1},x_{1}+\delta]} \phi_{1}(x^{\prime}) \mathrm{d}x^{\prime} = {\int}_{[x_{1},x_{1}+\delta]} \phi_{2}(x^{\prime}) \mathrm{d}x^{\prime} = \frac{\delta}{2}, $$

we obtain

$$ \frac{2}{\delta^{2}} u_{h}^{\texttt{N}}(x_{1},t) - \frac{2}{\delta^{2}} u_{h}^{\texttt{N}}(x_{1} + \delta,t) = b^{\texttt{N}}(x_{1},t). $$
(8.3)

We arrive at the critical interpolation step: What should the choice of bN(x1, t) be? Let us see why the choice of

$$ \begin{array}{@{}rcl@{}} b^{\texttt{N}}(x_{1},t) &:=& -\frac{\delta}{2} b_x^{\texttt{N}}(x_{1,t})\\ &=& -\frac{\delta}{2} \left( \frac{\mathrm{d}^{2} \alpha_{-}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + \frac{2}{\delta^{3}} c_{1} \alpha_{-}^{\texttt{N}}(t) \right) \end{array} $$
(8.4)

is suitable. After some algebra and with the choice in Eq. 8.4, the expression in Eq. 8.3 is equivalent to

$$ \frac{u_{h}^{\texttt{N}}(x_{1} + \delta,t) - u_{h}^{\texttt{N}}(x_{1},t)}{\delta} = \frac{\delta^{2}}{4}\frac{\mathrm{d}^{2} \alpha_{-}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + \alpha_{-}^{\texttt{N}}(t), $$

which leads to

$$ \frac{\partial u_{h}^{\texttt{N}}}{\partial x} (x_{1},t) = \lim_{\delta \to 0} \frac{u_{h}^{\texttt{N}}(x_{1} + \delta,t) - u_{h}^{\texttt{N}}(x_{1},t)}{\delta} = \alpha_{-}^{\texttt{N}}(t). $$

Consequently, the choice Eq. 8.4 guarantees that the BC is satisfied with \(\mathcal {O}(\delta ^{2})\) accuracy as δ → 0.

When the kernel function C1 is used, we reported the choice of bN in Eq. 8.4. When C2 is used, the choice becomes

$$ b^{\texttt{N}}(x_{1},t) := -\frac{\delta}{2} \left( \frac{s+1}{s+2} \right) \left( \frac{\mathrm{d}^{2} \alpha_{-}^{\texttt{N}}}{\mathrm{d} t^{2}}(t) + \frac{2}{\delta^{3}} c_{2} \alpha_{-}^{\texttt{N}}(t) \right), $$

where \(c_{2} := {\int \limits }_{\Omega } C_{2}(x) \mathrm {d} x = 2 \delta s/(s+1)\).

Implementation and Numerical Experiments

When a uniform grid is used, the structure of the kernel function does not permit the boundary data to enter correctly, i.e., without distortion, into the bulk. Hence, we are forced to use a fitted grid with spacing h and δ inside and outside the bulk, respectively. Hence, grid nodes are

$$ {\Omega}_{h} := \{-1,-1+\delta, -1+\delta+h, \ldots, -h, 0, h, \ldots, 1 -\delta -h, 1 -\delta, 1\}. $$

When the boundary data are homogeneous, one can use a uniform grid instead of a fitted one. Let us dwell on why fitted grid is essential for our numerical method. The boundary data reside in the first and last degrees of freedom (DOF). Hence, the first and last columns of the stiffness matrix are the most important columns for propagating data into the domain. For instance, consider the Dirichlet problem with the kernel C1. With a uniform grid, observe that the first/last column entries corresponding to DOF between the boundary and the bulk vanish simply due to the structure of the kernel function (see Fig. 2). As a result, boundary data cannot propagate into the domain. When we choose C2 as the kernel function, those entries in the first/last column become nonzero. But, this is a partial fix because now the rows corresponding to DOF do not satisfy the zero row sum property, which gives rise to wave distortion. Consequently, the fitted grid is a requirement of the inhomogeneous boundary data.

In order to maintain regularity assumptions, the boundary data choices satisfy \(\alpha _{\pm }^{\texttt {BC}}(t) \in \mathcal {C}^{3}([0,10])\). For time integration, we employ the Newmark scheme with Δt = 0.95 × 10− 3 and Δt = 0.50 × 10− 3 for known and unknown solutions, respectively.

Dirichlet and Neumann Problem with Known Exact Solution

The pointwise relative error between the exact and the approximate displacement is defined as

$$ e^{\texttt{BC}}(x_{i},t_{j}):= \frac{G^{\texttt{BC}}(x_{i},t_{j}) - u^{\texttt{BC}}(x_{i},t_{j}))}{\|G^{\texttt{BC}}(\cdot,t_{j})\|_{L^{2}({\Omega})}}, $$
(9.1)

where uBC denotes the approximate displacement. On the other hand, for the Neumann problem, the relative strain error is defined as

$$ e_{\text{strain}} (x_{i},t_{j}):= \frac{G_{x}^{\texttt{N}}(x_{i},t_{j}) - s(x_{i},t_{j})}{\|G_{x}^{\texttt{N}}(\cdot,t_{j})\|_{L^{2}({\Omega})}}, $$
(9.2)

where s(xi, tj) denotes the approximate strain computed by a central difference scheme.

We resort to the method of shifting the data presented in Section 6. The accuracy of the numerical solution is verified by setting up a test case in which the exact solution is identically equal to the shift function as indicated in Eq. 6.8. The shift functions are chosen as the practical ones given in Eqs. 6.2 and 6.3. We use the same boundary data for Dirichlet and Neumann problems:

$$ \alpha_{-}^{\texttt{BC}}(t) := \left\{ \begin{array}{llll} & \frac14(1 - \cos(\pi t))^{2} + 1, && t \in [0,2] \\ & 0, && t \in (2,10] \end{array} \right. \quad \text{and} \quad \alpha_{+}^{\texttt{BC}}(t):= 1, \quad t \in [0,10], $$
(9.3)

where

$$ u^{\texttt{D}}(\pm 1,t) = \alpha_{\pm}^{\texttt{D}}(t) \quad \text{and} \quad \frac{\partial u^{\texttt{N}}}{\partial x}(\pm 1,t) = \alpha_{\pm}^{\texttt{N}}(t). $$

Note that the boundary data in Eq. 9.3 is only different from that for unknown solutions in Eq. 9.4 by a shift of 1. The shift guarantees nonvanishing \(\|G^{\texttt {BC}}(\cdot ,t_{j})\|_{L^{2}({\Omega })}\) and \(\|G_{x}^{\texttt {N}}(\cdot ,t_{j})\|_{L^{2}({\Omega })}\) values to be able to report relative errors.

The forcing functions are chosen as

$$ b^{\texttt{BC}}(x,t) = G_{tt}^{\texttt{BC}}(x,t) + \mathcal{M}_{\texttt{BC}} G^{\texttt{BC}}(x,t), \quad x \in \overline{\Omega}, ~t \in [0,10], $$

with a time step of \({\Delta } t = h = \mathcal {O}(10^{-3})\) and a grid spacing of h = 2− 10. It can be seen that the computational solutions well approximate the exact solutions (see Fig. 3). For the Dirichlet problem, the relative error in displacement is computed using Eq. 9.1 and is \(e^{\texttt {D}}(x_{i},t_{j}) = \mathcal {O}(10^{-6}) = \mathcal {O}({\Delta } t^{2} + h^{2})\).

Fig. 3
figure3

Displacement of the Dirichlet problem with known exact solution with h = 2− 10, Δt = 0.95 × 10− 3, kernel function C1(x), and δ = 2− 8 (left) and δ = 2− 10 (right)

For the Neumann problem, the relative errors in displacement and strain are computed using Eqs. 9.1 and 9.2, respectively. With h = 2− 12, the error observed is \(e^{\texttt {N}}(x_{i},t_{j}) = \mathcal {O}(10^{-2})\) and \(e_{\text {strain}}^{\texttt {N}}(x_{i},t_{j}) = \mathcal {O}(10^{-3})\). Due to the large displacement error, a smaller grid spacing of h = 2− 14 was used, which gave rise to \(e^{\texttt {N}}(x_{i},t_{j}) = e_{\text {strain}}^{\texttt {N}}(x_{i},t_{j}) = \mathcal {O}(10^{-3}) = \mathcal {O}({\Delta } t + h)\) (see Figs. 4 and 5). The Neumann problem is less accurate and seems more sensitive to grid spacing than the Dirichlet problem.

Fig. 4
figure4

Displacement of the Neumann problem with known exact solution with δ = 2− 10, h = 2− 12, Δt = 0.50 × 10− 3, and kernel function C1(x)

Fig. 5
figure5

Displacement of the Neumann problem with known exact solution with δ = 2− 10, h = 2− 14, Δt = 0.50 × 10− 3, and kernel function C1(x)

Dirichlet and Neumann Problem with Unknown Exact Solution

In this section, we report experiments for the Dirichlet problem (1.1a) with unknown exact solution (numerical solution only). We choose zero initial data, i.e., \(u^{\texttt {D}}(x,0) = u_{t}^{\texttt {D}}(x,0) = 0\), and zero forcing function in the interior so that the wave propagation is initiated only by the boundary data.

The same boundary data are used for both Dirichlet and Neumann problems:

$$ \alpha_{-}^{\texttt{BC}}(t) := \left\{ \begin{array}{llll} & \frac14(1 - \cos(\pi t))^{2}, && t \in [0,2] \\ & 0, && t \in (2,10] \end{array} \right. \quad \text{and} \quad \alpha_{+}^{\texttt{BC}}(t):= 0, \quad t \in [0,10], $$
(9.4)

where

$$ u^{\texttt{D}}(\pm 1,t) = \alpha_{\pm}^{\texttt{D}}(t) \quad \text{and} \quad \frac{\partial u^{\texttt{N}}}{\partial x}(\pm 1,t) = \alpha_{\pm}^{\texttt{N}}(t). $$

Reflecting on Eq. 3.101 and the interpolation strategy in Section 8, more specifically Eq. 8.4, for the kernel C1 the forcing functions respectively become

$$ \begin{array}{@{}rcl@{}} b^{\texttt{D}}(\pm 1,t) &=& \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{D}}}{\mathrm{d} t^{2}}(t) + \frac{2}{\delta^{3}} c_{1} \alpha_{\pm}^{\texttt{D}}(t) \quad\quad\quad\qquad \text{ and} \quad b^{\texttt{D}}(x,t) = 0, \quad x \in {\Omega}, \\ b^{\texttt{N}}(\pm 1,t) &=& -\frac{\delta}{2} \left( \frac{\mathrm{d}^{2} \alpha_{\pm}^{\texttt{N}}} {\mathrm{d} t^{2}}(t) + \frac{2}{\delta^{3}} c_{1} \alpha_{\pm}^{\texttt{N}}(t) \right) \qquad \text{and} \quad b^{\texttt{N}}(x,t) = 0, \quad x \in {\Omega}. \end{array} $$

Wave patterns consisting of multiple reflections of opposite sign can be seen in Fig. 6 that are reminiscent of solutions to the classical wave equation. A grid spacing of h = 2− 10 was chosen. The cases of δ = h and δ = 4h,16h correspond to local and nonlocal computations, respectively. The results for both kernels are shown in Fig. 6. The wave speed with kernel function C2 is slower than that with kernel function C1. Furthermore, a larger δ size gives rise to a slower wave speed (see Fig. 6).

Fig. 6
figure6

Approximate displacement uD(x, t) of the Dirichlet problem with unknown solution with δ = 2− 6,2− 8,2− 10, h = 2− 10, Δt = 0.95 × 10− 3, and kernel function C1(x) (left) and C2(x) and s = 1 (right)

Using the same BC as in Eq. 9.4, a numerical experiment with nonzero initial displacement illustrates wave collision and superposition (see Fig. 7). For the kernel C1 case with δ = 2− 10, observe that the reflection of the initial displacement splits the boundary data. For the kernel C2 case, wave propagation is slower and reflection takes place at a time later than t = 2. As a result, reflection does not split the boundary data.

Fig. 7
figure7

Approximate displacement uD(x, t) of the Dirichlet problem with unknown solution with δ = 2− 6,2− 8,2− 10, h = 2− 10, Δt = 0.95 × 10− 3, kernel function C1(x) (left) and C2(x), s = 1 (right), and nonzero initial displacement

The strain is computed from the displacement data using a central difference approximation. The boundary data for the Neumann problem is chosen to be the same as the Dirichlet problem so that the strain profiles are identical to that of displacement of the Dirichlet problem. One can also rigorously show this equivalence, which we skip here. We simply use the equivalence to verify the validity of numerical experiments with Neumann BC. Note that strain profiles in Fig. 8 are identical to displacement profiles in Fig. 6. For the strain, a reflection pattern with opposite sign is observed, which agrees with the classical solution. The cascadic displacement profile also agrees with that of the classical problem (see Fig. 9).

Fig. 8
figure8

Approximate strain s(x, t) of the Neumann problem with unknown solution with δ = 2− 6,2− 8,2− 10, h = 2− 10, Δt = 0.95 × 10− 3, kernel function C1(x) (left) and C2(x) and s = 1 (right)

Fig. 9
figure9

Approximate displacement uN(x, t) of the Neumann problem with unknown solution with δ = 2− 6,2− 8,2− 10, h = 2− 10, Δt = 0.95 × 10− 3, kernel function C1(x) (left) and C2(x) and s = 1 (right)

Conclusion

A comprehensive treatment on how to enforce inhomogeneous local BC in nonlocal problems was presented. We explained methodically how to construct forcing functions to enforce local BC and their relationship to initial values. Exact solutions with both homogeneous and inhomogeneous BC were derived and used to verify numerical experiments. We explained the critical role of the Hilbert-Schmidt property in enforcing local BC rigorously. For the strain BC, an interpolation strategy was prescribed to find the appropriate value of the forcing function from its derivative.

Our ongoing work aims to extend these operators to vector valued problems which will help apply peridynamics to problems that require local BC. Furthermore, construction of higher order node based collocation in higher dimensions is work in progress. Our construction depends on the assumption of a rectangular/box geometry [7]. We are investigating the case of general geometry in higher dimensions.

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Acknowledgments

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-18-2-0090. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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Correspondence to Burak Aksoylu.

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Aksoylu, B., Gazonas, G.A. On Nonlocal Problems with Inhomogeneous Local Boundary Conditions. J Peridyn Nonlocal Model (2020). https://doi.org/10.1007/s42102-019-00022-w

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Keywords

  • Nonlocal wave equation
  • Nonlocal operator
  • Inhomogeneous local boundary condition
  • Peridynamics
  • Functional calculus