We develop a fastconvolution fully spectral method to solve the nonlinear peridynamic problem (6).
Without loss of generality, we assume \(\Omega =[1,1]\) and \(t\in [1,1]\), and we fix \(N+1>0\) as the total number of collocation points in both the space and time directions, and we take the Gauss–Lobatto points \((x_{n},t_{m})\) as grid points for the discretization.
We look for an approximation of \(u(x,t)\) in the form
$$\begin{aligned} u^{N}(x,t)=\sum_{j=0}^{N} \sum_{k=0}^{N} u_{jk} T_{j}(x) T_{k}(t). \end{aligned}$$
(21)
Substituting \(u^{N}(x,t)\) into (4), we find the full expression of the peridynamic operator
$$\begin{aligned} \begin{aligned} \mathcal{L} u^{N}(x,t) ={}& \bigl(C\ast \bigl(u^{N} \bigr)^{3} \bigr) (x,t)3 u^{N}(x,t) \bigl(C\ast \bigl(u^{N} \bigr)^{2} \bigr) (x,t) \\ &{} + 3 \bigl(u^{N}\bigr)^{2}(x,t) \bigl(C\ast u^{N}\bigr) (x,t)\beta \bigl(u^{N}(x,t) \bigr)^{3}. \end{aligned} \end{aligned}$$
(22)
If we evaluate \(u^{N}(x,t)\) at \((x_{n},t_{m})\), we obtain the discrete form of (22), namely
$$\begin{aligned} \begin{aligned} \mathcal{L} u^{N}_{nm} ={}& \bigl(\mathcal{F}_{N}^{1} \bigl( \mathcal{F}_{N}(C) \mathcal{F}_{N} \bigl( \bigl(u^{N}_{nm} \bigr)^{3} \bigr) \bigr) \bigr) \\ &{} 3 \bigl(\mathcal{F}_{N}^{1} \bigl( \mathcal{F}_{N} \bigl( \bigl(u^{N}_{nm} \bigr) \bigr)\ast \bigl(\mathcal{F}_{N}( C) \mathcal{F}_{N} \bigl( \bigl(u^{N}_{nm} \bigr)^{2} \bigr) \bigr) \bigr) \bigr) \\ &{} +3 \bigl(\mathcal{F}_{N}^{1} \bigl( \mathcal{F}_{N} \bigl( \bigl(u^{N}_{nm} \bigr)^{2} \bigr)\ast \bigl(\mathcal{F}_{N}( C) \mathcal{F}_{N} \bigl( \bigl(u^{N}_{nm} \bigr) \bigr) \bigr) \bigr) \bigr) \\ &{}  \beta \bigl( u^{N}_{nm}\bigr)^{3}, \end{aligned} \end{aligned}$$
(23)
where \(u_{nm}^{N}\) is equal to \(u^{N}(x_{n},t_{m})\).
Moreover, thanks to the differentiation theorem for the Chebyshev transform, we have
$$\begin{aligned} \partial ^{2}_{tt}u^{N}(x_{n},t_{m})= \frac{2}{N}\mathcal{F}_{N}^{1} \bigl( t_{m}^{2} \mathcal{F}_{N}\bigl(u^{N}_{nm} \bigr) \bigr), \end{aligned}$$
(24)
or equivalently,
$$\begin{aligned} \partial ^{2}_{tt}u^{N}(x_{n},t_{m})= \sum_{j=0}^{N}\sum _{k=0}^{N} \hat{u}_{kj} T_{j}(x_{n})T_{k}(t_{m}), \end{aligned}$$
(25)
with \(\hat{u}_{kj}\) as in (20), for \(k, j=0,\dots,N\).
Thus, we can consider the discrete form of the model (6), namely
$$\begin{aligned} \textstyle\begin{cases} \sum_{j=0}^{N}\sum_{k=0}^{N} \hat{u}_{kj} T_{j}(x_{n})T_{k}(t_{m})  \mathcal{L}(u^{N}_{nm},t_{m})=0, & n=0,\dots,N, m=1,\dots,N, \\ \sum_{j=0}^{N}\sum_{k=0}^{N} (1)^{k} u_{jk} T_{j}(x_{n})=u_{0}(x_{n}),& n=0,\dots,N, \\ \sum_{j=0}^{N}\sum_{k=0}^{N}\sum_{\ell =0}^{N} (1)^{k} D_{k\ell} u_{j \ell} T_{j}(x_{n})=v(x_{n}),& n=0,\dots,N, \end{cases}\displaystyle \end{aligned}$$
(26)
where in the peridynamic operator \(\mathcal{L}\) we have explicitly shown the time dependence.
After solving the above nonlinear system with respect to \(u^{N}_{nm}\), we find an approximate solution of (6) having the form as in (21). In practice, in the next section, we use the FSOLVE command implemented in MATLAB software to solve the system (26). It consists in a quasiNewton method, called Levenberg–Marquardt method.
We analyze the convergence of the proposed method in the space of functions which admit a modulus of continuity. We start by giving some definitions and recalling some standard results (see [3]).
Definition 1
A continuous function \(W:\mathbb{R}_{+}\to \mathbb{R}_{+}\) is called a modulus of continuity if it satisfies the following properties:

W is increasing,

\(\lim_{z\to 0} W(z)=0\),

\(W(z_{1}+z_{2})\le W(z_{1})+W(z_{2})\), for \(z_{1}\), \(z_{2}\in \mathbb{R}_{+}\),

there exists a constant \(c>0\) such that \(z\le c W(z)\), for all \(0< z\le 2\).
An example of a modulus of continuity is given by the functions \(W(z)=z^{\alpha}\), \(0<\alpha \le 1\).
Let \(B^{2}\) be the unit ball in \(\mathbb{R}^{2}\).
Definition 2
We say that a continuous function \(u(\cdot,\cdot )\) on \(B^{2}\) admits a modulus of continuity \(W(\cdot )\) if
$$\begin{aligned} \bigl\vert u(\cdot,\cdot ) \bigr\vert _{W}=\sup _{(\bar{x},\bar{t})\ne (\tilde{x}, \tilde{t})} \biggl\{ \frac{u(\bar{x},\bar{t})u(\tilde{x},\tilde{t})}{W( \vert \vert \vert (\bar{x},\bar{t})(\tilde{x},\tilde{t}) \vert \vert \vert )}, (\bar{x},\bar{t}), ( \tilde{x},\tilde{t})\in B^{2} \biggr\} \end{aligned}$$
(27)
is finite.
In (27), \((\bar{x},\bar{t})(\tilde{x},\tilde{t})=\max \{\bar{t} \tilde{t},\bar{x}\tilde{x}: (\bar{x},\bar{t}), (\tilde{x}, \tilde{t})\in B^{2}, (\bar{x},\bar{t})\ne (\tilde{x},\tilde{t})\}\).
We denote the class of all functions described in Definition 2 by \(\mathcal{C}^{0}_{W}(B^{2})\). Then, it is a Banach space with the norm
$$\begin{aligned} \bigl\Vert u(\cdot,\cdot ) \bigr\Vert _{0,W}= \bigl\Vert u(\cdot,\cdot ) \bigr\Vert _{\infty} + \bigl\vert u(\cdot,\cdot ) \bigr\vert _{W}. \end{aligned}$$
(28)
Moreover, we denote the class of ktimes differentiable functions on \(B^{2}\) whose kth derivatives admit W as a modulus of continuity by \(\mathcal{C}^{k}_{W}\). It is a Banach space with the norm
$$\begin{aligned} \bigl\Vert u(\cdot,\cdot ) \bigr\Vert _{k,W}=\sum _{ \vert s \vert \le k} \biggl\Vert \frac{\partial ^{s} u}{\partial t^{s}} \biggr\Vert _{\infty }+ \sum_{ \vert s \vert \le k} \biggl\Vert \frac{\partial ^{s} u}{\partial x^{s}} \biggr\Vert _{ \infty }+ \sum _{ \vert s \vert = k} \biggl\vert \frac{\partial ^{s} u}{\partial t^{s}} \biggr\vert _{W} + \sum_{ \vert s \vert = k} \biggl\vert \frac{\partial ^{s} u}{\partial x^{s}} \biggr\vert _{W}. \end{aligned}$$
(29)
We can extend the previous definition on \(\bar{\Omega}=[1,1]\times [1,1]\) as follows:
$$\begin{aligned} \mathcal{C}^{k}_{W}(\bar{\Omega})={}& \bigl\{ u \in \mathcal{C}^{k}( \bar{\Omega}): \text{for each $(\tilde{x}, \tilde{t})\in \bar{\Omega}$ there exists a map $\phi:B^{2}\to \bar{ \Omega}$} \\ & \text{such that $(\tilde{x},\tilde{t})\in \operatorname{int}\bigl(\phi \bigl(B^{2}\bigr)\bigr)$ and $f\circ \phi \in \mathcal{C}^{k}_{W} \bigl(B^{2}\bigr)$} \bigr\} \end{aligned}$$
(30)
Since the multiplication by a \(\mathcal{C}^{\infty}\) function and the composition with a \(\mathcal{C}^{\infty}\) function are continuous linear transformations, it is possible to show that if
$$\begin{aligned} \phi _{i}:B^{2}\to \bar{\Omega},\quad i=1,\dots,\ell, \end{aligned}$$
are a finite collection of maps with
$$\begin{aligned} \bar{\Omega}=\bigcup_{i=1}^{\ell }\operatorname{int} \bigl(\phi _{i}\bigl(B^{2}\bigr)\bigr), \end{aligned}$$
then \(u(\cdot,\cdot )\in \mathcal{C}^{k}_{W}(\bar{\Omega})\) if and only if \((u\circ \phi )(\cdot,\cdot )\in \mathcal{C}^{k}_{W}(B^{2})\) for each \(i=1,\dots,\ell \). Moreover, the space \(\mathcal{C}^{k}_{W}(\bar{\Omega})\) is a Banach space with the norm
$$\begin{aligned} \bigl\Vert u(\cdot,\cdot ) \bigr\Vert _{k,W}=\sum _{i=1}^{\ell} \bigl\Vert (u \circ \phi _{i}) (\cdot,\cdot ) \bigr\Vert _{k,W}. \end{aligned}$$
(31)
Additionally, any other choice of finitely many maps covering Ω̄ provides an equivalent norm for the Banach space (for more details, see [32]).
Let \(\mathcal{P}(N,N,\bar{\Omega})\) be the space of all polynomials of total degree at most 2N on Ω̄, namely
$$\begin{aligned} \mathcal{P}(N,N,\bar{\Omega})= \Biggl\{ p(\tilde{x},\tilde{t})=\sum _{i=0}^{N} \sum_{j=0}^{N} b_{ij}\tilde{x}^{i}\tilde{t}^{j}: (\tilde{x}, \tilde{t})\in \bar{\Omega}, b_{ij}\in \mathbb{R} \Biggr\} . \end{aligned}$$
The following result is a generalization of the Stone–Weierstrass theorem on the space \(\mathcal{C}^{k}_{W}(\bar{\Omega})\).
Theorem 1
(See [32])
For any \(u(\cdot,\cdot )\in \mathcal{C}^{k}_{W}(\bar{\Omega})\), there exists a polynomial \(p(\cdot,\cdot )\in \mathcal{P}(N,N,\bar{\Omega})\) such that
$$\begin{aligned} \bigl\Vert u(\cdot,\cdot )p(\cdot,\cdot ) \bigr\Vert _{\infty}\le \frac{L_{0} L_{1}}{(2N)^{k}} W \biggl(\frac{1}{(2N)^{k}} \biggr), \end{aligned}$$
(32)
where \(L_{1}= \Vert u(\cdot,\cdot ) \Vert _{k,W}\) and \(L_{0}\) is a constant depending on W, but independent of N.
In order to prove the convergence of the method and the existence of solutions of the system (26), we reformulate it as a system of algebraic inequalities in the following way:
$$\begin{aligned} \textstyle\begin{cases} \vert \sum_{j=0}^{N}\sum_{k=0}^{N} \hat{u}_{kj} T_{j}(x_{n})T_{k}(t_{m})  \mathcal{L}(u^{N}_{nm}, t_{m}) \vert \le \frac{\sqrt{N}}{(2N2)^{2}} W (\frac{1}{(2N2)^{2}} ),& n=0, \dots,N, \\ &m=1,\dots,N, \\ \vert \sum_{j=0}^{N}\sum_{k=0}^{N} (1)^{k} u_{jk} T_{j}(x_{n})u_{0}(x_{n}) \vert \le \frac{\sqrt{N}}{(2N2)^{2}} W (\frac{1}{(2N2)^{2}} ),& n=0,\dots,N, \\ \vert \sum_{j=0}^{N}\sum_{k=0}^{N}\sum_{\ell =0}^{N} (1)^{k} D_{k \ell} u_{j\ell} T_{j}(x_{n})v(x_{n}) \vert \le 0,& n=0,\dots,N, \end{cases}\displaystyle \end{aligned}$$
(33)
where N is sufficiently large and W is a given modulus of continuity.
We can notice that
$$\begin{aligned} \lim_{N\to \infty}\frac{\sqrt{N}}{(2N2)^{2}} W \biggl( \frac{1}{(2N2)^{2}} \biggr)=0, \end{aligned}$$
so any solution \(\bar{u}^{N}=(\bar{u}^{N}_{nm})\) for \(n, m=0,\dots,N\) of the system (33) is a solution of the system (26) when N tends to infinity. As a consequence, to prove the existence of solutions of (26), it is sufficient to prove the existence of solutions for the system (33).
The following lemmas are preliminary to the convergence theorem.
Lemma 1
Let \(u\in \mathcal{C}^{2}_{W}(\bar{\Omega})\) be a solution of the peridynamic model (6). Then there exists a function ũ such that
$$\begin{aligned} \bigl\vert u(\bar{x},\bar{t})\tilde{u}(\bar{x},\bar{t}) \bigr\vert \le \frac{2L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr), \quad ( \bar{x}, \bar{t})\in \bar{\Omega}, \end{aligned}$$
(34)
for some constant \(L>0\).
Proof
By Theorem 1, there exists \(p(\cdot,\cdot )\in \mathcal{P}(N2,N,\bar{\Omega})\) such that
$$\begin{aligned} \bigl\Vert u_{tt}(\bar{x},\bar{t})p(\bar{x},\bar{t}) \bigr\Vert _{\infty} \le \frac{L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr), \quad( \bar{x},\bar{t})\in \bar{\Omega}, \end{aligned}$$
(35)
for some constant \(L>0\) independent of N.
We define
$$\begin{aligned} \tilde{u}(\bar{x},\bar{t})=u(\bar{x},1)+u_{t}( \bar{x},1) (\bar{t}+1)+ \int _{1}^{\bar{t}} \int _{1}^{\tau }p(\bar{x},s) \,ds\,d\tau, \end{aligned}$$
(36)
and get
$$\begin{aligned} \begin{aligned} \bigl\vert u(\bar{x},\bar{t})\tilde{u}(\bar{x}, \bar{t}) \bigr\vert &= \biggl\vert \int _{1}^{\bar{t}} \int _{1}^{\tau } \bigl(u_{tt}( \bar{x},s)p( \bar{x},s) \bigr)\,ds\,d\tau \biggr\vert \\ &\le \int _{1}^{\bar{t}} \int _{1}^{\tau } \bigl\vert u_{tt}( \bar{x},s)p( \bar{x},s) \bigr\vert \,ds\,d\tau \\ &\le \frac{L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr) \int _{1}^{ \bar{t}} \int _{1}^{\tau }\,ds\,d\tau \\ &\le \frac{2L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr) \end{aligned} \end{aligned}$$
□
Lemma 2
Let \(u_{1}, u_{2}\in \mathcal{C}^{2}_{W}(\bar{\Omega})\) satisfy equation (34). Then, there is a positive constant L such that the following estimate holds:
$$\begin{aligned} \bigl\vert \mathcal{L}(u_{1},\bar{t}) \mathcal{L}(u_{2},\bar{t}) \bigr\vert \le \frac{4L\beta}{(2N2)^{2}} W \biggl( \frac{1}{(2N2)^{2}} \biggr). \end{aligned}$$
(37)
Proof
Thanks to Lemma 1 and the property of the peridynamic operator (3), we find that there exists a positive constant L such that
$$\begin{aligned} \begin{aligned} \bigl\vert \mathcal{L}(u_{1},\bar{t}) \mathcal{L}(u_{2}, \bar{t}) \bigr\vert \le {}& L \int _{B_{\delta}(x)} C\bigl(xx'\bigr) \bigl\vert u_{1}\bigl(x', \bar{t}\bigr)u_{2} \bigl(x',\bar{t}\bigr) \bigr\vert \,dx' \\ &{}+L \bigl\vert u_{1}(x,\bar{t})u_{2}(x,\bar{t}) \bigr\vert \int _{B_{\delta}(x)}C\bigl(xx'\bigr) \,dx' \\ \le{} & \frac{4L\beta}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr). \end{aligned} \end{aligned}$$
□
Now, we are able to prove that there exists at least one solution of (33).
Theorem 2
Let \(u\in \mathcal{C}^{2}_{W}(\bar{\Omega})\) be a solution of the peridynamic model (6). Then there exists a positive integer K such that, for any \(N\ge K\), the system (33) admits a solution \(\bar{u}^{N}=(\bar{u}^{N}_{nm})\) for \(n, m=0,\dots,N\) such that
$$\begin{aligned} \bigl\vert u(\bar{x}_{k},\bar{t}_{h}) \bar{u}^{N}_{nm} \bigr\vert \le \frac{L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr),\quad h, k=0, \dots,N, \end{aligned}$$
(38)
for some positive constant L independent of N.
Proof
We define
$$\begin{aligned} \bar{u}^{N}_{nm}=\tilde{u}(x_{n},t_{m}),\quad n, m=0,\dots,N, \end{aligned}$$
(39)
where ũ is defined in (36) and satisfies equation (34).
By the definition of ũ, we find that it is a polynomial of degree at most 2N. Thus, its second derivatives at Gauss–Lobatto nodes \((x_{n},t_{m})\), \(n, m=0,\dots,N\) are given by
$$\begin{aligned} \tilde{u}_{tt}(x_{n},t_{m})=\sum _{j=0}^{N}\sum_{k=0}^{N} \sum_{ \ell =0}^{N} \hat{D}_{k\ell}^{(t)} u_{j\ell} T_{j}(x_{n})T_{k}(t_{m}). \end{aligned}$$
(40)
Using the relations (6), (34), (37), and (40), we get
$$\begin{aligned} \begin{aligned} & \Biggl\vert \sum _{j=0}^{N}\sum_{k=0}^{N} \sum_{\ell =0}^{N} \hat{D}_{k \ell}^{(t)} u_{j\ell} T_{j}(x_{n})T_{k}(t_{m}) \mathcal{L}\bigl(\bar{u}^{N}_{nm},t_{m}\bigr) \Biggr\vert \\ &\quad= \bigl\vert \tilde{u}_{tt}(x_{n},t_{m}) \mathcal{L}\bigl(\bar{u}^{N}_{nm},t_{m}\bigr) \bigr\vert \\ &\quad\le \bigl\vert \tilde{u}_{tt}(x_{n},t_{m})u_{tt}(x_{n},t_{m}) \bigr\vert + \bigl\vert u_{tt}(x_{n},t_{m}) \mathcal{L}\bigl(\bar{u}^{N}_{nm},t_{m}\bigr) \bigr\vert \\ &\quad= \bigl\vert p(x_{n},t_{m})u_{tt}(x_{n},t_{m}) \bigr\vert + \bigl\vert \mathcal{L}\bigl(u(x_{n},t_{m}) \bigr)\mathcal{L}\bigl(\bar{u}^{N}_{nm},t_{m} \bigr) \bigr\vert \\ &\quad\le \frac{L(1+4\beta )}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr). \end{aligned} \end{aligned}$$
(41)
Moreover, we find an analogous estimate for the initial conditions:
$$\begin{aligned} \begin{aligned} \bigl\vert \tilde{u}(x_{n},1)u_{0}(x_{n}) \bigr\vert &\le \bigl\vert \tilde{u}(x_{n},1)u(x_{n},1) \bigr\vert + \bigl\vert u(x_{n},1)u_{0}(x_{n}) \bigr\vert \\ &\le \frac{2L}{(2N2)^{2}} W \biggl(\frac{1}{(2N2)^{2}} \biggr) \end{aligned} \end{aligned}$$
(42)
and, by equation (36),
$$\begin{aligned} \bigl\vert \tilde{u}_{t}(x_{n},1)v(x_{n}) \bigr\vert &\le \bigl\vert \tilde{u}_{t}(x_{n},1)u_{t}(x_{n},1) \bigr\vert + \bigl\vert u_{t}(x_{n},1)v(x_{n}) \bigr\vert \le 0. \end{aligned}$$
(43)
Therefore, if we choose K such that
$$\begin{aligned} \max \bigl\{ L(4\beta +1),2L\bigr\} \le \sqrt{N}, \end{aligned}$$
(44)
we have that \(\bar{u}^{N}_{nm}\), \(n, m=0,\dots,N\) defined in (39) satisfies (33) for \(N\ge K\), and this concludes the proof. □
Finally, we prove that the solution of the system (33) converges to the solution of the peridynamic model (6).
Theorem 3
Let K be the index defined in (44) and \(\bar{u}^{N}=(\bar{u}^{N}_{nm})_{n,m=0}^{N}\) for \(N\ge K\) be the sequence of solutions of (33) given by (39), and let \(u^{N}(\cdot,\cdot )\) for \(N\ge K\) be its interpolating polynomial
$$\begin{aligned} u^{N}(\bar{x},\bar{t})=\sum _{i=0}^{N}\sum_{j=0}^{N} \bar{a}^{N}_{ij} T_{i}(\bar{x})T_{j}( \bar{t}), \end{aligned}$$
(45)
with
$$\begin{aligned} \bar{a}^{N}_{ij}=\frac{1}{\gamma _{i}\gamma _{j}}\sum _{n=0}^{N}\sum_{m=0}^{N} \bar{u}^{N}_{nm} T_{i}(x_{n})T_{j}(t_{m}) w_{n} w_{m}. \end{aligned}$$
Assume that, for any \(\bar{x}\in [1,1]\), the sequence \(\{u^{N}(\bar{x},1),u^{N}_{t}(\bar{x},1),u^{N}_{tt}(\cdot,\cdot ) \}_{N=K}^{\infty}\) has a subsequence \(\{u^{N_{i}}(\bar{x},1),u^{N_{i}}_{t}(\bar{x},1),u^{N_{i}}_{tt}( \cdot,\cdot )\}_{i=0}^{\infty}\) uniformly converging to
$$\begin{aligned} \bigl(\varphi _{1}(\bar{x}),\varphi _{2}(\bar{x}),\varphi _{3}(\cdot, \cdot )\bigr), \end{aligned}$$
where \(\varphi _{1}\), \(\varphi _{2}\in \mathcal{C}^{2}([1,1])\) and \(\varphi _{3}\in \mathcal{C}^{2}(\bar{\Omega})\). Then
$$\begin{aligned} \lim_{i\to \infty} u^{N_{i}}(\bar{x},\bar{t})= \tilde{u}(\bar{x}, \bar{t}),\quad (\bar{x},\bar{t})\in \bar{\Omega} \end{aligned}$$
(46)
is a solution of the peridynamic model (6).
Proof
Due to our assumptions, we have
$$\begin{aligned} \tilde{u}(\bar{x},\bar{t})=\varphi _{1}(\bar{x})+ \varphi _{2}(\bar{x}) ( \bar{t}+1)+ \int _{1}^{\bar{t}} \int _{1}^{\tau }\varphi _{3}(\bar{x},s) \, ds \,d\tau. \end{aligned}$$
(47)
By contradiction, assume that there is an \(n\in \{1,\dots,N\}\) such that \(\tilde{u}(\bar{x}_{n},\cdot )\) does not satisfy (6). Hence, there is a \(y\in (1,1)\) such that
$$\begin{aligned} \tilde{u}_{tt}(\bar{x}_{n},y)\mathcal{L}\bigl(\tilde{u}( \bar{x}_{x},y), \bar{t}_{m}\bigr)\ne 0. \end{aligned}$$
Since the Gauss–Lobatto nodes \(\{\bar{t}_{m}\}_{m=0}^{N}\) are dense in \([1,1]\) for \(N\to \infty \), there is a subsequence \(\{\bar{t}_{\ell _{N_{i}}}\}_{i=1}^{\infty}\) such that \(\lim_{i\to \infty} \bar{t}_{\ell _{N_{i}}}=y\) and \(0<\ell _{N_{i}}<N_{i}\).
We have
$$\begin{aligned} 0&\ne \tilde{u}_{tt}(\bar{x}_{n},y)\mathcal{L}\bigl( \tilde{u}(\bar{x}_{n},y), \bar{t}_{m}\bigr) \\ &=\lim_{i\to \infty} \bigl(\tilde{u}_{tt}( \bar{x}_{n},\bar{t}_{\ell _{N_{i}}}) \mathcal{L}\bigl(\tilde{u}( \bar{x}_{n},\bar{t}_{\ell _{N_{i}}}),\bar{t}_{m}\bigr) \bigr) \\ &\le \lim_{i\to \infty}\frac{\sqrt{N_{i}}}{(2N_{i}2)^{2}} W \biggl( \frac{1}{(2N_{i}2)^{2}} \biggr)=0. \end{aligned}$$
Therefore, \(\tilde{u}(\bar{x},\bar{t})\) satisfies the model (6) for all \(\bar{t}\in [1,1]\) and \(\bar{x}=\bar{x}_{n}\), \(n=1,\dots,N\).
Using the same argument, we can prove that \(\tilde{u}(\bar{x}_{n},1)=u_{0}(\bar{x}_{n})\) and \(\tilde{u}_{t}(\bar{x}_{n},1)=v(\bar{x}_{n})\) for \(n=0,\dots,N\), and this completes the proof. □