1 Introduction

Let \(\Theta\subset R^{2}\) be a connected and bounded region with smooth boundary \(\Gamma: =\partial\Theta\), \(\Theta^{c}: =R^{2}\backslash {\overline{\Theta}}\), \(\boldsymbol{x}=(x_{1},x_{2})\), \(\vert \boldsymbol{x} \vert =\sqrt{x_{1}^{2}+x_{2}^{2}}\). For given time upper bound T, we consider the following Sobolev equation in the 2D unbounded domain:

Problem I

Find ϖ that satisfies

$$ \textstyle\begin{cases} \varpi_{t}-\varepsilon\Delta\varpi_{t}-\gamma\Delta\varpi =f(\boldsymbol {x},t), &(\boldsymbol {x},t)\in\Theta^{c}\times(0, T), \\ \frac{\partial\varpi}{\partial\boldsymbol {n}}=g(\boldsymbol {x},t), &(\boldsymbol {x},t)\in\Gamma\times(0, T), \\ \varpi(\boldsymbol {x},0)=u_{0}(\boldsymbol {x}), &\boldsymbol {x}\in \Theta^{c}, \end{cases} $$
(1)

where \(\varpi_{t}=\frac{\partial\varpi}{\partial t}\) represents the partial derivative for the unknown function \(\varpi(\boldsymbol {x},t)\) about time, ε and γ are positive constants, \(f(\boldsymbol {x},t)\), \(g(\boldsymbol {x},t)\), and \(u_{0}(\boldsymbol {x})\) are three known functions satisfying the appropriate conditions, \(\frac{\partial}{\partial\boldsymbol {n}}\) denotes the external normal derivative operator, and n represents a normal vector onto boundary Γ of domain \(\Theta^{c}\) toward the interior of domain Θ. In addition, we also assume the function \(\varpi(\boldsymbol {x},t)\) is bounded at infinity point (see [1, 2]).

Just like the heat equation (see [3]) and the reaction diffusion equation (see [4]), Problem I is an important system of equations with real-life application background. It has been widely applied to many practical engineering fields (see [5, 6]), for example, it is used to describe the procedure of fluid flow permeating rocks, the soils moisture migration, and the different media heat transfer. It usually includes complex computing domain, initial and boundary value functions, or source term for the Sobolev equation in the 2D unbounded domain in the real world so that it has no analytic solution. Hence, one has to depend on the numerical methods.

The natural boundary element (NBE), which was proposed by Feng and Yu at the end of 1970s (see [7–11]), is a novel type of boundary element method (BEM) and suitable for solving the unbound regional problem and is an attractive and promising numerical method. It has been effectively applied to complicated boundary and infinity regional problem. It has more specific advantages than the usual BEM, for instance, it has a unique form of boundary integral equation, remains the unchanged energy functional, holds very good numerical stability, adopts the same variation principle as the finite element method (FEM), can couple with FEM and we need not calculate a great many singular integrations in practice. The main idea of the NBE method consists in introducing an appropriate artificial boundary and then restricting the computation to an appropriate large finite spatial domain. It divides the domain into two subregions, a bounded inner region \(\Theta_{1}\) (bounded annular region by \(\Gamma_{0}\) and \(\Gamma _{R}\)) and a regular unbounded region \(\Theta_{2}\) (unbounded domain outside circle \(\Gamma_{R}\)) (see [1, 2]). One can obtain the natural integral equation on boundary \(\Gamma_{R}\) and corresponding Poisson integral formula of the subproblem over unbounded domain \(\Theta_{2}\) by the natural boundary reduction. Only the function itself, not its normal derivative at the common boundary \(\Gamma_{R}\), appears in the variational. It has been used to solve the second-order elliptic equations, standard parabolic equations, and hyperbolic equations (see [1, 2, 7, 9, 10]).

However, for all we know, up to now, the NBE method has yet not been used to solve the Sobolev equation. Especially, the Sobolev equation not only includes a first-order derivative term of time and two second-order derivative terms of spatial variables but also does two mixed derivative terms of spatial variables (second-order) and time (first-order) so that either the establishment of NBE format or the theoretical analysis needs more skills and is confronted with more difficulties than the second-order elliptic equations, standard parabolic equations, and hyperbolic equations as mentioned above, but it has certain special applications as mentioned above. Therefore, it is worth to study the NBE method for the Sobolev equation in the 2D unbounded domain.

The forthcoming contents are scheduled as follows. In Section 2, we establish a semi-discretized super-convergence format about time for the Sobolev equation in the 2D unbounded domain and deduce the super-convergence error estimates of the semi-discretized solutions about time. In the next Section 3, by the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and provide the error estimates between the exact solution and the fully discretized NBE solutions. In Section 4, we supply some numerical experimentations to validate that the numerical computational consequences are concordant with the theoretical ones. Section 5 summarizes main conclusions.

2 Semi-discretized formulation by the Newmark method and error estimate about time for the Sobolev equation in the 2D unbounded domain

By using Green’s formula, we can obtain the following variational form for the Sobolev equation.

Problem II

For \(t\in(0,T)\), find \(\varpi(t)\in H^{1}(\Theta^{c})\) that satisfies

$$\begin{aligned} (I-\varepsilon\Delta) (\varpi_{t},\upsilon)+\gamma( \nabla\varpi,\nabla\upsilon)=(f,\upsilon)+ \langle g,\upsilon \rangle,\quad \forall \upsilon\in H^{1}\bigl(\Theta^{c}\bigr), \end{aligned}$$
(2)

where \((\cdot,\cdot)\) represents the inner product in \(L^{2}(\Theta ^{c})\), but \(\langle\cdot,\cdot \rangle\) is the inner product in \(L^{2}(\Gamma)\).

The existence and uniqueness of the solution for the variational form, Problem II, are well known (see [5, 6, 12]).

Set τ is the time-step, \(t_{k}=k\cdot\tau\), \(\varpi^{k}=\varpi (\boldsymbol {x},t_{k})\), and \(\dot{\varpi}(\boldsymbol {x},t_{k})\)=\(\frac{\partial\varpi}{\partial t}(\boldsymbol {x},t_{k})\). By the Newmark method (see, e.g., [13]), we can establish the following iterative scheme:

$$\begin{aligned}& \dot{\varpi}^{k}(\boldsymbol {x})=\gamma(I-\varepsilon \Delta )^{-1}\Delta\varpi^{k}(\boldsymbol {x})+(I-\varepsilon \Delta )^{-1}f^{k}(\boldsymbol {x}),\quad k=1,2,\ldots, N, \end{aligned}$$
(3)
$$\begin{aligned}& \varpi^{k}(\boldsymbol {x})=\varpi^{k-1}( \boldsymbol {x})+\tau \bigl[(1-\beta)\dot{\varpi}^{k-1}(\boldsymbol {x})+ \beta\dot{\varpi }^{k}(\boldsymbol {x})\bigr],\quad k=1,2,\ldots, N, \end{aligned}$$
(4)

where \(\beta\in(0,1]\) and \(N=[T/\tau]\) (\([T/\tau]\) denotes the integer part of \(T/\tau\)).

By using \([(\varpi^{k}-\varpi^{k-1})/{\tau}-(1-\beta)\dot {\varpi}^{k-1} ]/\beta\) to approximate to \(\varpi_{t}^{k}\), we obtain the following semi-discretized format about time for the Sobolev equation.

Problem III

Find \(\varpi^{k}\in H^{1}(\Theta^{c})\) such that

$$\begin{aligned}& \bigl(\varpi^{k}-\varpi^{k-1},\upsilon\bigr)+(\varepsilon+ \beta\tau\gamma ) \bigl(\nabla\varpi^{k},\nabla\upsilon\bigr)+\bigl( \tau\gamma(1-\beta )-\varepsilon\bigr) \bigl(\nabla\varpi^{k-1},\nabla \upsilon\bigr) \\& \quad =\bigl(\beta\tau f^{k}+\tau(1-\beta)f^{k-1},\upsilon \bigr) + \bigl\langle (\beta\tau\gamma+\varepsilon)g^{k}+\bigl(\tau \gamma (1-\beta)-\varepsilon\bigr) g^{k-1},\upsilon \bigr\rangle , \\& \quad \forall\upsilon\in H^{1}\bigl(\Theta^{c}\bigr), k=1,2,\ldots, N, \end{aligned}$$
(5)

where \(\varpi^{0}=u_{0}(\boldsymbol {x})\), \(f^{k}=f(\boldsymbol {x},t_{k})\), and \(g^{k}=g(\boldsymbol {x},t_{k})\).

The solutions to Problem III possess the following conclusions.

Theorem 1

If \(f\in L^{2}(0,T; L^{2}(\Theta^{c}))\), \(g\in L^{2}(0,T; L^{2}(\Gamma))\), and \(u_{0}\in H^{1}(\Theta^{c})\), then Problem  III has a unique set of solutions \(\{\varpi^{k}\}_{k=1}^{N}\subset H^{1}(\Theta ^{c})\) satisfying

$$\begin{aligned} \bigl\Vert \varpi^{k} \bigr\Vert _{0}&\leq \Biggl(2 \Vert u_{0} \Vert _{0}^{2}+2(\varepsilon+\tau \beta\gamma) \Vert \nabla u_{0} \Vert _{0}^{2} \\ &\quad{}+16\sum_{i=1}^{k} \bigl(C_{0}\bigl(\beta\gamma+\varepsilon\tau ^{-1}\bigr) \bigl\Vert g^{i} \bigr\Vert _{0,\Gamma}^{2}+ \beta^{2}\tau \bigl\Vert f^{i} \bigr\Vert _{0,\Theta^{c}}^{2} \bigr) \Biggr)^{\frac{1}{2}}\exp (T/2), \end{aligned}$$
(6)

where \(C_{0}\) is the nonnegative constant in the trace theorem and \(k=1,2,\ldots, N\). This signifies that the solutions of Problem  III are stable and consecutively rely on the source function f, boundary value function g, and initial value function \(u_{0}\). In addition, when ϖ is sufficiently smooth about t, we have the following error estimates:

$$\begin{aligned} \bigl\Vert \nabla\bigl(\varpi(t_{k})- \varpi^{k}\bigr) \bigr\Vert _{0}= \bigl\Vert \nabla e^{k} \bigr\Vert _{0}\leq C\tau, \quad k=1,2,\ldots,N, \end{aligned}$$
(7)

where \(C^{2}=2\tau\varepsilon^{-1}\sum_{i=1}^{k}(4 \Vert (I-\varepsilon\Delta)\varpi_{tt}(\xi_{2}^{i}) \Vert _{0}^{2}+ \Vert (I-\varepsilon\Delta)\varpi_{tt}(\xi _{1}^{i}) \Vert _{0}^{2})\cdot\exp(T)\), \(t_{k-1}\leq\xi _{1}^{k}\), \(\xi_{2}^{k}\leq t_{k+1}\).

Proof

Because Problem III is a linear equation about unknown function ϖ, to prove the existence and uniqueness of solutions of Problem III is equivalent to demonstrating that it has only a zero solution when \(g=f=u_{0}=0\).

Choosing \(\upsilon=\varpi^{k}\) in Problem III together with utilizing the Hölder and Cauchy-Schwarz inequalities and the trace theorem (see [14]), we have

$$\begin{aligned}& \bigl\Vert \varpi^{k} \bigr\Vert _{0}^{2}+( \varepsilon+\beta\tau \gamma) \bigl\Vert \nabla\varpi^{k} \bigr\Vert _{0}^{2} \\& \quad\leq4 \bigl( \beta^{2}\tau \bigl\Vert f^{k} \bigr\Vert _{0}^{2}+C_{0}\tau^{-1}( \varepsilon+\beta\tau\gamma) \bigl\Vert g^{k} \bigr\Vert _{0,\Gamma}^{2} \bigr)+\frac{1}{2} \bigl( \bigl\Vert \varpi^{k} \bigr\Vert _{0}^{2}+ \bigl\Vert \varpi^{k-1} \bigr\Vert _{0}^{2} \bigr) \\& \qquad{} +\frac{\tau}{4} \bigl( \bigl\Vert \varpi^{k} \bigr\Vert _{0}^{2}+(\varepsilon+\beta\tau\gamma) \bigl\Vert \nabla\varpi ^{k} \bigr\Vert _{0}^{2} \bigr)+ \frac{(\varepsilon+\beta\tau \gamma)}{2} \bigl( \bigl\Vert \nabla\varpi^{k-1} \bigr\Vert _{0}^{2}+ \bigl\Vert \nabla\varpi^{k} \bigr\Vert _{0}^{2} \bigr), \end{aligned}$$
(8)

where \(C_{0}\) is the nonnegative constant in the trace theorem (see [14]). By summing (8) from 1 to k, we have

$$\begin{aligned}& \bigl\Vert \varpi^{k} \bigr\Vert _{0}^{2}+( \varepsilon+\beta\tau \gamma) \bigl\Vert \nabla\varpi^{k} \bigr\Vert _{0}^{2} \\& \quad\leq\frac{\tau}{2}\sum_{i=1}^{k} \bigl( \bigl\Vert u^{i} \bigr\Vert _{0}^{2}+( \varepsilon+\beta\tau\gamma) \bigl\Vert \nabla u^{i} \bigr\Vert _{0}^{2}\bigr)+ \Vert u_{0} \Vert _{0}^{2}+(\varepsilon+\beta\tau\gamma) \Vert \nabla u_{0} \Vert _{0}^{2} \\& \qquad{} +8\sum_{i=1}^{k} \bigl(C_{0}\tau^{-1}(\varepsilon+\beta\tau\gamma) \bigl\Vert g^{i} \bigr\Vert _{0,\Gamma}^{2}+\beta^{2} \tau \bigl\Vert f^{i} \bigr\Vert _{0}^{2} \bigr), \quad1\le k\le N. \end{aligned}$$
(9)

When τ is adequately small such that \({\tau}\leq1\), by Gronwall’s lemma (see [15, 16]), we have

$$\begin{aligned} &\bigl\Vert \varpi^{k} \bigr\Vert _{0} \\ &\quad \leq \Biggl[2 \Vert u_{0} \Vert _{0}^{2}+2(\varepsilon+\tau \beta\gamma) \Vert \nabla u_{0} \Vert _{0}^{2} \\ &\qquad{}+16\sum_{i=1}^{k} \bigl(C_{0}\bigl(\beta\gamma+\varepsilon\tau ^{-1}\bigr) \bigl\Vert g^{i} \bigr\Vert _{0,\Gamma}^{2}+ \beta^{2}\tau \bigl\Vert f^{i} \bigr\Vert _{0,\Theta^{c}}^{2} \bigr) \Biggr]^{\frac{1}{2}}\cdot \exp(T/2), \quad1\le k\le N. \end{aligned}$$
(10)

Thus, when \(f=g=u_{0}=0\), from (10), we obtain \(\Vert \varpi ^{k} \Vert _{0}=0\), implying \(\varpi^{k}=0\) (\(k=1, 2, \ldots, N\)). Therefore, Problem III has a unique set of solutions.

With Taylor’s expanding formula, we acquire

$$\begin{aligned} \varpi_{t}(t_{k}) =&\frac{1}{\beta} \biggl\{ \frac{1}{\tau} \bigl[\varpi(t_{k})-\varpi(t_{k-1}) \bigr]-(1-\beta)\varpi _{t}(t_{k-1}) \biggr\} \\ &{} +\frac{\tau}{2\beta}\varpi_{tt}\bigl(\xi^{k}_{1} \bigr)-\frac{(1-\beta)\tau }{\beta}\varpi_{tt}\bigl(\xi^{k}_{2} \bigr), \end{aligned}$$
(11)

where \(t_{k-1}\leq\xi_{1}^{k}, \xi_{2}^{k}\leq t_{k+1}\). From Problem II, we obtain

$$\begin{aligned}& (\varpi(t_{k})-\varpi(t_{k-1},\upsilon)+(\varepsilon+\beta \tau \gamma) \bigl(\nabla\varpi(t_{k}),\nabla\upsilon\bigr)+\bigl(\tau \gamma(1-\beta )-\varepsilon\bigr) \bigl(\nabla\varpi(t_{k-1}),\nabla \upsilon\bigr) \\& \quad=\bigl(\beta\tau f(t_{k})+\tau(1-\beta)f(t_{k-1}), \upsilon\bigr)+\tau ^{2}(1-\beta) \bigl((I-\varepsilon\Delta) \varpi_{tt}\bigl(\xi^{k}_{2}\bigr),\upsilon\bigr) \\& \qquad {}+\bigl\langle (\beta\tau\gamma+\varepsilon)g(t_{k})+\bigl(\tau\gamma (1-\beta)-\varepsilon\bigr) g(t_{k-1}),\upsilon \bigr\rangle - \frac {\tau^{2}}{2}\bigl((I-\varepsilon\Delta)\varpi_{tt}\bigl( \xi^{k}_{1}\bigr),\upsilon\bigr), \\& \quad\forall\upsilon\in H^{1}\bigl(\Theta^{c} \bigr). \end{aligned}$$
(12)

Let \(e^{k}=\varpi(t_{k})-\varpi^{k}\). By subtracting (5) from (12) taking \(t=t_{k}\), we obtain

$$\begin{aligned}& \bigl(e^{k}-e^{k-1},\upsilon\bigr)+(\varepsilon+\beta\tau) \bigl(\nabla e^{k},\nabla\upsilon\bigr)+\bigl(\tau\gamma(1-\beta)- \varepsilon\bigr) \bigl(\nabla e^{k-1},\nabla\upsilon\bigr) \\& \quad=\tau^{2}(1-\beta) \bigl((I-\varepsilon\Delta) \varpi_{tt}\bigl(\xi _{2}^{k}\bigr),\upsilon \bigr)-\frac{\tau^{2}}{2}\bigl((I-\varepsilon\Delta)\varpi _{tt}\bigl( \xi_{1}^{k}\bigr),\upsilon\bigr). \end{aligned}$$
(13)

By taking \(\upsilon=e^{k}\) in (13) and using the Cauchy and Hölder-Schwarz inequalities, we have

$$\begin{aligned} &\bigl\Vert e^{k} \bigr\Vert _{0}^{2}+( \varepsilon+\beta\tau\gamma ) \bigl\Vert \nabla e^{k} \bigr\Vert _{0}^{2} \\ &\quad \leq\frac{1}{2}\bigl( \bigl\Vert e^{k} \bigr\Vert _{0}^{2}+ \bigl\Vert e^{k-1} \bigr\Vert _{0}^{2}\bigr)+ \tau^{3} \bigl\Vert (I-\varepsilon\Delta)\varpi_{tt}\bigl( \xi _{1}^{k}\bigr) \bigr\Vert _{0}^{2} \\ &\qquad{}+\frac{\tau}{4} \bigl\Vert e^{k} \bigr\Vert _{0}^{2}+2\tau ^{3}\beta^{2} \bigl\Vert (I-\varepsilon\Delta)\varpi_{tt}\bigl(\xi _{2}^{k} \bigr) \bigr\Vert _{0}^{2}+\frac{(\varepsilon+\beta\tau\gamma )}{2}\bigl( \bigl\Vert \nabla e^{k-1} \bigr\Vert _{0}^{2}+ \bigl\Vert \nabla e^{k} \bigr\Vert _{0}^{2} \bigr). \end{aligned}$$
(14)

By summing (14) from 1 to k, when τ is adequately small such that \(\tau/2\leq1/2\), we have

$$\begin{aligned} \bigl\Vert e^{k} \bigr\Vert _{0}^{2}+2( \varepsilon+\beta\tau\gamma ) \bigl\Vert \nabla e^{k} \bigr\Vert _{0}^{2}&\leq\tau\sum_{i=1}^{k-1} \bigl\Vert e^{i} \bigr\Vert _{0}^{2}+2 \tau^{3}\sum_{i=1}^{k}\bigl( \bigl\Vert (I-\varepsilon\Delta)\varpi_{tt}\bigl(\xi _{1}^{i} \bigr) \bigr\Vert _{0}^{2} \\ &\quad{}+4(1-\beta)^{2} \bigl\Vert (I-\varepsilon\Delta)\varpi _{tt}\bigl(\xi_{2}^{i}\bigr) \bigr\Vert _{0}^{2}\bigr). \end{aligned}$$
(15)

By Gronwall’s lemma (see [15, 16]), we have

$$\begin{aligned} \bigl\Vert e^{k} \bigr\Vert _{0}^{2}+2( \varepsilon+\beta\tau\gamma ) \bigl\Vert \nabla e^{k} \bigr\Vert _{0}^{2}&\leq2\tau^{3}\sum _{i=1}^{k}\bigl(4(1-\beta)^{2} \bigl\Vert (I-\varepsilon\Delta)\varpi _{tt}\bigl(\xi_{2}^{i} \bigr) \bigr\Vert _{0}^{2} \\ &\quad{}+ \bigl\Vert (I-\varepsilon\Delta)\varpi_{tt}\bigl(\xi _{1}^{i}\bigr) \bigr\Vert _{0}^{2} \bigr)\cdot\exp(T). \end{aligned}$$
(16)

From (16), we obtain

$$\begin{aligned} \bigl\Vert \nabla\bigl(\varpi(t_{k})- \varpi^{k}\bigr) \bigr\Vert _{0}= \bigl\Vert \nabla e^{k} \bigr\Vert _{0}\leq C\tau, \quad k=1,2,\ldots,N, \end{aligned}$$
(17)

where \(C^{2}=2\tau\varepsilon^{-1}\sum_{i=1}^{k}(4 \Vert (I-\varepsilon\Delta)\varpi_{tt}(\xi_{2}^{i}) \Vert _{0}^{2}+ \Vert (I-\varepsilon\Delta)\varpi_{tt}(\xi _{1}^{i}) \Vert _{0}^{2})\cdot\exp(T)\). This finishes the proof of Theorem 1. □

3 Natural boundary reduction on the outside circle area and error estimate for the fully discretized NBE solutions

We define \(\mu:=(\sqrt{(\varepsilon+\tau\beta\gamma)} )^{-1} \), \(\widetilde{\varpi}^{k}:=(I-\varepsilon\Delta)\varpi^{k-1}+\tau (I-\varepsilon\Delta)(1-\beta)\dot{\varpi}^{k-1}\), \(\widetilde {f}:= -\widetilde{\varpi}^{k}-\tau\beta f^{k}\).

The procedure for solving the above semi-discretized Problem III is given as follows.

(1) Prediction

$$\begin{aligned} \widetilde{\varpi}^{k}:= (I-\varepsilon\Delta) \varpi^{k-1}+\tau (I-\varepsilon\Delta) (1-\beta)\dot{ \varpi}^{k-1}. \end{aligned}$$
(18)

(2) Solve the problem

$$\begin{aligned}& \Delta\varpi^{k}-\mu^{2} \varpi^{k}=\mu^{2}\widetilde{f}^{k}, \quad \boldsymbol {x}\in\Theta^{c}, \end{aligned}$$
(19)
$$\begin{aligned}& \frac{\partial\varpi^{k}}{\partial n}=g^{k},\quad \boldsymbol {x}\in \Gamma, \end{aligned}$$
(20)
$$\begin{aligned}& \bigl\vert \varpi^{k} \bigr\vert < +\infty, \quad \vert \boldsymbol {x} \vert \rightarrow+\infty. \end{aligned}$$
(21)

(3) Update value

$$\begin{aligned} \dot{\varpi}^{k}= \frac{1}{\tau\beta}\bigl( \varpi^{k}-\widetilde {\varpi}^{k}\bigr). \end{aligned}$$
(22)

From the above procedure, it is not difficult to find that our main task is to solve elliptic boundary value problems at each time level.

Let \(I_{n}(x)\) and \(K_{n}(x)\) (\(n=0,1,2,\ldots\)) be, severally, the first and second type of modified Bessel functions (see [17]), and \(\Re_{\mu}\) and \(\Im_{\mu}\) be, respectively, the natural and Poisson integral operators (see [1, 10]). From the NBE method (see [1, 18, 19]), the Dirichlet boundary \(\hat{\varpi}_{0}^{k}\) and the Neumann boundary \(\frac{\partial\varpi^{k}}{\partial\boldsymbol {n}}\) satisfy the following relationship:

$$\begin{aligned} \frac{\partial\varpi^{k}}{\partial\boldsymbol {n}}+\mathcal{ N}\bigl(\mu,r; \widetilde{f}^{k}, \theta\bigr)=\Re_{\mu}\hat{\varpi}_{0}^{k}, \end{aligned}$$
(23)

and the relationship between the solution \(\varpi^{k}\) of Problem III with its Dirichlet boundary value \(\hat{\varpi}_{0}^{k}\) is as follows:

$$\begin{aligned} \varpi^{k}=\Im_{\mu}\hat{ \varpi}_{0}^{k}+\mathcal{F}\bigl(\mu,r;\widetilde{f}^{k},R, \theta\bigr), \end{aligned}$$
(24)

where

$$\begin{aligned}& \mathcal{N}\bigl(\mu,r;\tilde{f}^{k},\theta\bigr) = \frac{\mu^{2}}{2}\sum_{n=0}^{+\infty} \xi_{n} \int_{r}^{+\infty}\bar{G}_{n}(\mu,r;\sigma) \bigl[\tilde{f}_{n}^{k,c}(\sigma)\cos n\theta+\tilde {f}_{n}^{k,s}(\sigma)\sin n\theta \bigr]\,\mathrm{d}\sigma, \\& \quad\xi_{0}=1; \xi_{n}=2, n=1,2, \ldots, \\& \bar{G}_{n}(\mu,r;\sigma)=-\frac{K_{n}(\mu\sigma)}{K_{n}(\mu r)}\cdot\frac{\sigma}{r},\quad n=0,1,2,\ldots, \\& \mathcal{F}\bigl(\mu,r;\tilde{f}^{k},R,\theta\bigr)= \frac{\mu^{2}}{2}\sum_{n=0}^{+\infty} \xi_{n} \int_{r}^{+\infty}\sigma^{2}G_{n}(R, \sigma ) \bigl[\tilde{f}_{n}^{k,c}(\sigma)\cos n\theta+\tilde {f}_{n}^{k,s}(\sigma)\sin n\theta \bigr]\,\mathrm{d}\sigma, \\& \sigma^{2}G_{n}(R,\sigma)= \textstyle\begin{cases} \frac{\phi_{n}(\sigma)\psi_{n}(R)}{E_{n}(\sigma)}, & R\leq\sigma, \\ \frac{\psi_{n}(\sigma)\phi_{n}(R)}{E_{n}(\sigma)}, &R\geq\sigma, \end{cases}\displaystyle \quad n=0,1,2,\ldots, \\& \phi_{n}(\sigma)=K_{n}(\mu\sigma), \qquad \psi_{n}(\sigma)=I_{n}(\mu\sigma)K_{n}(\mu r)-K_{n}(\mu\sigma )I_{n}(\mu r),\quad n=0,1,2,\ldots, \\& E_{n}(\sigma)=\psi_{n}(\sigma)\phi'_{n}( \sigma)-\phi_{n}(\sigma )\psi'_{n}(\sigma),\quad n=0,1,2,\ldots, \\& \tilde{f}_{n}^{k,c}(\sigma)=\frac{1}{\pi} \int_{0}^{2\pi}\tilde {f}^{k}(\sigma, \theta)\cos n\theta\,\mathrm{d}\theta,\quad n=0,1,2,\ldots, \\& \tilde{f}_{n}^{k,s}(\sigma)=\frac{1}{\pi} \int_{0}^{2\pi}\tilde {f}^{k}(\sigma, \theta)\sin n\theta\, \mathrm{d}\theta,\quad n=1,2, \ldots. \end{aligned}$$

The above (23) and (24) are named the natural and the Poisson integral equations, respectively. Thus, (23) is equivalent to the following variation form:

Find \(\varpi_{0}^{k}\in H^{\frac{1}{2}}(\Gamma)\) (\(1\le k\le N\)) that satisfy

$$\begin{aligned} \hat{B}\bigl(\varpi_{0}^{k}, \nu^{k}\bigr)= \bigl\langle g^{k}(r,\theta )+\mathcal{N} \bigl(\mu,r;\widetilde{f^{k}},\theta\bigr),\nu^{k} \bigr\rangle , \quad \forall\nu^{k}\in H^{\frac{1}{2}}(\Gamma), \end{aligned}$$
(25)

where \(\hat{B}(\varpi_{0}^{k},\nu^{k})= \langle\Re_{\mu}\hat {\varpi}_{0}^{k},\nu^{k} \rangle=:\int_{\Gamma}(\Re_{\mu }\hat{\varpi}_{0}^{k})\nu^{k}\,\mathrm{d}s\) and \(\langle\omega,\ell \rangle=:\int_{\Gamma}\omega\ell\,\mathrm{d}s\).

3.1 Natural boundary reduction on the external circle area

Now, let the domain Θ be a circle with radius r and center at origin. For convenience, we also suppose that the solutions \(\varpi ^{k}\) of Problem III are appropriately smooth. Under the polar coordinates, \(\Gamma=\{(R, \theta): R=r, \theta\in[0,2\pi]\} \), \(\Theta^{c}= \{(R, \theta): R= \vert \boldsymbol{x} \vert >r,\theta\in[0,2\pi] \}\), and the external normal derivative operator onto Γ satisfies \(\frac{\partial }{\partial\boldsymbol {n}}=-\frac{\partial}{\partial\boldsymbol {R}}\). The solution of equations (19)-(21) can be expressed with the following form in the polar coordinates:

$$\begin{aligned} \varpi^{k}(R,\theta)=\frac{1}{2}a_{0}(R)+ \sum_{n=1}^{+\infty} \bigl[a_{n}(R)\cos n\theta+b_{n}(R)\sin n\theta \bigr], \end{aligned}$$
(26)

where

$$\begin{aligned}& a_{n}(R)=\frac{1}{\pi} \int_{0}^{2\pi}\varpi^{k}(R,\theta)\cos n \theta\mathrm{d}\theta, \quad n=0,1,2,\ldots; \\& b_{n}(R)=\frac{1}{\pi} \int_{0}^{2\pi}\varpi^{k}(R,\theta)\sin n \theta\mathrm{d}\theta, \quad n=1,2,\ldots. \end{aligned}$$

By calculating, we get the solution \(\varpi^{k}(R,\theta)\) of equations (19)-(21) as follows.

$$\begin{aligned}& \begin{aligned} \varpi^{k}(R,\theta)&=\frac{1}{2\pi} \sum_{n=0}^{+\infty}\xi _{n} \int_{0}^{2\pi}\frac{K_{n}(\mu R)}{K_{n}(\mu r)}\cos n\bigl(\theta - \theta'\bigr)\varpi^{k}\bigl(r,\theta' \bigr) \,\mathrm{d}\theta' \\ &\quad {}+\mathcal{F}\bigl(\mu,r;\tilde{f}^{k},R,\theta\bigr),\quad R>r, \end{aligned} \end{aligned}$$
(27)
$$\begin{aligned}& \frac{\partial\varpi^{k}(r,\theta)}{\partial\boldsymbol {n}}+\mathcal{N}\bigl(\mu,r;\tilde{f}^{k}, \theta\bigr)=\frac{\mu}{2\pi} \int _{0}^{2\pi}\bar{K}_{n}\bigl(\mu,r; \theta-\theta'\bigr)\varpi^{k}\bigl(r,\theta '\bigr)\,\mathrm{d}\theta', \end{aligned}$$
(28)

where \(\bar{K}_{n}(\mu,r;\theta-\theta')=-\sum_{n=0}^{+\infty}\xi _{n}\cos n(\theta-\theta')\cdot{K'_{n}(\mu r)}/{K_{n}(\mu r)}\).

Remark 1

The formats (27) and (28) are, respectively, the Poisson and the natural integral equations. We can attain the solution \(\varpi^{k}(r,\theta)\) from the natural integral equation (28) and then obtain the solution of the original boundary value, i.e., Problem I, by the Poisson integral formula (27). But the solution of Problem I can be acquired directly by the Poisson integral formula (27) for the Cauchy-Dirichlet initial boundary value problem, because the function \(\varpi^{k}(r,\theta)\) is known.

Remark 2

In numerical computations, we can limit \(r\leq R<+\infty\) on the \(r\leq R \leq R_{\mathrm{max}}\) and use the corresponding numerical integral to calculate the integral calculation of the \(\mathcal{N}(\mu,r;\tilde{f}^{k},\theta)\) and \(\mathcal {F}(\mu,r;\tilde{f}^{k},R,\theta)\). Meanwhile, the expressions of an infinite series summation can be substituted with a finite series summation in the practical applications.

3.2 Error analysis of NBE solutions

We divide the circumference Γ into some finite elements, which satisfy regular conditions. For the sake of simplicity, we take uniform subdivision. Let \(\mathcal{S}_{h}(\Gamma) \subset H^{1/2}(\Gamma)\) be a finite element subspace spanned by appropriate basis functions. Thus, the NBE approximation to Problem II is as follows.

Problem IV

Find \(\varpi^{k}_{0h}\in\mathcal {S}_{h}(\Gamma)\) (\(1\le k\le N\)) that satisfy

$$\begin{aligned} \hat{B}\bigl(\varpi^{k}_{0h}, \nu^{k}\bigr)= \bigl\langle g^{k}(r,\theta )+\mathcal{N} \bigl(\mu,r;\widetilde{f^{k}},\theta\bigr),\nu^{k} \bigr\rangle , \quad \forall\nu^{k}\in\mathcal{S}_{h}(\Gamma), \end{aligned}$$
(29)

and

$$\begin{aligned} \varpi^{k}_{h}(R,\theta) =&\frac{1}{2\pi}\sum _{n=0}^{+\infty}\xi _{n} \int_{0}^{2\pi}\frac{K_{n}(\mu R)}{K_{n}(\mu r)}\cos n\bigl(\theta - \theta'\bigr)\varpi^{k}_{0h}\bigl(r, \theta'\bigr)\,\mathrm{d}\theta' \\ &{}+\mathcal{F}\bigl(\mu,r;\tilde{f}^{k},R,\theta\bigr),\quad R>r. \end{aligned}$$
(30)

The above formula (30) is the approximate expression of Poisson integral formula (27). In order to analyze the errors of NBE solutions of Problem IV, it is necessary to introduce the following \(L^{2}\)-projection and its property (see [16]).

Definition 1

An operator \(P_{h}: L_{2}(\Gamma )\rightarrow\mathcal{S}_{h}(\Gamma)\) (where \(\mathcal{S}_{h}(\Gamma )\subset L^{2}(\Gamma)\) is a finite element space) is known as an \(L^{2}\)-projection if, for any \(\upsilon\in L^{2}(\Gamma)\), there exists unique \(P_{h}\upsilon\in\mathcal{S}_{h}(\Gamma)\) satisfying

$$\begin{aligned} \langle\upsilon-P_{h}\upsilon, \upsilon_{h} \rangle =0, \quad\forall\upsilon_{h}\in\mathcal{S}_{h}(\Gamma). \end{aligned}$$

Lemma 1

If \(\mathcal{S}_{h}(\Gamma)\) is a subspace spanned by piecewise linear polynomials and \(\upsilon\in H^{2}(\Gamma)\), then the \(L^{2}\)-projection \(P_{h}\) satisfies

$$\begin{aligned} \Vert \upsilon-P_{h}\upsilon \Vert _{s}\le Ch^{2-s} \Vert \upsilon \Vert _{2,\Gamma}, \quad s=-1,0,1, \end{aligned}$$

where C used next represents a generic positive real independent of τ and h.

Problem IV possesses the following conclusion.

Theorem 2

Let \(\varpi_{0}^{k}\in H^{\frac{1}{2}}(\Gamma)\) and \(\varpi _{0h}^{k}\) be, respectively, solutions to (25) together with (29), \(\tau=O(h^{2})\), and \(\mathcal{S}_{h}(\Gamma)\) be the piecewise linear polynomial subspace. Then we have the following error estimates:

$$\begin{aligned} \big\Vert \varpi_{0}^{k}- \varpi_{0h}^{k}\big\Vert _{0,\Gamma}\leq Ch^{2},\quad k=1,2, \ldots, N. \end{aligned}$$
(31)

Proof

By subtracting (29) from (25) taking \(\nu ^{k}=\nu^{k}_{h}\), we obtain

$$\begin{aligned} &\hat{B}\bigl(\varpi_{0}^{k}-\varpi^{k}_{0h}, \nu_{h}^{k}\bigr) \\ &\quad = \Biggl\langle \frac{\mu^{2}}{2\pi}\sum _{n=0}^{+\infty}\xi_{n} \int_{r}^{+\infty}\bar {G}_{n}(\mu,r;\sigma) \biggl[ \int_{0} ^{2\pi}\bigl[-\bigl(\varpi_{0}^{k-1}- \varpi ^{k-1}_{0h}\bigr) \\ &\qquad {}+\varepsilon\Delta\bigl(\varpi_{0}^{k-1}- \varpi^{k-1}_{0h}\bigr)-\tau (1-\beta)\gamma\Delta\bigl( \varpi_{0}^{k-1}-\varpi^{k-1}_{0h}\bigr) \bigr]\cos n \hat{\theta}\,\mathrm{d}\hat{\theta}\cos n\theta \\ &\qquad {}+ \int_{0}^{2\pi}\bigl[-(I-\varepsilon\Delta) \bigl( \varpi_{0}^{k-1}-\varpi ^{k-1}_{0h}\bigr)- \tau\gamma\Delta\bigl(\varpi_{0}^{k-1}-\varpi^{k-1}_{0h} \bigr) \\ &\qquad {} +\tau\beta\gamma\Delta\bigl(\varpi_{0}^{k-1}- \varpi^{k-1}_{0h}\bigr)\bigr]\sin n\hat{\theta}\,\mathrm{d}\hat{ \theta}\sin n\theta \biggr]\,\mathrm {d}\sigma,\nu_{h}^{k} \Biggr\rangle ,\quad \forall\nu^{k}\in\mathcal {S}_{h}( \Gamma). \end{aligned}$$
(32)

Due to \(\hat{B}(\cdot,\cdot)\) being positive definite on \(H^{\frac {1}{2}}(\Gamma)\times H^{\frac{1}{2}}(\Gamma)\) (see [1]), by using the Hölder inequality, we have

$$\begin{aligned} &M \bigl\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \bigr\Vert ^{2}_{0,\Gamma }\\ &\quad \leq \bigl\vert \hat{B}\bigl( \varpi_{0}^{k}-\varpi_{0h}^{k},\varpi _{0}^{k}-\varpi_{0h}^{k}\bigr) \bigr\vert \\ &\quad \leq \bigl\vert \hat{B}\bigl(\varpi_{0}^{k}- \varpi_{0h}^{k},\varpi _{0}^{k}-P_{h} \varpi_{0}^{k}\bigr) \bigr\vert + \bigl\vert \hat{B} \bigl(\varpi _{0}^{k}-\varpi_{0h}^{k},P_{h} \varpi_{0}^{k}-\varpi_{0h}^{k}\bigr) \bigr\vert \\ &\quad \leq \bigl\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \bigr\Vert _{0,\Gamma} \bigl\Vert \varpi_{0}^{k}-P_{h} \varpi_{0}^{k} \bigr\Vert _{0,\Gamma}+ \Biggl\vert \Biggl\langle \frac{\mu^{2}}{2\pi}\sum_{n=0}^{+\infty} \xi_{n} \int_{r}^{+\infty}\bar{G}_{n}(\mu,r;\sigma) \\ &\qquad {}\times \biggl[ \int_{0} ^{2\pi}\bigl[-\bigl(\varpi_{0}^{k-1}- \varpi ^{k-1}_{0h}\bigr)+\varepsilon\Delta\bigl( \varpi_{0}^{k-1}-\varpi ^{k-1}_{0h}\bigr) \\ &\qquad {}-\tau(1-\beta)\gamma\Delta\bigl(\varpi_{0}^{k-1}- \varpi ^{k-1}_{0h}\bigr)\bigr]\cos n \hat{\theta}\,\mathrm{d} \hat{\theta}\cos n\theta \\ &\qquad {}+ \int_{0}^{2\pi} \bigl[-(I-\varepsilon\Delta) \bigl( \varpi _{0}^{k-1}-\varpi^{k-1}_{0h}\bigr) \\ &\qquad {}-\tau(1-\beta)\gamma\Delta\bigl(\varpi _{0}^{k-1}-\varpi^{k-1}_{0h}\bigr) \bigr] \sin n\hat{\theta}\,\mathrm{d}\hat {\theta}\sin n\theta \biggr]\,\mathrm{d} \sigma,P_{h}\varpi _{0}^{k}-\varpi_{0h}^{k} \Biggr\rangle \Biggr\vert \\ &\quad \leq \bigl\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \bigr\Vert _{0,\Gamma} \bigl\Vert \varpi_{0}^{k}-P_{h} \varpi_{0}^{k} \bigr\Vert _{0,\Gamma}+C \bigl[ \bigl\Vert \varpi_{0}^{k-1}-\varpi _{0h}^{k-1} \bigr\Vert _{0,\Gamma} \bigl\Vert P_{h}\varpi _{0}^{k}-\varpi^{k}_{0} \bigr\Vert _{0,\Gamma} \\ &\qquad {}+\bigl(\varepsilon \bigl\Vert \varpi_{0}^{k-1}- \varpi_{0h}^{k-1} \bigr\Vert _{0,\Gamma}+\tau \bigl\Vert \nabla\bigl(\varpi_{0}^{k-1}-\varpi _{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma}\bigr) \bigl\Vert P_{h}\varpi _{0}^{k}-\varpi_{0}^{k} \bigr\Vert _{0,\Gamma} \\ &\qquad {}-\tau\beta \bigl\Vert \nabla\bigl(\varpi_{0}^{k-1}- \varpi _{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \bigl\Vert P_{h}\varpi _{0}^{k}-\varpi_{0}^{k} \bigr\Vert _{0,\Gamma} \\ &\qquad {}+(1+\varepsilon) \bigl\Vert \varpi_{0}^{k-1}-\varpi _{0h}^{k-1} \bigr\Vert _{0,\Gamma} \bigl\Vert \varpi_{0}^{k}-\varpi ^{k}_{0h} \bigr\Vert _{0,\Gamma} \\ &\qquad {}+\tau\gamma(1-\beta) \bigl\Vert \nabla\bigl(\varpi _{0}^{k-1}-\varpi_{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \bigl\Vert \varpi _{0}^{k}- \varpi^{k}_{0h} \bigr\Vert _{0,\Gamma} \bigr]. \end{aligned}$$

Further, by the Cauchy-Schwarz inequality and Lemma 1 as well as the inverse estimating theorem (see [16]), we acquire

$$\begin{aligned} \big\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \big\Vert _{0,\Gamma}^{2}\leq C\bigl( \bigl\Vert \varpi_{0}^{k-1}-\varpi_{0h}^{k-1} \bigr\Vert _{0,\Gamma }^{2}+h^{4}\bigr). \end{aligned}$$

Further, we acquire

$$\begin{aligned} \big\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \big\Vert _{0,\Gamma}\leq Ch^{2}. \end{aligned}$$

This finishes the proof of Theorem 2. □

The solutions for approximate expression (30) have the following error estimates.

Theorem 3

Let \(\varpi^{k}\) and \(\varpi_{h}^{k}\) be, respectively, the solutions of (27) and (30) and \(\tau=O(h^{2})\). Then we have the error estimates:

$$\begin{aligned} \big\Vert \varpi^{k}-\varpi_{h}^{k} \big\Vert _{0,\infty,\Theta ^{c}}\leq Ch^{2}. \end{aligned}$$
(33)

Proof

By the literature [17], we can immediately derive

$$\begin{aligned} K_{n}(x)=\sqrt{\frac{\pi}{2n}} \biggl(\frac{2n}{ex} \biggr)^{n}\bigl[1+o\bigl(n^{-1}\bigr)\bigr], \quad n \rightarrow+\infty. \end{aligned}$$

Therefore, we have \({K_{n}(\mu R)}/{K_{n}(\mu r)} \leq1\) (\(r < R\)). Thus, there is real \(\tilde{M}\geq0\) that satisfies

$$\begin{aligned} \Biggl\vert \sum_{n=0}^{+\infty} \xi_{n}\frac{K_{n}(\mu R)}{K_{n}(\mu r)} \Biggr\vert \leq\tilde{M}. \end{aligned}$$

By using (18), we have \(\tilde{f}^{k}=[-(I-\varepsilon\Delta )-\tau(1-\beta)\gamma\Delta]\varpi^{k-1}-\tau\beta f^{k}-\tau (1-\beta)f^{k-1}\). Thus, from (27) and (30), we have

$$\begin{aligned} &\bigl\vert \varpi^{k}-\varpi_{h}^{k} \bigr\vert \\ &\quad \leq\frac {1}{2\pi} \Biggl\vert \sum_{n=0}^{+\infty} \xi_{n} \int_{0}^{2\pi }\sum_{n=0}^{+\infty} \frac{K_{n}(\mu R)}{K_{n}(\mu r)}\cos n\bigl(\theta -\theta'\bigr)\cdot\bigl( \varpi_{0}^{k}-\varpi_{0h}^{k}\bigr) \,\mathrm{d}\theta ' \Biggr\vert \\ &\qquad {}+\frac{\mu^{2}}{2\pi}\sum_{n=0}^{+\infty} \xi_{n} \int _{r}^{+\infty}\sigma^{2}G_{n}(R, \sigma) \biggl\{ \biggl[ \int _{0}^{2\pi} \bigl\vert -(I-\varepsilon\Delta) \bigl(\varpi_{0}^{k-1} -\varpi_{0h}^{k-1} \bigr)\cos n\hat{\theta} \bigr\vert \,\mathrm{d}\hat {\theta} \\ &\qquad {}+ \int_{0}^{2\pi} \bigl\vert -\tau(1-\beta)\gamma \Delta \bigl(\varpi_{0}^{k-1}-\varpi_{0h}^{k-1} \bigr)\cos n\hat{\theta} \bigr\vert \,\mathrm{d}\hat{\theta} \biggr]\cos n\theta \\ &\qquad {}+ \biggl[ \int_{0}^{2\pi} \bigl\vert -(I-\varepsilon\Delta) \bigl(\varpi _{0}^{k-1}-\varpi_{0h}^{k-1} \bigr)\sin n\hat{\theta} \bigr\vert \,\mathrm{d}\hat{\theta} \\ &\qquad {}+ \int_{0}^{2\pi} \bigl\vert -\tau(1-\beta)\gamma \Delta\bigl(\varpi_{0}^{k-1}-\varpi_{0h}^{k-1} \bigr)\sin n\hat{\theta} \bigr\vert \,\mathrm{d}\hat{\theta} \biggr]\sin n\theta \biggr\} \,\mathrm{d}\sigma \\ &\quad \leq \Biggl\vert \frac{1}{2\pi}\sum_{n=0}^{+\infty} \xi_{n}\frac {K_{n}(\mu R)}{K_{n}(\mu r)} \Biggr\vert \biggl( \int_{0}^{2\pi}\cos ^{2} n\bigl(\theta- \theta'\bigr)\,\mathrm{d}\theta' \biggr)^{\frac{1}{2}} \cdot \bigl\Vert \varpi_{0}^{k}-\varpi_{0h}^{k} \bigr\Vert _{0,\Gamma} \\ &\qquad {}+\frac{\mu^{2}}{2\pi}\sum_{n=0}^{+\infty} \xi_{n} \int _{r}^{+\infty}\sigma^{2}G_{n}(R, \sigma)\cdot \biggl\{ \biggl[ \bigl\Vert \varpi_{0}^{k-1}- \varpi_{0h}^{k-1} \bigr\Vert _{0,\Gamma} \biggl( \int_{0}^{2\pi}\cos^{2} n\hat{\theta} \,\mathrm{d}\hat{\theta} \biggr)^{\frac{1}{2}} \\ &\qquad {}+\varepsilon \bigl\Vert \bigl(\varpi_{0}^{k-1}- \varpi_{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \Vert \Delta\cos n\hat{\theta} \Vert _{0} \\ &\qquad {}+\tau(1-\beta)\gamma \bigl\Vert \nabla\bigl(\varpi _{0}^{k-1}-\varpi_{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \Vert \nabla \cos n\hat{\theta} \Vert _{0} \biggr]\cos n\theta \\ &\qquad {}+ \biggl[ \bigl\Vert \varpi_{0}^{k-1}- \varpi_{0h}^{k-1} \bigr\Vert _{0,\Gamma} \biggl( \int_{0}^{2\pi}\sin^{2} n\hat{\theta } \,\mathrm{d}\hat{\theta} \biggr)^{\frac{1}{2}} \\ &\qquad {}+\varepsilon \bigl\Vert \bigl(\varpi_{0}^{k-1}- \varpi _{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \Vert \Delta\sin n\hat {\theta} \Vert _{0} \\ &\qquad {}+\tau(1-\beta)\gamma \bigl\Vert \nabla\bigl(\varpi _{0}^{k-1}-\varpi_{0h}^{k-1}\bigr) \bigr\Vert _{0,\Gamma} \Vert \nabla \sin n\hat{\theta} \Vert _{0} \biggr]\sin n\theta \biggr\} \,\mathrm{d}\sigma \\ & \quad \leq Ch^{2}+Ch^{2}\frac{\mu^{2}}{2\pi}\sum _{n=0}^{+\infty }\varepsilon_{n} \int_{r}^{+\infty}\sigma^{2}G_{n}(R, \sigma) (\sin n\theta+\cos n\theta)\,\mathrm{d}\sigma \\ &\quad \leq Ch^{2}. \end{aligned}$$
(34)

From (34), we immediately gain (33). This finishes the proof of Theorem 3. □

For the fully discretized NBE format, Problem IV, we have the following conclusion.

Theorem 4

If \(f^{k}\in L^{2}(\Theta^{c})\), \(g\in L^{0}(\Gamma)\), and \(\vartheta _{0}\in H^{1}(\Theta^{c})\), Problem  IV has a unique solution \(\varpi^{k}_{0h} \in\mathcal{S}_{h}(\Gamma)\) satisfying

$$\begin{aligned} \bigl\Vert \varpi_{0h}^{k} \bigr\Vert _{0}\leq C \Biggl[ \bigl\Vert u_{h}^{0} \bigr\Vert _{0,\Gamma}+\sum_{i=1}^{k}\bigl( \bigl\Vert g^{i} \bigr\Vert _{0,\Gamma}+\tau\beta \bigl\Vert f^{i} \bigr\Vert _{0,\Theta ^{c}}\bigr) \Biggr]\cdot\exp\bigl((1+ \varepsilon+\beta\gamma)T\bigr), \end{aligned}$$
(35)

where \(u_{h}^{0}=P_{h}u_{0}(\boldsymbol{x})\). This signifies that the solutions of Problem  IV are steady and consecutively rely on the source function f, boundary value function g, and initial value function \(\vartheta_{0}\). Furthermore, when \(\tau=O(h^{2})\), we have the error estimates.

$$\begin{aligned} \big\Vert \varpi(t_{k})-\varpi_{h}^{k} \big\Vert _{0,\Theta^{c}}\leq C\bigl(\tau+h^{2}\bigr), \quad k=1,2, \ldots,N. \end{aligned}$$
(36)

Proof

Due to \(\hat{B}(\cdot,\cdot)\) being symmetrical, continuous, and positive definitive on \(H^{\frac{1}{2}}(\Gamma)\times H^{\frac{1}{2}}(\Gamma)\) (see [1, 10]), by the Lax-Milgram theorem (see [1, 10, 16]), we know that Problem IV has a unique set of solutions.

Then, by taking \(\nu_{h}^{k}=\varpi_{0h}^{k}\) in (29) and using the Hölder inequality, we have

$$\begin{aligned} M \bigl\Vert \varpi_{0h}^{k} \bigr\Vert _{0,\Gamma}^{2} \leq& \bigl\vert \hat{B}\bigl( \varpi_{0h}^{k},\varpi_{0h}^{k}\bigr) \bigr\vert = \bigl\vert \bigl\langle g^{k}(r,\theta)+\mathcal{N} \bigl(\mu,r;\tilde{f}^{k},\theta\bigr),\varpi _{0h}^{k} \bigr\rangle \bigr\vert \\ \leq& \Biggl[ \bigl\Vert g^{k} \bigr\Vert _{0,\Gamma}+\Bigg\Vert \frac {\mu^{2}}{2\pi}\sum_{n=0}^{+\infty} \xi_{n} \int_{r}^{+\infty}\bar {G}_{n}(\mu,r;\sigma) \biggl\{ \int_{0}^{2\pi} \bigl[\bigl(-(I-\varepsilon\Delta ) \\ & {}-\tau(1-\beta)\gamma\Delta\bigr)\varpi_{0h}^{k-1}- \tau(1-\beta )f^{k-1}-\tau\beta f^{k} \bigr]\cdot\cos n\hat{ \theta}\,\mathrm {d}\hat{\theta}\cos n\theta \\ &{} + \int_{0}^{2\pi}\bigl[\bigl(-(I-\varepsilon\Delta)- \tau(1-\beta)\gamma\Delta \bigr)\varpi_{0h}^{k-1} \\ & {} -\tau(1-\beta)f^{k-1}-\tau\beta f^{k} \bigr]\sin n\hat{\theta}\,\mathrm {d}\hat{\theta}\sin n\theta\biggr\} \,\mathrm{d} \sigma \Bigg\Vert _{0,\Gamma} \Biggr] \big\Vert \varpi_{0h}^{k} \big\Vert _{0,\Gamma} \\ \leq& C \bigl[(1+\varepsilon+\beta\tau) \bigl\Vert \varpi _{0h}^{k-1} \bigr\Vert _{0,\Gamma}+\tau\beta \bigl\Vert f^{k} \bigr\Vert _{0,\Theta^{c}}+ \bigl\Vert g^{k} \bigr\Vert _{0,\Gamma} \bigr] \bigl\Vert \varpi_{0h}^{k} \bigr\Vert _{0,\Gamma}. \end{aligned}$$
(37)

Furthermore, we have

$$\begin{aligned} \bigl\Vert \varpi_{0h}^{k} \bigr\Vert _{0}\leq C\bigl((1+\varepsilon+\beta \tau) \bigl\Vert \varpi_{0h}^{k-1} \bigr\Vert _{0}+ \bigl\Vert g^{k} \bigr\Vert _{0,\Gamma}+\tau\beta \bigl\Vert f^{k} \bigr\Vert _{0,\Theta^{c}}\bigr). \end{aligned}$$
(38)

By summing (38) from 1 to k and using Gronwall’s lemma (see [15, 16]), we get

$$\begin{aligned} \bigl\Vert \varpi_{0h}^{k} \bigr\Vert _{0}\leq C \Biggl[ \Vert u_{0} \Vert _{0,\Gamma}+\sum _{i=1}^{k}\bigl( \bigl\Vert g^{i} \bigr\Vert _{0,\Gamma}+\tau\beta \bigl\Vert f^{i} \bigr\Vert _{0,\Theta ^{c}}\bigr) \Biggr]\cdot\exp\bigl((1+ \varepsilon+\beta\gamma)T\bigr). \end{aligned}$$
(39)

By using the triangle inequality,

$$\begin{aligned} \big\Vert \varpi(t_{k})-\varpi_{h}^{k} \big\Vert _{0,\Theta^{c}}\leq \bigl\Vert \varpi(t_{k})- \varpi^{k} \bigr\Vert _{0,\Theta^{c}}+C \bigl\Vert \varpi^{k}-\varpi_{h}^{k} \bigr\Vert _{0,\infty,\Theta^{c}}, \end{aligned}$$
(40)

and combining (7) and (33) with (40), we can acquire (36). This finishes the proof of Theorem 4. □

4 Some numerical experiments

At the moment, we utilize some numerical experimentations to validate that the numerical computational conclusions coincide with the theoretical ones and that the NBE format is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

Let \(\Theta^{c}\) be the external region outside the unit circle. The source term is chosen as

$$\begin{aligned} f(\boldsymbol {x},t) =&3\bigl[\bigl(R^{-1}-2.25\pi^{2}R^{-1}-R^{-3} \bigr)\sin(1.5\pi R)+1.5\pi^{2}R^{-2}\cos(1.5\pi R)\bigr] \cos(3t) \\ &{}+ \bigl[R^{-3}-2.25\pi^{2}R^{-1}\sin(1.5\pi R)-1.5\pi R^{-2}\cos (1.5\pi R) \bigr]\sin(3t), \end{aligned}$$

where \(R= \vert \boldsymbol {x} \vert =\sqrt{x^{2}+y^{2}}\ge1\) and take \(\varepsilon=\gamma=1\). The boundary and initial functions are, respectively, chosen as \(g(\boldsymbol {x},t)=-\sin(3t)\) and \(u_{0}(\boldsymbol {x})=0\) as in [1]. We approximately replace \(\sum_{n=1}^{+\infty}\) with \(\sum_{n=1}^{M}\) and adopt the numerical integral to calculate \(\mathcal{N}(\mu,r;\tilde{f}^{k},\theta)\) and \(\mathcal{F}(\mu,r;\tilde{f}^{k},R,\theta)\) in our numerical experimentations.

We divide the circumference Γ into 64 arc paragraphes with side length \(\triangle\theta={\pi}/{32}\), which satisfies usual regular conditions, and take \(M=120\) and the time step size \(\tau =0.0125\). We find exact solution \(\varpi_{0}^{k}\) and numerical solutions \(\varpi_{0h}^{k}\) at time \(t= 1\), 10, 30, 60, 90, 120, 180, which are shown in Photo (a)’s and (b)’s of Figs. 1-7, severally. Whereas the errors between the exact solution and numerical solutions at \(t= 1\), 10, 30, 60, 90, 120, 180 are exhibited graphically in Photo (c)’s of Figs. 1-7, severally. From each group of photos in Figs. 1-6, we can clearly see that the exact solutions are basically the same as the numerical solutions. In particular, the error photos indicated that the numerical computing consequences are consistent with the theoretical ones since both theoretical and numerical errors are \(O(10^{-4})\). Especially, it is super-convergent about time accuracy. Even if \(t=300\), the numerical solution still converges and maintains accuracy \(O(10^{-4})\). These sufficiently signify that the NBE method is very effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

Figure 1
figure 1

The exact and NBE solutions and the error between them at \(\pmb{t=1}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 2
figure 2

The exact and NBE solutions and the error between them at \(\pmb{t=10}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 3
figure 3

The exact and NBE solutions and the error between them at \(\pmb{t=30}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 4
figure 4

The exact and NBE solutions and the error between them at \(\pmb{t=60}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 5
figure 5

The exact and NBE solutions and the error between them at \(\pmb{t=90}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 6
figure 6

The exact and NBE solutions and the error between them at \(\pmb{t=120}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image
Figure 7
figure 7

The exact and NBE solutions and the error between them at \(\pmb{t=180}\) . Photos (a) and (b) are severally the exact and NBE solutions and Photo (c) is the error between them.

Full size image

5 Conclusions

In this article, we have established the semi-discretized format about time for the Sobolev equation in the 2D unbounded domain by the Newmark method and gained the error estimates of super-convergence of the semi-discretized solutions about time. Especially, we have built the fully discretized NBE format and analyzed the errors between the analytical solution and the fully discretized NBE solutions. We have also provided some numerical experiments to validate that our method is effective and feasible. The most important thing is that the NBE method applied to solve the Sobolev equation in the 2D unbounded domain is first presented, it is new and original. Moreover, the method can also solve many practical problems.