Born approximation and sequence for hyperbolic equations

Abstract

The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are $C^{\infty}$ , and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations.

Keywords

Born approximation Born sequence smoothing hyperbolic equation

1. Introduction

The Born approximation and Born series/sequence are well known for the Lippman–Schwinger equation associated to the Schrödinger equation given as $\begin{matrix} (1.1) & - \frac{h^{2}}{2 μ} Δ ψ + V (x) ψ = E ψ in R^{3} \end{matrix}$ for the wave function ψ with reduced mass μ and its potential energy $V (x) \in L^{\infty} (R^{3})$ , state of energy E, and the Planck constant h. This is equivalent to $\begin{matrix} (1.2) & (Δ + κ^{2}) ψ = \frac{2 μ}{h^{2}} V (x) ψ in R^{3} \end{matrix}$ with $κ^{2} = \frac{2 μ E}{h^{2}}$ . To derive the Lippman–Schwinger equation, let $G (x - y)$ be the outgoing Green function of the Helmholtz operator $Δ + κ^{2}$ which is given by $\begin{matrix} (1.3) & G (x - y) = - \frac{1}{4 π} \frac{e^{i κ | x - y |}}{| x - y |}, x, y \in R^{3}, x \neq y . \end{matrix}$ Then for a given incident wave $φ^{i} (x) : = e^{i κ d \cdot x}$ with incident direction d $(| d | = 1)$ , ψ is given as the solution to the Lippman–Schwinger equation $\begin{matrix} (1.4) & ψ (x) = φ^{i} (x) + (\hat{G} V ψ) (x), \end{matrix}$ which is nothing but $\begin{matrix} (1.5) & (I - \hat{G} V) ψ = φ^{i}, \end{matrix}$ where $\hat{G}$ is the integral operator with kernel $2 μ h^{- 2} G (x - y)$ and V is the multiplication operator multiplying $V (x)$ . Suppose the energy potential V is small in its norm. Then the Born approximation/series are $\begin{matrix} (1.6) & \begin{matrix} ψ = \sum_{n = 0}^{\infty} {(\hat{G} V)}^{n} φ^{i} (Born series), \\ ψ \sim φ^{i} + \hat{G} V φ^{i} ((1st) Born approximation) \end{matrix} \end{matrix}$ and they are used to describe multiple scattering due to a small perturbation of the energy state by the energy potential.

A very quick argument to derive the Born series is as follows. Introduce an artificially small parameter $ϵ > 0$ in (1.2). That is consider $\begin{matrix} (1.7) & (Δ + κ^{2}) ψ = ϵ \frac{2 μ}{h^{2}} V (x) ψ in R^{3} \end{matrix}$ and look for ψ in the form: $\begin{matrix} (1.8) & ψ = \sum_{n = 0}^{\infty} ϵ^{n} ψ_{n} . \end{matrix}$ Substitute this into (1.7), and then assembling the terms with the same power of ϵ and equating them zero, we have the recursion formula: $\begin{matrix} (1.9) & \{\begin{matrix} (Δ + κ^{2}) ψ_{0} = 0, \\ (Δ + κ^{2}) ψ_{n} = \frac{2 μ}{h^{2}} V (x) ψ_{n - 1}, n \in N . \end{matrix} \end{matrix}$ If V is already small we can set $ϵ = 1$ in (1.8). Hence by taking $ψ_{0} = φ^{i}$ , we can have the Born series in (1.6). In general this kind of formal asymptotic argument with such a small parameter ϵ can be used to derive Born series.

If V is compactly supported, then by taking $χ \in C_{0}^{\infty} (R^{3})$ with $χ = 1$ on $supp V$ , the Born series can be written in the form $\begin{matrix} (1.10) & φ^{i} + \hat{G} V φ^{i} + (\hat{G} V) \sum_{n = 1}^{\infty} {(χ \hat{G} V)}^{n} φ^{i} . \end{matrix}$ Hence it converges if the operator norm $‖ χ \hat{G} V ‖_{L^{2} (R^{3}) \to L^{2} (R^{3})}$ of $χ \hat{G}$ on $L^{2} (R^{3})$ is smaller than 1. Almost exactly in the same way, we can argue about Born approximation/series for Shrödinger equation in $R^{n}$ . In the rest of this paper, for our convenience of description, we only take $R^{3}$ as an ambient space for the space.

The Helmholtz equation $(Δ + ω^{2}) ψ = 0$ appears when we look for a time harmonic solution $u (t, x) = e^{i ω t} ψ (x)$ with frequency $ω > 0$ for the wave equation $(\partial^{2} - Δ) u = 0$ . If we perturb this wave equation by adding a potential $V \in L^{\infty} (R^{3})$ , we have $\begin{matrix} (1.11) & (\partial_{t}^{2} - Δ + V) u = 0 . \end{matrix}$ There are wave groups ${W_{0} (t) : t \in R}$ and ${W (t) : t \in R}$ associated to the Cauchy problems for the wave equation and (1.11) on the space $H^{1} (R^{3}) \oplus L^{2} (R^{3})$ with the generators A and $A + V$ , respectively (see [7]). Here A and $V$ are defined as $\begin{matrix} (1.12) & A = (\begin{matrix} 0 & 1 \\ Δ & 0 \end{matrix}), V = (\begin{matrix} 0 & 0 \\ V & 0 \end{matrix}) . \end{matrix}$ Both of them map the Cauchy data $(ϕ_{0}, ϕ_{1})$ to $(u (t, \cdot), \partial_{t} u (t, \cdot))$ , where $u (t, \cdot)$ is the corresponding solution of the Cauchy problem for each of their equations. Since $\begin{matrix} (\frac{d}{d t} - A) U (t) = - V U (t), \end{matrix}$ we have $\begin{matrix} (1.13) & U (t) - (K U) (t) = U_{0}, \end{matrix}$ where $\begin{matrix} (1.14) & (K U) (t) : = - \int_{0}^{t} W_{0} (t - s) V U (s) d s, U_{0} = (ϕ_{0}, ϕ_{1}), U (t) = W (t) U_{0} . \end{matrix}$ Because (1.13) is a Volterra equation of the second kind, $U (t)$ can be given as the following Neumann series $\begin{matrix} (1.15) & U (t) = \sum_{n = 0}^{\infty} (K^{n} U_{0}) (t), \end{matrix}$ which is the Born series for the wave equation with potential. Further assuming V has a compact support, the scattering kernel for the scattering of incoming free wave scattered by the potential V was also computed as a Born series directly by using the Lax–Phillips Radon transform [6].

In the previous example of the Born approximation/series for (1.2) and (1.11), their respective perturbations $- (2 μ) / (h^{2}) V$ and V were in the lower order term. As an example of Born approximation/series for an operator with perturbation in the leading term of an operator, we will consider a boundary value problem for a reduced wave equation in a bounded domain Ω with smooth boundary $\partial Ω$ . Here the regularity assumption on $\partial Ω$ can be weakened.

To begin with let $\begin{matrix} (1.16) & a = c + γ \in C^{1} (\overline{Ω}) with 0 < c (x) on \overline{Ω}, ‖ γ ‖_{C^{1} (\overline{Ω})} = : {\hat{γ}}_{1} ≪ 1 . \end{matrix}$ Define $L_{a}$ by $\begin{matrix} (1.17) & L_{a} ∙ = \nabla \cdot (a \nabla ∙) or a Δ ∙ . \end{matrix}$ Then consider the boundary value problem for the reduced wave equation $\begin{matrix} (1.18) & \{\begin{matrix} (L_{a} + ω^{2}) u = f \in H^{- 1} (Ω) : = {(H_{0}^{1} (Ω))}^{*} in Ω, \\ u = 0 on \partial Ω, \end{matrix} \end{matrix}$ where $H_{0}^{1} (Ω) : = {ϕ \in H^{1} (Ω) : ϕ = 0 on \partial Ω}$ with the standard Sobolev space $H^{1} (Ω)$ in Ω and ${(H_{0}^{1} (Ω))}^{*}$ is the dual space of $H_{0}^{1} (Ω)$ . For the case $L_{a} = a Δ$ , $a^{- 1}$ describes the density of a medium and ω is the angular frequency of time harmonic vibration of a medium.

If (1.18) with $a = c$ and $f = 0$ only has a trivial solution in $H_{0}^{1} (Ω)$ , the Born approximation/series are similar as before. More precisely replace $\hat{G}$ , V by $\hat{G} : = - {(L_{c} + k^{2})}^{- 1}$ , $V : = L_{γ}$ and there is no need to use χ in (1.10). Here $\hat{G}$ includes the boundary condition i.e. $\hat{G}$ is considered as a mapping $H^{- 1} (Ω) \to H_{0}^{1} (Ω)$ . The Born series can be expressed as $u_{0} + \sum_{n = 1}^{\infty} (u_{n} - u_{n - 1})$ using the Born sequence $u_{n}, n \in Z_{+} : = N \cup {0}$ defined by $\begin{matrix} (1.19) & \{\begin{matrix} u_{0} = - \hat{G} f, \\ u_{n} = - \hat{G} f + \hat{G} V u_{n - 1}, n \in N . \end{matrix} \end{matrix}$ Note that the perturbation γ of c for the reduced wave equation is in the leading part of the equation and there is no regularity loss of the operator $\hat{G} V : H_{0}^{1} (Ω) \to H_{0}^{1} (Ω)$ .

Even for the parabolic equation obtained by replacing $ω^{2}$ of (1.18) by $- \partial_{t}$ we can have the convergence of Born sequence/series. Then a natural question to ask is how about for hyperbolic equations. This question was asked and discussed in [4, Remark 4.5] very briefly with a negative speculation to this question. Even though, since the Born approimation and the convergence of Born series/sequence are so important, we would like to give a remedial study to this question.

For simplicity we only consider this question for $\begin{matrix} (1.20) & (MP) \{\begin{matrix} □_{a} u : = (\partial_{t}^{2} - L_{a}) u = f in Ω_{T} : = (0, T) \times Ω, \\ u = 0 on {(\partial Ω)}_{T} : = (0, T) \times \partial Ω, \\ u = \partial_{t} u = 0 at t = 0 \end{matrix} \end{matrix}$ in the above bounded domain Ω under the assumption (1.16) for an appropriate f. With the notation $\begin{matrix} (1.21) & □ = □_{c} : = \partial_{t}^{2} - L_{c}, \end{matrix}$ the equation of (1.20) can be rewritten as $\begin{array}{l} □ u = f + L_{γ} u . \end{array}$ Further, let $□^{- 1}$ denote the solution operator of (1.20) with $a (x)$ replaced by $c (x)$ .

If γ is small, it is natural to consider the following Born sequence $u_{0}, u_{1}, \dots$ defined by $\begin{matrix} (1.22) & \{\begin{matrix} u_{0} = □^{- 1} f, \\ u_{k} = □^{- 1} (f + L_{γ} u_{k - 1}), k \in N . \end{matrix} \end{matrix}$ As k increases, we expect $u_{k}$ to be a better and better approximation of u. Nevertheless, the number of $u_{k}$ that can be obtained is in general limited due to the so called regularity loss or a failure of compatibility (see Definition A.4 and Theorem A.6). More precise explanations of these are as follows by looking (1.22) at $k = 1$ . Suppose $f + L_{γ} u_{0}$ satisfies the compatibility condition. Then, due to the fact that $□^{- 1}$ gains only one regularity for the solution from the source term and $L_{γ}$ loses two, $u_{0}$ is one higher regular than f, while $u_{1}$ is one lower regular than $u_{0}$ , If $f + L_{γ} u_{0}$ does not satisfy the compatibility condition, then $u_{1}$ could be more lower regular than $u_{0}$ . Similar situations can happen at any $k ⩾ 2$ . Thus (1.22) has to stop after a finite number of iterations.

This is a very bad news to consider about any convergence of Born sequence for hyperbolic equations. One could try to apply iterative schemes developed for solving nonlinear hyperbolic equations such as Nash–Moser iteration scheme (see [1,3,5]) and an iteration scheme using an elliptic-hyperbolic coupled system (see [8,9]). For the former scheme the result given in [1] is very interesting. They used the scheme to solve an initial boundary value problem for a second order hyperbolic equation degenerating at $t = 0$ with Dirichlet boundary condition assuming that the data are smooth and they satisfy the compatibility condition of infinite order. Even the degeneracy of the second order hyperbolic part may cause some regularity loss, their perturbation is the semilinear part which has lower order derivatives than those of the hyperbolic part. This could be different from our situation where the perturbation $L_{γ}$ has the same order of derivatives compared with those of the non-perturbed part $L_{c}$ . Further, since the solution obtained by the Nash–Moser iteration scheme is local in its nature, the time interval over which a solution exists as a limit function of the iteration scheme becomes smaller than $[0, T]$ . As for the latter scheme, the sequence in the iteration cannot be defined due to the regularity loss coming from $L_{γ}$ applied to previous term of sequence for the hyperbolic equation in the iteration. Nevertheless, we could come up with some results which we want to present in this paper are as follows.

To begin with, for $m \in Z_{+} : = N \cup {0}$ , let $H^{m} (Ω_{T})$ be the standard $L^{2}$ Sobolev space of order m in $Ω_{T}$ . If $f \notin H^{\infty} (Ω_{T}) : = ⋂_{m \in Z_{+}} H^{m} (Ω_{T})$ with enough regularity and compatibility condition of some order, finite terms Born sequence (abbreviated by finite Born sequence) should be enough in practice. Then we have to provide an estimate such as $\begin{matrix} (1.23) & ‖ u - u_{n} ‖_{H^{1} (Ω_{T})} ⩽ C_{N} {\hat{γ}}_{1}^{n}, 0 ⩽ n ⩽ N \end{matrix}$ for given $N \in Z_{+}$ . For this kind of estimate, see Theorem 2.1 for more precise statement of the result we got. By numerical simulations, we will see that the approximation of u by $u_{n}$ can be better than this estimate (see Section 4).

Now concerning the convergence of the Born sequence for the case $f \notin H^{\infty} (Ω_{T})$ , we not only have to smooth each $u_{k}$ but we also need to resolve the compatibility issue for the smoothed $u_{k - 1}$ . In order to handle these, we will appy the following operators to $L_{γ} u_{k - 1}$ in (1.22). They are the 0-extension to $t < 0$ , translation in time, mollification in time and the restriction to $t ⩾ 0$ , which are applied successively. We refer this Born sequence ${u_{k}}$ obtained by having these operations as smoothed Born sequence. Needless to say the major questions for the smoothed Born sequence are its convergence and to know the equation which its limit satisfies. The key for this is to have an improved version of energy estimate for the solution operator $□^{- 1}$ (see the proof of Theorem 3.1 and Remark A.7). We will see that the smoothed Born sequence does converges, but the limit function of this sequence does not satisfy the original equation. The equation for the limit function is a wave equation with a memory term which is the scalar version of viscoelastic equation with memory term. The error estimate between this limit function and u will be given (see Theorem 3.1). By numerical simulations, we will see how $u_{k}$ approximates u (see Section 4).

It should be remarked here that we should also have to consider about the convergence of the Born sequence for the case that $f \in H^{\infty} (Ω)$ satisfies the compatibility condition of infinite order. In this case we do not have to worry about the regularity loss and lack of having the compatibility condition. Nevertheless, we will show in this case that the constant $C_{N}$ will in general becomes large as N becomes large (see Remark 2.2, (iii)).

The rest of this paper is organized as follows. In the next section we will give the error estimate of finite Born sequence, i.e. (1.23). The convergence of the smoothed Born sequence and how its limit approximates the solution of (1.20) will be discussed in Section 3. Section 4 is devoted to the numerical performance of the smoothed Born sequence and finite terms Born approximation. The numerical results given here are quite interesting and very suggestive. We believe that those results can lead to further theoretical studies on the Born approximation for hyperbolic equations. In Section 5 we give some conclusions of our study on the Born approximation for hyperbolic equations. After all these sections there is an Appendix in which we gave a very important improved energy estimate of solutions to the initial boundary value problem and the numerical aspect of mollification.

2. Finite Born sequence

In this section we will discuss about the error estimate of the finite Born sequence. To begin with we introduce some notations. Denote $[0, T]$ by I and let ${Y_{j}}_{j \in Z_{+}}$ be a scale of Hilbert spaces, i.e. $Y_{j}$ , $j \in Z_{+}$ are Hilbert spaces such that $\begin{matrix} Y_{j + 1} \subset Y_{j}, ‖ \cdot ‖_{j} ⩽ ‖ \cdot ‖_{j + 1}, j \in Z_{+}, \end{matrix}$ where each $‖ \cdot ‖_{j}$ denotes the norm defined in $Y_{j}$ . For $0 ⩽ j ⩽ p$ , let $H^{j} (I; Y_{p - j})$ denote the $L^{2}$ Sobolev space of order j for $Y_{p - j}$ valued functions defined on I and denotes its norm by $‖ \cdot ‖_{H^{j} (I; Y_{p - j})}$ (see Sections 24, 25 of [11]). Then define $X_{p}$ and $X_{p}^{'}$ by $\begin{array}{l} (2.1) & X_{p} = ⋂_{j = 0}^{p} H^{j} (I; Y_{p - j}), X_{p}^{'} = H^{0} (I; Y_{p - 1}) \cap ⋂_{j = 1}^{p} H^{j} (I; Y_{p - j}), p \in N \end{array}$ with natural norms $‖ \cdot ‖_{X_{p}}$ and $‖ \cdot ‖_{X_{p}^{'}}$ given as $\begin{matrix} (2.2) & \{\begin{matrix} ‖ u ‖_{X_{p}} = \sum_{j = 0}^{p} ‖ u (t, \cdot) ‖_{H^{j} (I; Y_{p - j})}, \\ ‖ u ‖_{X_{p}^{'}} = ‖ u (t, \cdot) ‖_{Y_{p - 1}} + \sum_{j = 1}^{p} ‖ u (t, \cdot) ‖_{H^{j} (I; Y_{p - j})} \end{matrix} \end{matrix}$ which are equivalent to $\begin{matrix} (2.3) & \{\begin{matrix} \sum_{j = 0}^{p} ‖ \partial_{t}^{j} u (t, \cdot) ‖_{L^{2} (I; Y_{p - j})}, \\ ‖ u (t, \cdot) ‖_{Y_{p - 1}} + \sum_{j = 1}^{p} ‖ \partial_{t}^{j} u (t, \cdot) ‖_{L^{2} (I; Y_{p - j})}, \end{matrix} \end{matrix}$ where $L^{2} (I; Y_{p - j}) = H^{0} (I; Y_{p - j})$ is the set of $Y_{p - j}$ valued $L^{2}$ functions over I. We will abuse the notations $‖ \cdot ‖_{X_{p}}$ and $‖ \cdot ‖_{X_{p}^{'}}$ to denote these equivalent norms. Note that we have $\begin{matrix} ‖ u ‖_{X_{p}^{'}} ≲ ‖ u ‖_{X_{p}}, p \in N, \end{matrix}$ where the notation ≲ means that the inequality ⩽ is true up to a multiplicative positive general constant.

Henceforth, we will take $Y_{p - j} = H^{p - j} (Ω)$ in (2.1) which is the standard $L^{2}$ Sobolev space of order $p - j$ in Ω. It should be remarked here that a rather unnatural way of defining $X_{p}^{'}$ with this choice of $Y_{p - j}$ is coming from an energy estimate for the solution operator $□^{- 1}$ . We will see later in Section 3 that this function space will become very important for the convergence of smoothed Born sequence with smoothing in t. The energy estimate is given in the Appendix and it is derived by carefully reexamining the argument of deriving an energy estimate for the wave equation with variable coefficients in [2]. Also for simplifying notations, we will use the following abbreviations:

abbreviate, for example, $H^{j} (I; Y_{p - j}) = H^{j} (I; H^{p - j} (Ω))$ as $H_{t}^{j} H_{x}^{p - j}$ ,

sometimes write $L_{t}^{2} H_{x}^{m}$ instead of $H_{t}^{0} H_{x}^{m}$ ,

sometime write $v^{'}, v^{″}, \dots, v^{(k)}$ instead of $v_{t}, v_{t t}, \dots, \partial_{t}^{k} v$ .

Now for the initial boundary value problem (1.20), and for $2 ⩽ m \in N$ , define the following regularity and compatibility condition $P (m, a)$ for f.

Regularity condition $R^{'} (m)$ : $\begin{matrix} (2.4) & f \in L^{2} (I; H^{m - 1} (Ω)), \frac{\partial^{k} f}{\partial t^{k}} \in L^{2} (I; H^{m - k} (Ω)), k = 1, \dots, m . \end{matrix}$

Compatibility condition $P^{'} (m)$ for $□_{a}$ ; $\begin{array}{l} (2.5) & \{\begin{matrix} g_{0} : = u |_{t = 0} = 0, h_{1} : = \partial_{t} u |_{t = 0} = 0 \in H_{0}^{1} (Ω) \\ g_{2 ℓ} : = \frac{\partial^{2 ℓ - 2} f}{\partial t^{2 ℓ - 2}} (0, \cdot) + L_{a} g_{2 ℓ - 2} \in H_{0}^{1} (Ω), ℓ \in N, 2 ℓ ⩽ m \\ h_{2 ℓ + 1} : = \frac{\partial^{2 ℓ - 1} f}{\partial t^{2 ℓ - 1}} (0, \cdot) + L_{a} h_{2 ℓ - 1} \in H_{0}^{1} (Ω), ℓ \in N, 2 ℓ + 1 ⩽ m . \end{matrix} \end{array}$

By combining these conditions together, we have the specialized version of mth order compatibility condition

P (m, a)

given in Definition A.4 for the case the boundary datum and initial datum are zeroes.

Then we have the following error estimate for the finite Born sequence.

Theorem 2.1.
Fix $m \in N$ and let f satisfy the mth order compatibility conditions for $□_{a}$ and $□_{c}$ . Denote ${\hat{γ}}_{m} : = ‖ γ ‖_{W^{max (1, m - 1), \infty} (Ω)} < \infty$ with $‖ γ ‖_{W^{s, \infty} (Ω)} : = \sum_{| α | ⩽ s} ‖ \partial_{x}^{α} γ ‖_{L^{\infty} (Ω)}$ , $s \in Z_{+}$ , where $L^{\infty} (Ω)$ is the set of bounded measurable functions in Ω. Then, we can iteratively solve ( 1.22 ) for $u_{k} \in X_{m + 1 - k}$ for $k = 0, \dots, m - 1$ . Furthermore, we have the estimate $\begin{matrix} (2.6) & ‖ u - u_{m - 1} ‖_{X_{1}} ⩽ C {\hat{γ}}_{m}^{m} ‖ f ‖_{X_{m}^{'}}, \end{matrix}$ where the constant C is uniformly bounded for small $‖ γ ‖_{L^{\infty} (Ω)}$ .
Proof.
Fix $m \in N$ . We first show $u_{k} \in X_{m + 1 - k}$ , $0 ⩽ k ⩽ m - 1$ by induction on k. By Theorem A.6 we have $u_{0} \in X_{m + 1}$ . Then we suppose that we have obtained $u_{k} \in X_{m + 1 - k}$ for some $k < m - 1$ . Again by Theorem A.6, it is enough to show that $f + L_{γ} u_{k}$ satisfies $R^{'} (m - 1 - k)$ and $P^{'} (m - 1 - k)$ . It obviously satisfies $R^{'} (m - 1 - k)$ . Further, by $u_{k} (0, \cdot) = \frac{\partial u_{k}}{\partial t} (0, \cdot) = 0$ and the compatibility conditions for f, $P^{'} (m - 1 - k)$ is also satisfied.

Now for our convenience let $u_{- 1}$ be the zero function on $I \times Ω$ . Then we have $\begin{matrix} (2.7) & □_{a} (u - u_{m - 1}) = L_{γ} (u_{m - 1} - u_{m - 2}), \end{matrix}$ and, if $m ⩾ 2$ and $1 ⩽ s ⩽ m - 1$ , $\begin{matrix} (2.8) & □ (u_{m - s} - u_{m - s - 1}) = L_{γ} (u_{m - s - 1} - u_{m - s - 2}) . \end{matrix}$ Applying Theorem A.3 to (2.7), we obtain $\begin{array}{l} ‖ u - u_{m - 1} ‖_{X_{1}} & ⩽ C_{1} (a) {‖ L_{γ} (u_{m - 1} - u_{m - 2}) ‖}_{L_{t}^{2} L_{x}^{2}} \\ (2.9) & ≲ C_{1} (a) ‖ γ ‖_{W^{1, \infty} (Ω)} ‖ u_{m - 1} - u_{m - 2} ‖_{X_{2}}, \end{array}$ where $C_{1} (a) = C_{1} (c + γ)$ is uniformly bounded for small $‖ γ ‖_{L^{\infty} (Ω)}$ .

If $m = 1$ , (2.9) gives what we want to show. If $m ⩾ 2$ , then we apply Theorem A.6 to (2.8), and obtain $\begin{array}{l} ‖ u_{m - s} - u_{m - s - 1} ‖_{X_{s + 1}} & ⩽ C_{s + 1} (c) {‖ L_{γ} (u_{m - s - 1} - u_{m - s - 2}) ‖}_{X_{s}^{'}} \\ (2.10) & ≲ C_{s + 1} (c) ‖ γ ‖_{W^{s, \infty} (Ω)} ‖ u_{m - s - 1} - u_{m - s - 2} ‖_{X_{s + 2}} . \end{array}$ Combining (2.9) and (2.10) with $s = 1, \dots, m - 1$ , we have $\begin{matrix} ‖ u - u_{m - 1} ‖_{X_{1}} ≲ C_{1} (a) C_{2} (c) \dots C_{m} (c) ‖ γ ‖_{W^{m - 1, \infty} (Ω)}^{m} ‖ u_{0} ‖_{X_{m + 1}} . \end{matrix}$ Hence by Theorem A.6, this immediately implies (2.6). □
Remark 2.2.

Under the assumption of the theorem, we can acturally obtain $u_{m} \in X_{1}$ , but we cannot estimate $u - u_{m}$ . This is due to the fact that the x-regularity of the source term of $□_{a} (u - u_{m}) = L_{γ} (u_{m} - u_{m - 1})$ is only of $H^{- 1} (Ω)$ .

If $γ \in C_{0}^{\infty} (Ω)$ , we do not need to assume that f satisfies the compatibility condition for $□_{a}$ .

Note that even in the case $γ \in C_{0}^{\infty} (Ω)$ and $(f, g, h) \in P (m, c)$ for any $m \in N$ , we cannot establish the convergence of $u_{k}$ to u (in any sense) for a fixed small γ, since the constant C in (2.6) may go to infinity as $m \to \infty$ .

3. Approximation by smoothed Born sequence and its convergence

In this section we will consider the smoothed Born sequence and its convergence. To begin with, we will define an extension operator $E$ , a restriction operator R, a translation operator τ and a mollifier $S_{θ}$ as follows. Let $E$ be the extension operator extending any function defined on $[0, T] \times Ω$ to $R \times Ω$ which consists of the following two extensions such that the extended function is compactly supported with respect to $t \in R$ , say it is supported in $[- 2 T, 2 T]$ . The one is the zero extension to ${t < 0} \times Ω$ and the other is a regularity preserving extension to ${t > T} \times Ω$ . Also let R be the restriction operator restricting any function defined on $R \times Ω$ to $[0, T] \times Ω$ , and let τ be the translation operator defined by $(τ h) (t, x) = h (t - t_{0}, x)$ with a fixed small $t_{0} > 0$ for any function $h (t, x)$ defined on $R \times Ω$ . Further define the mollifier $S_{θ}$ with respect to t for a large parameter $θ > 0$ as follows. $\begin{matrix} (3.1) & (S_{θ} h) (t, x) = \int_{R} η_{θ} (s) h (t - s, x) d s, \end{matrix}$ where $η_{θ} (t) = θ η (θ t)$ with $η (t) \in C_{0}^{\infty} (R)$ such that $\int_{R} η (t) d t = 1$ , $supp η \subset {| t | ⩽ 1}$ and $0 ⩽ η (t) ⩽ 1$ for $t \in R$ . Take θ such that $θ^{- 1} < t_{0}$ .

Let $m \in N$ , $m ⩾ 2$ and let $f \in X_{m}^{'}$ satisfy the compatibility condition of order $m - 2$ and m for $□_{a}$ and $□_{c}$ , respectively. We further let f satisfy the following condition: $\begin{matrix} (3.2) & \partial_{t}^{j} f |_{t = 0} = 0, 0 ⩽ j ⩽ m - 6 if m ⩾ 6 . \end{matrix}$ Define a sequence $u_{n} \in X_{m + 1}$ , $n \in Z_{+}$ by $\begin{matrix} (3.3) & \{\begin{matrix} u_{0} = □^{- 1} f, \\ u_{n} = □^{- 1} (f + R S_{θ} τ E L_{γ} u_{n - 1}), n \in N . \end{matrix} \end{matrix}$ Note that $u_{n} \in X_{m + 1}$ , $n \in Z_{+}$ can be shown inductively by using the assumptions on f and Theorem A.6.

Then we have the following theorem.

Theorem 3.1.
If ${\hat{γ}}_{m - 1} θ$ is small, then $u_{n} \to u^{θ} (n \to \infty)$ in $X_{m + 1}$ , where $u^{θ}$ is the solution to $\begin{matrix} (3.4) & {(MP)}_{θ} \{\begin{matrix} □ u^{θ} = f + R S_{θ} τ E L_{γ} u^{θ}, \\ u^{θ} = 0 on {(\partial Ω)}_{T}, \\ u^{θ} = \partial_{t} u^{θ} = 0 at t = 0 . \end{matrix} \end{matrix}$ Further we have the following estimates $\begin{matrix} (3.5) & \{\begin{matrix} ‖ u^{θ} ‖_{X_{m + 1}} ≲ ‖ f ‖_{X_{m}^{'}} uniformly for small {\hat{γ}}_{1} θ, \\ ‖ u - u^{θ} ‖_{X_{m - 1}} ⩽ ‖ (1 - R S_{θ} τ E) L_{γ} u^{θ} ‖_{X_{m - 2}} \\ ≲ {\hat{γ}}_{m - 2} (θ^{- 1} + t_{0}) ‖ f ‖_{X_{m}} uniformly for small γ_{1} θ . \end{matrix} \end{matrix}$

The proof of this theorem will be given after the following remark. Remark 3.2.

$(u_{n} =) u_{0} + (u_{1} - u_{0}) + \dots + (u_{n} - u_{n - 1})$ corresponds to the truncated Born series and the following estimate holds for $u_{n} - u_{n - 1} = □^{- 1} R S_{θ} τ E L_{γ} (u_{n - 1} - u_{n - 2})$ $\begin{matrix} (3.6) & ‖ u_{n} - u_{n - 1} ‖_{X_{m + 1}} ≲ {\hat{γ}}_{m} θ ‖ u_{n - 1} - u_{n - 2} ‖_{X_{m + 1}}, n ⩾ 2 . \end{matrix}$

The use of τ is to have the compatibility condition satisfied even after mollification.

The equation of $u^{θ} (t, x)$ in (3.4) for $0 ⩽ t ⩽ T$ can be rewritten in the form $\begin{matrix} (3.7) & \partial_{t}^{2} u^{θ} (t, x) - L_{c} u^{θ} (t, x) - \int_{0}^{t} η_{θ} (t - s - t_{0}) L_{γ} u^{θ} (s, x) d s = f (t, x), \end{matrix}$ where we have used that $θ^{- 1} < t_{0}$ and $η_{θ} (t - s) = 0$ for $t + θ^{- 1} < t ⩽ t_{0}$ . This has a simple form of viscoelastic equation (see [10]). This observation suggests that the convergence of the Born sequence for the viscoelastic equations.

Proof.
For simplicity denote $z_{n} : = u_{n} - u_{n - 1}$ , $H : = R L_{γ} S_{θ} τ E = R S_{θ} L_{γ} τ E$ . Then we clearly have $\begin{matrix} (3.8) & z_{n + 1} = H z_{n}, n \in N . \end{matrix}$ Let ${\tilde{X}}_{m}$ , ${\tilde{X}}_{m}^{'}$ be defined as $X_{m}$ , $X_{m}^{'}$ over $[- 3 T, 3 T]$ .

First of all since $z_{n} \in X_{m + 1}$ and by the definition of H, we have $H z_{n} \in X_{m + 1}$ and it is 0 near $t = 0$ . Also by the definitions of H and $S_{θ}$ , we can easily have the estimate: $\begin{matrix} (3.9) & ‖ H z_{n} ‖_{X_{m}^{'}} ≲ ‖ S_{θ} L_{γ} τ E z_{n} ‖_{{\tilde{X}}_{m}^{'}} ≲ θ ‖ L_{γ} τ E z_{n} ‖_{{\tilde{X}}_{m - 1}} ≲ {\hat{γ}}_{m} θ ‖ z_{n} ‖_{X_{m + 1}} . \end{matrix}$ Then by using (3.9) and Theorem A.6, we have $\begin{matrix} (3.10) & ‖ z_{n + 1} ‖_{X_{m + 1}} ⩽ C {\hat{γ}}_{m} ‖ z_{n} ‖_{X_{m + 1}}, \end{matrix}$ where $C > 0$ is a general constant.

Now let $m = C {\hat{γ}}_{m} θ < 1$ . Then it is easy to see that for any $M, N \in N$ with $M < N$ , we have $\begin{matrix} (3.11) & ‖ u_{N} - u_{M} ‖_{X_{m + 1}} ⩽ \frac{m^{M}}{1 - m} ‖ u_{1} - u_{0} ‖_{X_{m + 1}} . \end{matrix}$ This yields the convergence of ${u_{n}}_{n \in Z_{+}}$ in $X_{m + 1}$ to a unique $u^{θ} \in X_{m + 1}$ and it becomes a solution of (3.4). Further from $‖ u_{1} - u_{0} ‖_{X_{m + 1}} ⩽ m ‖ u_{0} ‖_{X_{m + 1}}$ and (3.11), we have $\begin{matrix} (3.12) & ‖ u_{n + 1} ‖_{X_{m + 1}} ⩽ {\frac{m}{1 - m} + 1} ‖ u_{0} ‖_{X_{m + 1}} ⩽ \frac{1}{1 - m} ‖ u_{0} ‖_{X_{m + 1}}, n \in Z_{+} \end{matrix}$ which implies $\begin{matrix} (3.13) & {‖ u^{θ} ‖}_{X_{m + 1}} ⩽ \frac{1}{1 - m} ‖ u_{0} ‖_{X_{m + 1}}, \end{matrix}$ which immediately implies the first estimate in (3.5). By the equation of (1.20) and the equation of (3.4), we have $\begin{matrix} (3.14) & □_{a} (u - u^{θ}) = L_{γ} u^{θ} - R S_{θ} τ E L_{γ} u^{θ} = R (I - S_{θ}) τ E L_{γ} u^{θ} + R (I - τ) E L_{γ} u^{θ} . \end{matrix}$ To estimate the right hand side of (3.14), we will use the following well known estimate for the mollifier and a simple estimate for the operator τ: $\begin{matrix} (3.15) & {‖ (I - S_{θ}) h ‖}_{{\tilde{X}}_{m - 2}} ≲ θ^{- 1} ‖ h ‖_{{\tilde{X}}_{m - 1}}, {‖ (I - τ) h ‖}_{{\tilde{X}}_{m - 2}} ≲ t_{0} ‖ h ‖_{{\tilde{X}}_{m - 1}}, h \in {\tilde{X}}_{m - 1} . \end{matrix}$ Then we can estimate $‖ R (I - S_{θ}) τ E L_{γ} u^{θ} ‖_{X_{m - 2}^{'}}$ as follows. $\begin{array}{rcl} {‖ R (I - S_{θ}) τ E L_{γ} u^{θ} ‖}_{X_{m - 2}^{'}} & ≲ & {‖ (I - S_{θ}) τ E L_{γ} u^{θ} ‖}_{{\tilde{X}}_{m - 2}} \\ (3.16) & ≲ & θ^{- 1} ‖ τ E L_{γ} u^{θ} ‖ {\tilde{X}}_{m - 1} ≲ θ^{- 1} {‖ u^{θ} ‖}_{X_{m + 1}} . \end{array}$ Similarly we have $\begin{matrix} (3.17) & {‖ R (I - τ) E L_{γ} u^{θ} ‖}_{X_{m - 2}^{'}} ≲ t_{0} {‖ u^{θ} ‖}_{X_{m + 1}} . \end{matrix}$ Hence we have $\begin{matrix} (3.18) & {‖ L_{γ} u^{θ} - R S_{θ} τ E L_{γ} u^{θ} ‖}_{X_{m - 2}^{'}} ≲ (θ^{- 1} + t_{0}) {‖ u^{θ} ‖}_{X_{m + 1}} . \end{matrix}$ Here $R S_{θ} τ E L_{γ} u^{θ}$ clearly satisfies the compatibility condition of infinite order for $□_{a}$ . As for $L_{γ} u^{θ}$ , note that $u^{θ} = u_{0} + \sum_{n = 1}^{\infty} z_{n}$ , $u_{0} = □^{- 1} f$ and each $z_{n}$ is 0 near a fixed neighborhood in $[0, T]$ of $t = 0$ . Further by (3.2), $L_{γ} u_{0}$ satisfies the compatibility condition of order $m - 2$ for $□_{a}$ . Hence $L_{γ} u^{θ}$ satisfies the compatibility condition of order $m - 2$ for $□_{a}$ . Therefore by Theorem A.6 and (3.18), we have the second estimate in (3.5). □
Remark 3.3.
The proof of Theorem 3.1 is totally depending on the energy estimate given in Theorem A.6. The advantage of the estimate is as follows. Compare the norms of $X_{m + 1}$ and $X_{m}^{'}$ given in the forms of (2.3). Especially compare each term of these norms without t derivative. Then the difference of regularities with respect to x variable of these two terms are exactly order two and the regularity loss between the adjacent terms in Born sequence is also order two with respect to x variable. This is why the smoothing with respect to t variable works for the smoothed Born sequence.

4. Numerical experiments

In this section we will test the performance of finite Born approximation without smoothing and with smoothing by numerical simulations. For simplicity we consider $\begin{matrix} (4.1) & \{\begin{matrix} \partial_{t}^{2} u (t, x) - (1 + γ (x)) Δ u (t, x) = f (t, x) in (0, 4) \times (0, 4), \\ u (0, x) = \partial_{t} u (0, x) \equiv 0, \\ u (t, 0) = u (t, 4) \equiv 0, \end{matrix} \end{matrix}$ where $γ (x)$ is a smooth function on $[0, 4]$ such that $‖ γ ‖_{L^{\infty} (Ω)} < 1$ . We will compare the solution of (4.1) and the finite Born sequence obtained by truncating its associated Born sequence defined by $\begin{matrix} (4.2) & \{\begin{matrix} u_{0} = □^{- 1} f, \\ u_{k} = □^{- 1} (f + γ Δ S u_{n - 1}), k \in N, \end{matrix} \end{matrix}$ where $S$ is either (i) the identity operator (i.e. no smoothing) or (ii) a smoothing operator in t or in x. We will refer these smoothing operators by t-smoothing operator and x-smoothing operator.

We apply the standard explicit finite difference method to solve (4.1) with γ and without it. The step sizes in t and in x are 0.025 and 0.05 respectively. For the smoothing operators we use the following 3-points method: $\begin{matrix} {(S v)}^{j} = \frac{v^{j - 1} + 2 v^{j} + v^{j + 1}}{4}, \end{matrix}$ with respect to t or x, and apply it several times if we need more smoothing. Smoothing with respect to t is easier for any space dimension. For the one space dimensional case, the smoothings with respect to t and x should have the same effect. However we will see in the simulations given below, the smoothing with respect to x is better than the smoothing with respect to t.

In the following subsections, we will provide some numerical results on the numerical performance of Born sequence such as approximation of true solution and its convergence. All the results are for the one space dimensional case.

4.1. Simulation 1

Let $\begin{array}{l} γ = 0.8 sin (π x / 2), \\ f = 0.5. \end{array}$ The graphs of γ and f are shown in Fig. 1-0. We first examine the case without smoothing. Figure 1-1 shows u, $u_{0}$ , $u_{1}$ , $u_{2}$ , $u_{3}$ at the final time $t = 4$ without applying any smoothing operator. This figure shows that $u_{3}$ is already highly oscillatory.

Fig. 1.

Figures for simulation 1.

Next we examine the case with several kinds of smoothing. Figures 1-2 and 1-3 show the graph of $u_{3}$ when t-smoothing is applied and double t-smoothing is applied, i.e. applying the 3-points smoothing twice, respectively. One sees that smoothing in t does make $u_{3}$ more regular, but the effect is not great.

On the other hand Figs 1-4 and 1-5 show that we can have a much better approximation result when we use x-smoothing. The smoothing effect of x-smoothing is really good. In fact, by using the x-smoothing, we can go on to compute $u_{k}$ successively as much as possible, and it looks like that $u_{k}$ converges to some limit which is very close to u (see Fig. 1-5). We will refer such a limit by limit of Born sequence.

There are two points to be remarked for our simulation:

The “perturbation” γ is not really small compared to the background value 1.

The source term f, though extremely simple, does not satisfy the second order compatibility condition: $f |_{t = 0} = 0$ at $\partial Ω$ .

Even with these two points, it is interesting to know that the Born approximation with x-smoothing works so well.

4.2. Simulation 2

Now consider slightly irregular situations for γ and f: $\begin{array}{l} γ = γ_{0} cos (2 x^{2}), \\ f = \{\begin{matrix} 0.4 + 0.4 x & for 0 ⩽ x ⩽ 1, \\ 0.8 - 1.6 (x - 1) & for 1 ⩽ x ⩽ 2, \\ - 0.8 + 1.2 (x - 2) & for 2 ⩽ x ⩽ 3, \\ 0.4 - 0.8 (x - 3) & for 3 ⩽ x ⩽ 4, \end{matrix} \end{array}$ with $γ_{0} = 0.4$ and $γ_{0} = 0.8$ . The corresponding graphs of γ with the graph of f are shown in Figs 2-0 and 2-0′, respectively. Again f does not satisfy the compatibility condition of order 2. The case without smoothing are shown in Figs 2-1 and 2-1′. While the results for t-smoothing case are given in Figs 2-2 and 2-2′, and the results for the x-smoothing case are given in Figs 2-3 and 2-3′.

Fig. 2.

Figures for simulation 2.

Fig. 2.

(Continued.)

Fig. 3.

Figures for simulation 3.

Figures 2-4 and 2-4′ show the limits of Born sequences for $γ_{0} = 0.4$ and $γ_{0} = 0.8$ , respectively. The former case is basically the same as that in Simulation 1. However, for the latter case, even though $u_{k}$ still approaches to a limit when x-smoothing is applied, the limit is not close to the true solution u.

4.3. Simulation 3

Let $\begin{array}{l} γ = 0.8 cos (2 x^{2}), \\ f = 0.6 sin (2 t) cos (2 t) . \end{array}$ This γ is the same as in Simulation 2 with $γ_{0} = 0.8$ , while f is a smooth function depending on both t and x. Note that f satisfies the second order compatibility condition. What we can say from the simulations given in Figs 3-1 and 3-2. Without smoothing, anomalous oscillations occur earlier. Also with x-smoothing, Figs 3-1 and 3-2 show that the limit with smoothing is not a good approximation of the true solution.

4.4. Simulation 4

From the previous simulations, the influences of compatibility conditions are not significant. However we will see that they may still help in some situations. In fact let $\begin{matrix} γ = 0.7 cos (10 x), \end{matrix}$ and consider the following two cases: $\begin{array}{l} f = f_{1} = 0.5 (1 + cos (π x / 2)), \\ f = f_{2} = 0.5 (1 - cos (π x / 2)) . \end{array}$ The graphs of γ, $f_{1}$ and of γ, $f_{2}$ are shown in Figs 4-0 and 4-0′. Note that $f_{2}$ satisfies the 3rd order compatibility condition for $\partial_{t}^{2} - \partial_{x}^{2}$ . Without smoothing, Fig. 4-1 shows $u_{3}$ for $f = f_{1}$ . Like most examples shown before, one sees that it is already very irregular, and we are not able to obtain reasonable approximations by computing $u_{k}$ for $k ⩾ 4$ . However, also without smoothing, Fig. 4-1′ shows $u_{6}$ for the case $f = f_{2}$ , which is a very good approximation of u. We remark that anomalous oscillations still occurred in $u_{7}$ . Despite the above examples, it is hard to make sure the range of application without smoothing. In practice, it is always preferred to apply x-smoothing.

Fig. 4.

Figures for simulation 4.

5. Conclusions

We gave some study of Born sequence (i.e. the sequence obtained by truncating the Born series) for hyperbolic equations when the leading part of the equation has a small perturbation. For elliptic equations or parabolic equations, the Born sequence converges. However for the hyperbolic equations, it does not converge in general due to the so called regularity loss or the failure for the data to satisfy the compatibility condition. In this paper we studied the performance of approximating the solution by the Born sequence and the convergence issue for the smoothed Born sequence both from the theoretical and numerical point of views. Here the smoothed Born sequence is a modification of usual Born sequence by applying the following operations to its each term. They are the 0-extension to $t < 0$ , translation in time, mollification in time and the restriction to $[0, T] \times Ω$ , which are applied successively. The novelties of the paper are the following two results:

We numerically showed that a smoothing is important for the finite Born sequence to improve and stabilize the performance of approximating the true solution and stabilizing for the finite Born sequence. Comparing the smoothing with respect the time variable and smoothing with respect to the space variables, we showed that the latter one is more effective.

By improving the usual energy estimate, we showed that this smoothed Born series converges to a scalar version of viscoelastic equation with a memory term and also gave an estimate of approximating the true solution.

Despite the importance of Born sequence in practice, we could not find almost any detail study on the Born sequence for hyperbolic equations. We believe that our study is very suggestive for further studies on the Born series for hyperbolic equations.

Footnotes

Acknowledgements

The third author was supported by Grant-in-Aid for Scientific Research (15K21766 and 15H05740) of the Japan Society for the Promotion of Science.

Appendix

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