The Cauchy problem for linear inhomogeneous wave equations with variable coefficients

Accepted Manuscript The cauchy problem for linear inhomogeneous wave equations with variable coefficients Kai Liu, Wei Shi PII: DOI: Reference:

S0893-9659(18)30219-2 https://doi.org/10.1016/j.aml.2018.06.036 AML 5570

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Applied Mathematics Letters

Received date : 25 May 2018 Accepted date : 30 June 2018 Please cite this article as: K. Liu, W. Shi, The cauchy problem for linear inhomogeneous wave equations with variable coefficients, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.06.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The cauchy problem for linear inhomogeneous wave equations with variable coefficients Kai Liua , Wei Shib,∗ a College

of Applied Mathematics, Nanjing University of Finance& Economics, Nanjing 210023, P.R. China of Mathematical Sciences, Nanjing Tech University, Nanjing 211816, P.R.China

b College

Abstract In this paper, we present a new analytical formula for the Cauchy problem of the linear inhomogeneous wave equation with variable coefficients. The formula gives a much simpler solution than that given by the classical Poisson formula. The derivation is based on Duhamel’s Principle and the theory of pseudodifferential operator. An example is solved by using the formula to illustrate the feasibility. Keywords: Cauchy problem, inhomogeneous wave equation, variable coefficient 2010 Mathematics Subject Classification. 35L15, 35S10, 65L05, 65L06

1. Introduction The history of partial differential equations can date back to the 18th century when the first one-dimensional wave equation utt = u xx was introduced by d’Alembert as a model of a vibrating string [1]. Giving its initial displacement u(x, 0) = ϕ(x) and velocity ut (x, 0) = ψ(x), it is known that an explicit solution to the Cauchy, or initial value, problem of the wave equation is Z 1 1 x+t u(x, t) = (ϕ(x − t) + ϕ(x + t)) + ψ(ξ)dξ. 2 2 x−t

If an external force f (x, t) (per unit mass) applied on the string, then the equation becomes to an inhomogeneous wave equation utt = u xx + f (x, t). In this case, the solution which can be derived by using Duhamel’s Principle is Z Z Z 1 1 x+t 1 t x+(t−τ) u(x, t) = (ϕ(x − t) + ϕ(x + t)) + ψ(ξ)dξ + f (σ, τ)dσdτ. 2 2 x−t 2 0 x−(t−τ) Various methods have been utilized to solve the Cauchy problem for higher dimensional wave equation such as the Hadamard method of descent for n = 2, the method of spherical means for n = 3 [2], the analytic continuation of an integral of fractional order [3] and the finite part of a divergent integral [4], etc. A formal series solution has been derived by using a classical power series in [5] and the same formula is obtained in [6] by using the Adomian decomposition method. A direct derivation of the spherical means solution for arbitrary dimension n is presented using Fourier transform in [7]. Consider the Cauchy problem for the following linear inhomogeneous wave equation with variable coefficients in Rn  utt + Lu = f (x, t), (x, t) ∈ Rn × R+ ,      u(x, 0) = ϕ(x), x ∈ Rn , (1)      u (x, 0) = ψ(x), x ∈ Rn , t I The research was supported in part by the Natural Science Foundation of China under Grant 11501288, 11701271 and 11671200, by the Natural Science Foundation of Jiangsu Province under Grant BK20150934, and by the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant 16KJB110010 and 14KJB110009. ∗ Corresponding author Email addresses: [email protected] (Kai Liu), [email protected] (Wei Shi)

1

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K. Liu, W. Shi

where L is a linear differential operator of order m with variable coefficients and L has the following form X L ≡ L(x, D) = aα (x)Dα . (2) |α|6m

Here, the multi-index α = (α1 , α2 , . . . , αn ) ∈ Zn+ is an n-tuple of nonnegative integers. The length |α| of α is n P |α| = α j = α1 + α2 + · · · + αn , for x = (x1 , x2 , . . . , xn ) ∈ Rn , xα = x1α1 x2α2 · · · xnαn , the differential operator j=1

Dα is defined as

Dα h(x) =

1 ∂α1 ∂α2 ∂αn h(x). α1 α2 · · · |α| i ∂x1 ∂x2 ∂xnαn

It is noted that wave equation (1) has the same form as the following oscillatory ordinary differential equation    qtt (t) + Mq(t) = f (q(t)), (3)   q(0) = q , q (0) = p , 0 t 0

where q ∈ Rn , M ∈ Rn×n and M is a symmetric and positive semi-definite matrix. In the past few decades, much effort has been spent developing numerical methods to solve oscillatory system (3). In 1961, Gautschi [8] proposed the Gautschi method based on trigonometric polynomials. After that, many researchers have been constructing and analyzing Gautschi-type methods [9, 10]. In [11], Gonzalez et al. designed a new family of explicit Runge-Kutta type methods with G-functions for the numerical integration of highly oscillatory system. Recently, Wu et al. [12] formulated a standard form of the multi-frequency and multidimensional ERKN (extended Runge-Kutta-Nystr¨om) integrators in which both the internal stages and updates are incorporated with the structure brought by Mq. The ERKN integrators naturally integrate exactly the unperturbed linear equation qtt + Mq = 0. For more details on numerical methods for oscillatory system (3), we refer the reader to [13, 14, 15, 16]. It is noted that all the numerical methods mentioned above are essentially related with the following variation-of-constants formula which is established in many literatures with different notations (e.g., [10, 11, 17]). It gives the exact solution and its derivative for the oscillatory system (3) (following the setup in [17]): Z t 
 2 2   q(t) = φ (t M)q + tφ (t M)p + (t − s)φ1 ((t − s)2 M) f q(s) ds,  0 0 1 0    0 (4) Z t   
 2 2 2   q (t) = −tMφ (t M)q + φ (t M)p + φ ((t − s) M) f q(s) ds,  t 1 0 0 0 0 0

where the analytical functions φ j (·), j = 0, 1, . . . are defined as φ j (s) :=

∞ X (−1)k sk , (2k + j)! k=0

j = 0, 1, . . . .

(5)

In this paper, we will show that the linear inhomogeneous wave equation (1) also admits solution of the form (4). The rest of the paper is organized as follows. Some preliminaries and notation are introduced in Section 2. The new analytical formula for the wave equation is derived in Section 3. In Section 4, the feasibility of the formula is illustrated by an example. 2. Preliminaries and some notation First we will recall some basic definitions. Denote the Schwartz space or space of rapidly decreasing functions on Rn as S(Rn ) = { f ∈ C∞ (Rn ) : k f kα,β < ∞, ∀α, β ∈ Zn+ } with k f kα,β = sup x∈Rn |xα Dβ f |. The Fourier transform of a function ϕ ∈ S(Rn ) [18, 19] is given by Z b ϕ(ξ) = ϕ(x)e−ix·ξ dx, Rn

(6)

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K. Liu, W. Shi n

where x · ξ is the inner product of x and ξ in R . It follows immediately from integral by parts that α ϕ(ξ) = ξ αb d D ϕ(ξ) and

α ϕ(ξ). Dαb ϕ(ξ) = xd

Therefore, if ϕ(x) ∈ S(Rn ), then b ϕ(ξ) ∈ S(Rn ). Furthermore, the Fourier inversion formula holds Z 1 b ϕ(ξ)eiξ·x dξ. ϕ(x) = (2π)n Rn And we have

\ (ϕ ∗ ψ)(ξ) = b ϕ(ξ)b ψ(ξ),

(7)

(8) (9)

where ∗ denotes the convolution operation defined as Z (ϕ ∗ ψ)(x) = ϕ(x − y)ψ(y)dy. Rn

According to equations (7) and (9), we have, for ϕ(x) ∈ S(Rn ) Z 1 iξ·x Lϕ(x) = L(x, D)ϕ(x) = L(x, ξ)ϕ(ξ)e ˆ dξ, (10) (2π)n Rn X where L(x, ξ) = aα (x)ξα is a multivariate polynomial of degree m with respect to ξ. Not only for |α|6m

polynomials, the right-hand side of equation (10) is also well defined for general functions φ(x, ξ). To be more precise, if φ(x, ξ) ∈ C∞ (Rn × Rn ) satisfies α β m−|α| , ∀α, β ∈ Zn+ (11) Dξ D x φ(x, ξ) 6 Cα,β (1 + |ξ|) for some m ∈ R+ , then we have the following definition . Definition 2.1. ([20]) (Φϕ)(x) =

1 (2π)n

Z

Rn

iξ·x φ(x, ξ)ϕ(ξ)e ˆ dξ.

(12)

is said to be pseudodifferential operator. The function φ(x, ξ) is called symbol of the pseudodifferential operator Φ. 3. Solution to the wave equation It can be verified that the first three φ-functions (5) are √ √ sin( s) φ0 (s) = cos( s), φ1 (s) = √ , s

φ2 (s) =

√ sin2 ( s/2) . s/2

1 , j = 0, 1, . . . , the function φ j (s), j = 0, 1, . . . is well defined at s = 0. j! Now we are in a position to present our main results. Assume that for the linear differential operator L in wave equation (1) , the symbol L(x, ξ) > 0, for ∀x, ξ ∈ Rn . With reasonable restrictions on the smoothness and growth of coefficients aα (x), |α| 6 m such that (11) holds for L(x, ξ), the following two pseudodifferential operators, with symbols φ0 (t2 L(x, ξ)) and φ1 (t2 L(x, ξ)), respectively, are well defined for ∀t ∈ R+ . Z 1 iξ·x φ0 (t2 L(x, ξ))ϕ(ξ)e ˆ dξ, (φ0 (t2 L)ϕ)(x) = (2π)n Rn Z (13) 1 2 iξ·x (φ1 (t2 L)ϕ)(x) = φ (t L(x, ξ)) ϕ(ξ)e ˆ dξ 1 (2π)n Rn Since we have lim+ φ j (s) = s→0

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K. Liu, W. Shi

First, we consider the corresponding homogeneous wave equation of (1)  utt + Lu = 0,      u(x, 0) = ϕ(x),      u (x, 0) = ψ(x), t

(x, t) ∈ Rn × R+ , x ∈ Rn , x ∈ Rn .

(14)

Based on the two pseudodifferential operators defined by (13), the following theorem for the homogeneous wave equation (14) can be obtained. Theorem 3.1. Assume that the initial data ϕ, ψ ∈ S(Rn ). Then the solution to the Cauchy problem for the wave equation (14) and its derivative with respect to t are given as u(x, t) = (φ0 (t2 L)ϕ)(x) + t(φ1 (t2 L)ψ)(x),

(15)

ut (x, t) = −t(Lφ1 (t2 L)ϕ)(x) + (φ0 (t2 L)ψ)(x), Proof. Differentiating both sides of the first equation in (15) with respect to t twice and noticing that  d  tφ1 (t2 L) = φ0 (t2 L), dt

d φ0 (t2 L) = −tLφ1 (t2 L), dt we have and

ut (x, t) = −t(Lφ1 (t2 L)ϕ)(x) + (φ0 (t2 L)ψ)(x) utt (x, t) = −(Lφ0 (t2 L)ϕ)(x) − t(Lφ1 (t2 L)ψ)(x) = −Lu(x, t).

The proof is completed. @

Remark 3.1. Substituting φ0 and φ1 by their series expansions in (15) gives a formal series solution to the Cauchy problem of homogeneous wave equation (14) u(x, t) =

∞ ∞ X X t2k t2k+1 (−1)k Lk ϕ(x) + (−1)k Lk ψ(x), (2k)! (2k + 1)! k=0 k=0

(16)

where Lk ϕ is interpreted as L(Lk−1 ϕ). If L is the negative Laplacian operator −∆, this solution is coincided with the formula obtained previously in [5, 6]. Generally speaking, the closed form solutions for concrete problems are not easily obtainable. In this case, truncated series in the form of the sequence of partial sum uK (x, t) =

K K X X t2k t2k+1 (−1)k Lk ϕ(x) + (−1)k Lk ψ(x) (2k)! (2k + 1)! k=0 k=0

can be used to approximate the solution. Now we consider the inhomogeneous wave equation (1). Duhamel’s Principle states that if S (t) is the solution operator for the first-order initial-value problem Ut + AU = 0, U(0) = Φ, then the R t solution of the inhomogeneous problem Ut + AU = F, U(0) = Φ should be given by U(t) = S (t)Φ + 0 S (t − s)F(s)ds. Introducing a new function v(x, t) = ut (x, t), (1) can be rewritten as                 

u(x, t) v(x, t) u(x, 0) v(x, 0)

!0

!t

+ =

0 L

−1 0 ! ϕ(x) ψ(x)

!

u(x, t) v(x, t)

!

=

0 f (x, t)

!

,

Theorem 3.1 implies that the solution operator S (t) associated with (17) is given by

(17)

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K. Liu, W. Shi

S (t)

ϕ(x) ψ(x)

!

=

φ0 (t2 L) tφ1 (t2 L) 2 −tLφ1 (t L) φ0 (t2 L)

!

ϕ(x) ψ(x)

!

Applying Duhamel’s Principle to (17), the solution is given by ! ! ! u(x, t) φ0 (t2 L) tφ1 (t2 L) ϕ(x) = 2 2 v(x, t) −tLφ1 (t L) φ0 (t L) ψ(x) ! Z t 2 φ0 ((t − s) L) (t − s)φ1 ((t − s)2 L) + −(t − s)Lφ1 ((t − s)2 L) φ0 ((t − s)2 L) 0

(18)

0 f (x, s)

!

ds

Noticing that the solution of (1) is given by the first component of the solution of (17), we have the following theorem. Theorem 3.2. Assume that the initial data ϕ, ψ ∈ S(Rn ) and the source term f (x, t) ∈ S(Rn ) for all t > 0. Then the solution to the Cauchy problem for the wave equation (1) is given by Z t u(x, t) = (φ0 (t2 L)ϕ)(x) + t(φ1 (t2 L)ψ)(x) + (t − s)(φ1 ((t − s)2 L) f )(x, s)ds. (19) 0

If the right-hand side of (1) f (x, t) ≡ f (x) is independent of time variable t, then the solution (20) can be reduced to u(x, t) = (φ0 (t2 L)ϕ)(x) + t(φ1 (t2 L)ψ)(x) + t2 (φ2 (t2 L) f )(x).

(20)

Remark 3.2. Substituting φ0 , φ1 and φ2 by their series expansions in (20) gives a formal series solution to the Cauchy problem of inhomogeneous wave equation (1) u(x, t) =

∞ ∞ ∞ X X X t2k t2k+1 t2k+2 (−1)k Lk ϕ(x) + (−1)k Lk ψ(x) + (−1)k Lk f (x). (2k)! (2k + 1)! (2k + 2)! k=0 k=0 k=0

(21)

If the closed form solutions for concrete problems are not obtainable. Truncated series in the form of the sequence of partial sum uK (x, t) =

K K K X X X t2k t2k+1 t2k+2 (−1)k Lk ϕ(x) + (−1)k Lk ψ(x) + (−1)k Lk f (x) (2k)! (2k + 1)! (2k + 2)! k=0 k=0 k=0

can be used to approximate the solution. 4. Example It is worth noting that the formulas (20) and (21) are independent of the space dimension. For computation, only differentiation instead of surface integration as in classical poisson formula is involved, which makes the calculation much simpler. Now, we illustrate our formula through an example. Example. Consider the linear inhomogeneous wave equation with N = 3  ∂2 u ∂2 u ∂2 u ∂2 u    utt (x, t) − (x, t) − sin2 (x1 + x2 + x3 ) (x, t) − (x, t) − cos2 (x1 + x2 + x3 ) (x, t) = e−2x1 −8x2 −4x3 ,    ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x3 ∂x3     u(x, 0) = e x1 +4x2 +2x3 ,       u (x, 0) = (x − x )2 − x2 . t

1

2

3

The linear differential operator in space is L=

1 ∂2 1 ∂2 1 ∂2 1 ∂2 + sin2 (x1 + x2 + x3 ) 2 + 2 + cos2 (x1 + x2 + x3 ) 2 . 2 i ∂x1 ∂x1 i ∂x1 ∂x2 i ∂x2 ∂x2 i ∂x3 ∂x3

(22)

K. Liu, W. Shi

6

Its symbol is given by L(x, ξ) = ξ12 + sin2 (x1 + x2 + x3 )ξ1 ξ2 + ξ22 + cos2 (x1 + x2 + x3 )ξ32 , which satisfies that L(x, ξ) > 0, ∀x, ξ ∈ R3 . Therefore, formula (20) is applicable. It can be seen that     L e x1 +4x2 +2x3 = −21e x1 +4x2 +2x3 , Lk e x1 +4x2 +2x3 = (−21)k e x1 +4x2 +2x3 , k = 2, 3, . . . ,     2 2 k 2 2 L (x1 − x2 ) − x3 = −2, L (x1 − x2 ) − x3 = 0, k = 2, 3, . . . ,     −2x1 −8x2 −4x3 −2x1 −8x2 −4x3 k −2x1 −8x2 −4x3 k −2x1 −8x2 −4x3 L e = −86e , L e = (−86) e , k = 2, 3, . . . . Therefore,

  √ φ0 (t2 L) e x1 +4x2 +2x3 = cosh( 21t)e x1 +4x2 +2x3 , !    t2  φ1 (t2 L) (x1 − x2 )2 − x32 = 1 + (x1 − x2 )2 − x32 , 6 √  cosh( 86t) − 1  φ2 (t2 L) e−2x1 −8x2 −4x3 = e−2x1 −8x2 −4x3 , 86t2 The solution to the cauchy problem for the wave equation (22) is given by √ !  cosh( 86t) − 1 √ t3  u(x, t) = cosh( 21t)e x1 +4x2 +2x3 + t + (x1 − x2 )2 − x32 + e−2x1 −8x2 −4x3 . 6 86 References

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