Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

Journal Pre-proof Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

J.E. Macías-Díaz

PII:

S0168-9274(19)30372-1

DOI:

https://doi.org/10.1016/j.apnum.2019.12.021

Reference:

APNUM 3735

To appear in:

Applied Numerical Mathematics

Received date:

19 September 2019

Revised date:

19 December 2019

Accepted date:

20 December 2019

Please cite this article as: J.E. Macías-Díaz, Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system, Appl. Numer. Math. (2019), doi: https://doi.org/10.1016/j.apnum.2019.12.021.

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Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system J. E. Mac´ıas-D´ıaz∗ Departamento de Matem´aticas y F´ısica, Universidad Aut´onoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico

Abstract In the article [Appl. Num. Math. 146:245–29 (2019)], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations. Keywords: existence of discrete solutions, conservative finite-difference scheme, Klein–Gordon–Zakharov equation 2010 MSC: 65Mxx, 65Qxx

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1. Introduction In the published article [1], the authors presented a finite-difference model for a space-fractional Klein–Gordon– Zakharov (KGZ) equation [2]. Unfortunately, a theorem on the existence of solutions was not provided at all and the authors simply omitted it under the assumption that the proof was straightforward. It turns out that the proof is not an easy task. In this work, we establish the proof in order to provide the readers with a nontrivial tool for similar problems. Moreover, some important comments will be provided in this short communication. The mathematical model under investigation in [1] was the space-fractional KGZ system ∂2 u(x, t) ∂α u(x, t) − + u(x, t) + m(x, t)u(x, t) + |u(x, t)|2 u(x, t) = 0, ∀(x, t) ∈ Ω, ∂|x|α ∂t2   2 2 ∂2 m(x, t) ∂2 m(x, t) ∂ |u(x, t)| − = , ∀(x, t) ∈ Ω, 2 ∂x2 ∂x2 ⎧ ∂t ⎪ (x), m(x, 0) = m (x), ∀x ∈ B, u(x, 0) = u ⎪ 0 0 ⎪ ⎪ ⎪ ⎨ ∂u(x, 0) ∂m(x, 0) subject to ⎪ ⎪ = u1 (x), = m1 (x), ∀x ∈ B, ⎪ ⎪ ⎪ ⎩ u(x∂t, t) = u(x , t) = 0, m(x∂t, t) = m(x , t) = 0, ∀t ∈ [0, T ]. L

8 9

R

L

(1.1)

R

In this model, the fractional derivatives are understood in the Riesz sense [3, 4, 5]. Moreover, the authors of [1] proposed the following finite-difference model to solve the continuous problem (1.1):   (1)   (1) (1) n n (α) n n (1) n n2 μt U nj = 0, ∀( j, n) ∈ I δ(2) t U j − δ x U j + μt U j + M j μt U j + μt |U j | (2) n (2) n (2) n 2 δt M j − δ x M j − δ x |U j | = 0, ∀( j, n) ∈ I ⎧ 0 0 ⎪ (1.2) U = u (x ), M = m (x ), ∀ j ∈ IJ, ⎪ 0 j 0 j j ⎪ ⎪ ⎨ (1)j 0 (1) 0 such that ⎪ U = u (x ) δ M = m (x ), ∀ j ∈ I , δ 1 j 1 j J−1 t t ⎪ j j ⎪ ⎪ ⎩ U n = U n = 0, M0n = M Jn = 0, ∀n ∈ I N . J 0 ∗ Corresponding

author. Tel.: +52 449 910 8400; fax: +52 449 910 8401. Email address: [email protected] (J. E. Mac´ıas-D´ıaz)

Preprint submitted to Applied Numerical Mathematics

December 30, 2019

Note that the initial conditions of the numerical model (1.2) require the knowledge of the fictitious U −1 and M −1 . In order to eliminate them, we require for the difference equations of (1.2) to hold also for n = 0. Using then the initial data, we readily obtain that for each j ∈ I J−1 , the following identities hold: τ2

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 1 12 1 2 − − τu1 (x j )) 1 + + |U j | + |U j − 2τu1 (x j )| , 2 2   τ M 0j + |U 0j |2 . M 1j =m0 (x j ) + τm1 (x j ) + δ(2) x 2

2U 1j − 2u0 (x j ) − 2τu1 (x j )

0 =δ(α) x Uj

(U 1j

M 0j

(1.3) (1.4)

Using standard arguments with Taylor approximations, it can be checked that this formula will still preserve the quadratic consistency of the finite-difference method (1.2). ˚ h such that In what follows, we will let (V n )n∈I N be a sequence in V n n δ(2) x V j = δt M j ,

∀( j, n) ∈ I J−1 × I N−1 ,

(1.5)

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Under these circumstances, (U, M) will denote a solution of (1.2), and V = (V n )n∈I N will satisfy (1.5). Note that the numerical model (1.2) is a three-step technique. Indeed, notice that the first equation of that system yields an expression with complex parameters in which the only unknown is U n+1 j . On the other hand, the second equation of (1.2) is a fully explicit difference equation which can be easily solved for M n+1 j , for each ( j, n) ∈ I. Obviously, this ‘semi-explicit’ character of the method has computational advantages over other methodologies [6].

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2. Nomenclature

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In this section, we briefly recall the discrete notation used to define the finite-difference scheme 1.2. Let Iq = {1, . . . , q} and I q = Iq ∪ {0}, for each q ∈ N. Throughout, we let J, N ∈ N, and define h = (xR − xL )/J and τ = T/N. We consider uniform partitions of the intervals [xL , xR ] and [0, T ], of the forms xL = x0 < x1 < . . . < x j < . . . < x J = xR , 0 = t0 < t1 < . . . < tn < . . . < tN = T, 19 20 21 22

∀ j ∈ IJ,

∀n ∈ I N ,

(2.1) (2.2)

respectively. For convenience, let I = I J−1 × IN−1 and I = I J × I N . For each ( j, n) ∈ I, we will use U nj and M nj to represent numerical approximations to the values of unj = u(x j , tn ) and mnj = m(x j , tn ), respectively. In this manuscript, we let Rh = {x j : j ∈ I J }, and represent the set of all complex functions on Rh by Vh . If V ∈ Vh then we set V j = V(x j ) for each j ∈ I J . Let U n = (U nj ) j∈I J and M n = (M nj ) j∈I J , and set U = (U n )n∈I N and M = (M n )n∈I N . Let V represent any of the functions U or M. In our work, we will employ the first-order difference operators δ x V nj = δt V nj =

V nj+1 − V nj h V n+1 − V nj j τ

,

∀( j, n) ∈ I J−1 × I N ,

(2.3)

,

∀( j, n) ∈ I J × I N−1 ,

(2.4)

the second-order difference operators n δ(1) t Vj =

V n+1 − V n−1 j j

, ∀( j, n) ∈ I J × IN−1 , 2τ n n − 2V j + V j−1 n , ∀( j, n) ∈ I J−1 × I N , δ(2) x Vj = h2 V n+1 − 2V nj + V n−1 j j n δ(2) V = , ∀( j, n) ∈ I J × IN−1 , t j τ2 V nj+1

2

(2.5) (2.6) (2.7)

the average operators μt V nj =

V n+1 + V nj j

, ∀( j, n) ∈ I J × I N−1 , 2 V n+1 + V n−1 j j n , ∀( j, n) ∈ I J × IN−1 , μ(1) t Vj = 2 23

(2.8) (2.9)

and the fractional centered difference operator (see [7]) n δ(α) x Uj = −

1 (α) n g j−k Uk . hα

(2.10)

k∈I J

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3. Existence theorem Throughout this section, we will employ extensively the discrete nomenclature introduced in [1]. In the way, we will also employ Lemma 3.2 of that paper. For the sake of convenience, recall that n μ(1) t,Υ V =

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29 30 31

32 33

Υ + V n−1 , 2

˚ h and V = U, M. for any Υ ∈ V

Lemma 3.1 (Young’s inequality). Let a, b ∈ R+ ∪ {0}, and let p, q ∈ (1, ∞) be such that the following inequality holds: |a| p |b|q + . ab ≤ p q

(1) n n2 n−1 (b) 4 Re (μ(1)

= Φ44 − U n−1 44 . t,Φ |U | )(μt,Φ U ), Φ − U

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In addition, the following inequalities hold for each λ ∈ [0, 1]:

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41 42 43

44 45

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1 q

= 1. For each  > 0, (3.2)

˚ h , and assume that U is a sequence of complex Lemma 3.2. Let U = (U n )n∈I N and M = (M n )n∈I N be sequences in V ˚ h: functions while the functions of V are real. The following hold, for each n ∈ IN−1 and Φ ∈ V

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+

˚ h then V ± W2 ≤ 2V2 + 2W2 . (i) If V, W ∈ V 2 2 2 ˚ h (see [8]). (ii) It is also well known that δ x W22 ≤ h42 W22 , for each W ∈ V

n n−1 (a) 2 Re μ(1)

= Φ22 − U n−1 22 . t,Φ U , Φ − U

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1 p

In the proof of the next lemma, one occasionally uses the following elementary facts:

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(3.1)

1 n n−1 (c) λτ2 Re M n μ(1)

≥ − 12 λτ2 Φ44 − C1 . t,Φ U , Φ − U  n n−1 n−1 (d) Re Φ − 2λU + λU , Φ − U ≥ 1 − 16 (3λ + 1) Φ22 − C2 . n n−1 (e) Re −λτ2 δ(α)

≥ − 16 λτ2 Φ22 − C3 . x U ,Φ − U

Here, the constant C1 depends only on U n−1 , U n and M n . Meanwhile, C2 depends only on U n−1 and U n . Proof. The proofs of these relations are similar to those given in [6, Lemma 4.3]. We need only mention that, to reach 2 2 the inequalities, we applied Young’s theorem with  = 3, and the constants are C1 = 3τ4 M n 22 + τ2 | M n , |U n−1 |2 |, (α) n n 2 2 n−1 C2 = 4U n−1 22 + 3U n 22 + 2| U n , U n−1 | and C3 = 32 τ2 δ(α)

|. x U 2 + τ | δ x U , U Lemma 3.3 (Leray–Schauder fixed-point theorem [9]). Let X be a Banach space, and let F : X → X be continuous and compact. If the set S = {x ∈ X : λF(x) = x for some λ ∈ [0, 1]} is bounded then F has a fixed point. The following is the proof of the existence of discrete solutions of the finite-difference method of [1]. Theorem 3.4 (Solubility). The numerical model (1.2) is solvable for any set of initial conditions 3

Proof. Beforehand, note that M n+1 is defined explicitly in terms of M n , M n−1 , U n and U n−1 , for each n ∈ IN−1 , so we only need to establish the solubility of U. Let n ∈ IN−1 , and suppose that M n , M n−1 , U n and U n−1 have been ˚h →V ˚ h be defined for each j ∈ I J−1 and Φ ∈ V ˚ h by calculated. Let G : V  (1)   (1)  n 2 (1) n 2 n (1) n 2 n2 G j (Φ) = 2U nj − U n−1 μt,Φ U nj . (3.3) + τ2 δ(α) x U j − τ μt,Φ U j − τ M μt,Φ U j − τ μt,Φ |U j | j 48 49 50 51

In the case when j ∈ {0, J}, we let G j (Φ) = 0. It is obvious that G is a continuous and compact map from the Banach ˚ h into itself. We will prove next that S of Lemma 3.3 is a bounded subset of X = V ˚ h . Let Φ ∈ V ˚ h and space V λ ∈ [0, 1] satisfy λG(Φ) = Φ. Take the real part of the inner product of both sides of the equation 0 = Φ − λG(Φ) with Φ − U n−1 , use the results of Lemma 3.2, combine terms, simplify and bound from below using that λ ∈ [0, 1] to obtain 

λτ2 4λ + 1 λτ2 τ2 τ2 1 2 Φ2 − C ≤ 1 + − Φ22 + Φ44 − C1 − C2 − C3 − U n−1 22 − U n−1 44 ≤ 0, (3.4) 6 2 6 6 2 4

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where C = C1 +C2 +C3 + 12 τ2 U n−1 22 + 14 τ2 U n−1 44 . As a consequence, the set S of Lemma 3.3 is bounded, whence it follows that F has a fixed point. In this way, the existence of U n+1 is established. The existence of the approximation U 1 is carried out in similar fashion. The conclusion of the theorem follows now using induction.

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4. Conclusion

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In this note, we establish the existence of solutions of the discrete model reported in [1]. This result corrects the limitations of our published manuscript, inherent to the lack of completeness in that sense. It is worth mentioning that the actual proof to guarantee the existence of solutions requires some standard arguments. More precisely, we use the Leray–Schauder fixed-point theorem with the help of a suitable function G from a Banach space into itself in order to guarantee the existence of solutions. The author would like to apologize for any convenience caused by the missing information in [1]. Before closing this note, we would like to point out that the proofs of the existence of numerical solutions of [6, Theorem 5.3] is wrongful. The errors can be found in equations (5.6) and (5.7): the second and the third terms on the right-hand sides of those equations should have been multiplied by λ. Indeed, the use of the Leray–Schauder fixed-point theorem requires to consider λF(Φ, Ψ) = (Φ, Ψ), which is equivalent to the set of equations  (1)   (1)   (1)   (1)  (1) n 2 (1) n 2 n n 2 n2 μt,Φ U nj , (4.1) 0 = Φ j − 2λU nj + λU n−1 − λτ2 δ(α) x μt,Φ U j + λτ μt,Φ U j + λτ μt,Ψ M j μt,Φ U j + λτ μt,Φ |U j | j (1) n 2 (2) (1) n2 0 = Ψ j − 2λM nj + λM n−1 − λτ2 δ(2) x μt,Ψ M j − λτ δ x μt,Φ |U j | , j

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(4.2)

and not to (5.6) and (5.7) of that work. The proof of [6, Theorem 5.3] can be corrected now following an argument similar to the proof of Theorem 3.4.

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Acknowledgments. The author wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928. Finally, the author wishes to thank the anonymous reviewers and the editor in charge of handling this paper for all their comments and criticisms. All the suggestions were taken into account, obtaining thus a substantial improvement in the overall quality of this note.

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