Gselmann, Eszter

Aequationes Mathematicae

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On a discrete version of the wave equationEszter GselmannDedicated to Professor Ja´nos Acze´l on his 90th birthdayAbstract. The main purpose of this paper is to determine the general(and also the continuous) solutions of the discrete wave equation, thatis, to solve the following partial difference equation∆21(x)u(x, y) = ∆21(y)u(x, y).Mathematics Subject Classification (2010). Primary 39A14; Secondary43A45, 39B40.Keywords. partial difference equation, wave equation, exponential poly-nomial, spectral analysis and synthesis.1. IntroductionNumerical partial differential equations is a rather large area of mathematics.There exist many different approaches to solve partial differential equations,the so-called finite difference method is one of them.This is nothing but a numerical method for approximating the solutionsto (ordinary or partial) differential equations using finite difference equationsto approximate derivatives. More precisely, the method is the following.Assume u is a function of the two independent variables x and y. Subdi-vide the x− y plane into sets of equal rectangles of sides δx = h and δy = k,and let the co-ordinates (x, y) of the representative mesh point P bex = ih and y = jh,where i, j ∈ Z. Denote the value of u at P byuP = u(ih, jk) = ui,j .This research has been supported by the Hungarian Scientific Research Fund (OTKA)Grant NK 81402 and by the TA´MOP 4.2.4.A/2-11-1-2012-0001 (Nemzeti Kiva´lo´sa´g Pro-gram – Hazai hallgato´i, illetve kutato´i szeme´lyi ta´mogata´st biztos´ıto´ rendszer kidolgoza´sae´s mu˝ko¨dtete´se konvergencia program) project implemented through the EU and Hungaryco-financed by the European Social Fund.2 E. Gselmann                                                           xykh(i, j + 1)(i − 1, j)(i, j − 1)P = (i, j) (i + 1, j)Assuming that u is two times continuously differentiable, due to Taylor’stheorem (∂2u∂x2)P≈ u((i+ 1)h, jk)− 2u(ih, jk) + u((i− 1)h, jk)h2,that is, (∂2u∂x2)P≈ ui+1,j − 2ui,j + ui−1,jh2with a leading error of order h2.Obviously, while applying the finite difference method, partial differenceequations arise rather naturally. Among the most important of all hyperbolicpartial differential equations is undoubtedly the wave equation, i.e.,∂2u∂t2=∂2u∂x2.The main aim of this paper is to consider its discrete counterpart, that is,the following partial difference equation∆21(x)u(x, y) = ∆21(y)u(x, y).In other words, we would like to solve equationu(x+ 2, y)− 2u(x+ 1, y) = u(x, y + 2)− 2u(x, y + 1)at first on Z× Z and also on R× R.2. A method based on spectral analysis and synthesisIn what follows we will shortly summarize a method (based on spectral ana-lysis and synthesis) developed by L. Sze´kelyhidi (see [8, 9] and also [4, 5]).Roughly speaking, spectral analysis and synthesis deal with the descrip-tion of translation invariant function spaces over locally compact Abeliangroups. Such type of subspaces naturally arise e.g., in the theory of ordinaryand partial differential, resp. difference equations. The main problem is tofind an appropriate class of basic functions (they are the so-called exponen-tial monomials). If in addition, their linear hull is dense, we say that spectralsynthesis holds for this subspace.On a discrete version of the wave equation 3Let G be a locally compact Abelian group and denote C (G) the setof all complex valued continuous functions defined on G equipped with thetopology of uniform convergence on compact sets. On a variety we mean aproper translation invariant closed subspace of C (G).One of the earliest result in the theory of spectral analysis and synthesisis due to L. Schwartz [6].Theorem 2.1 (Schwartz). In C (R) any variety is the closed linear hull ofexponential monomials which are contained in it.Let C (Zn) denote the linear space of all complex valued continuousfunction defined on Zn. Concerning C (Zn) M. Lefranc in [3] proved the fol-lowing.Theorem 2.2 (Lefranc). In C (Zn) any variety is the closed linear hull ofexponential monomials which are contained in it.As a consequence of Lefranc’s result, the following statement was provedin Sze´kelyhidi [7].Theorem 2.3. Spectral synthesis holds for any finitely generated discrete Abe-lian groups.Let Γ be a nonempty set, cγ : Zk → C be a finitely supported functionfor all γ ∈ Γ. Then all solutions f : Zk → C of the system of linear differenceequations ∑y∈Zkcγ(y)f(x+ y) = 0(x ∈ Zk, γ ∈ Γ) (∗)form a variety, which is proper if and only if at least one of the functions cγare non identically zero.Theorem 2.4. The spectrum of (∗) is the set of all λ ∈ Ck with nonzerocomponents of the system of algebraic equations∑y∈Zkcγ(y)f(x+ y) = 0(x ∈ Zk) .Theorem 2.5. The pair (λ, p) is in the spectral set of equation (∗) and p is anonzero polynomial, then λ is in the spectrum of (∗).Theorem 2.6. The annihilator ideal of∑y∈Zkcγ(y)λyp(x+ y) = 0is generated by the differential operators∑y∈Zkcγ(y)λy〈y, ∂〉jfor all γ ∈ Γ and j = 0, 1, 2, . . ..4 E. Gselmann3. Solutions of the discrete wave equationOur main results concern the general as well as the regular solutions of thediscrete wave equation, that is, our main purpose in this section is solvepartial difference equationu(x+ 2, y)− 2u(x+ 1, y) = u(x, y + 2)− 2u(x, y + 1) (W )at first on Z× Z and after that on R× R.Theorem 3.1. Assume that for the function u : Z×Z→ C functional equationu(x+ 2, y)− 2u(x+ 1, y) = u(x, y + 2)− 2u(x, y + 1)is fulfilled for any (x, y) ∈ Z × Z. Then and only then, there exist functionsα, β : Z→ C such that β is a generalized second order polynomial andu(x, y) = α(x+ y) + β(x− y) (x, y ∈ Z) .Proof. Due to the method of L. Sze´kelyhidi, presented in the previous section,at first it is enough to determine the exponential polynomial solutions ofequation (W ). The corresponding system of characteristic equations is[λ2 (2 · ∂1 + 0∂2)j − 2λ (1 · ∂1 + 0 · ∂2)j]p(x, y)=[µ2 (0 · ∂1 + 2 · ∂2)j − 2µ (0 · ∂1 + 1 · ∂2)j]p(x, y)for any j = 0, 1, 2, . . .. In case j = 0 we get the algebraic equationλ2 − 2λ = µ2 − 2µ,from which we get that either µ = λ or µ = 2− λ.Suppose that µ = λ is fulfilled. Then the above system of equation forj = 1 yields thatλ(λ− 1)∂1p(x, y) = λ(λ− 1)∂2p(x, y)Again we have two alternatives, either λ 6= 1 or λ = 1. In case λ 6= 1, thenfor the polynomial p, we get the partial differential equation∂1p(x, y) = ∂2(x, y).From this, we derive thatp(x, y) = α(x+ y)with a certain function α.In case λ = 1, then the characteristic equation with j = 1 is inexpressive.On the other hand, for j = 2, we get that∂21p(x, y) = ∂22p(x, y).This yields however thatp(x, y) = α(x+ y) + β(x− y)is fulfilled with certain functions α and β.On a discrete version of the wave equation 5Suppose finally, that µ = 2 − λ is satisfied. In this case the system acharacteristic equations with j = 1 means thatλ(λ− 1)∂1p(x, y) = (λ− 2)(λ− 1)p(x, y).Clearly, λ 6= 2, otherwise µ would be zero and we are only interested inthe nontrivial exponential polynomial solutions. If so, we get after some re-arrangement thatλ∂1p(x, y) = (λ− 1)∂2p(x, y),that is,p(x, y) = α ((λ− 1)x+ λy)holds with a certain function α. Furthermore, the system of characteristicequations for j = 2 is(2λ2 − λ) ∂21 = (2λ2 − 9λ+ 10) ∂22Inserting the above form of p into this, we get that(2λ2 − λ) · (λ− 1)2p(x, y) = (2λ2 − 9λ+ 10) · λ2p(x, y).Therefore, either p is identically zero or λ is a solution of the algebraic equa-tion (2λ2 − λ) · (λ− 1)2 = (2λ2 − 9λ+ 10) · λ2.The solutions of this equation areλ1 = 0 λ2 =3 +√134λ3 =3−√134.Clearly, the only interesting cases are λ2,3 =3±√134. For j = 3 the systemof characteristic equations is(8λ2 − 2λ) ∂31p(x, y) = (8(2− λ)2 − 2(2− λ)) ∂23p(x, y),that is,(8λ2 − 2λ) · (λ− 1)3p(x, y) = (8(2− λ)2 − 2(2− λ)) · λ3p(x, y)has to be fulfilled. This however means that either λ is a solution of thealgebraic equation(8λ2 − 2λ) · (λ− 1)3 = (8(2− λ)2 − 2(2− λ)) · λ3or p is identically zero. It is not too hard to see (after a simple substitution)that neither λ2 nor λ3 fulfills this equation.All in all, the exponential polynomial form solution of equation (W ) areu(x, y) = α(x+ y) + β(x− y).However, substituting this form into (W ) we get that β must be a generalizedsecond order polynomial.The pointwise limits of linear combinations of such functions are also ofthe formu(x, y) = α(x+ y) + β(x− y) (x, y ∈ Z) ,6 E. Gselmannwhere α : Z→ C is any function and β : Z→ C is a generalized second orderpolynomial.Conversely, an easy computation shows that the above function satisfiesequation (W ) for arbitrary (x, y) ∈ Z× Z. Theorem 3.2. Suppose that the function u : R × R → C satisfies functionalequationu(x+ 2, y)− 2u(x+ 1, y) = u(x, y + 2)− 2u(x, y + 1)for all (x, y) ∈ R×R. Then and only then, there exist functions α, β : R→ Csuch that β is a generalized second order polynomial, further, there is anantisymmetric and biadditive function B : R× R→ C such thatu(x, y) = α(x+ y) + β(x− y) +B(x, y) (x, y ∈ R)is fulfilled.Proof. After a similar argument that was presented in the proof of Theorem3.1, we get that the exponential polynomial solution of equation (W ) on R×Rare of the formp(x, y) = α(x+ y) + β(x− y),where β is a generalized second order polynomial. The pointwise limits oflinear combinations of such functions also has this form. Additionally, themappingR× R 3 (x, y) 7−→ u(x, y)− u(x, 0)− u(y, 0)fulfills the so-called cocycle equation, that isF (x+ y, z) + F (x, y) = F (x, y + z) + F (y, z) (x, y, z ∈ R) .Thus applying a result of Hosszu´ [2] finally we get thatu(x, y) = α(x+ y) + β(x− y) +B(x, y) (x, y ∈ R) ,with an appropriate function α : R → C, a generalized second order polyno-mial β : R→ C and an antisymmetric and biadditive function B : R×R→ C.The converse is an easy computation. In view of the results of the previous theorem we are also able to de-termine the continuous solutions of equation (W ) that is contained in thefollowing.Corollary 3.3. Let us assume that the continuous function u : R × R → Csatisfies functional equationu(x+ 2, y)− 2u(x+ 1, y) = u(x, y + 2)− 2u(x, y + 1)for all (x, y) ∈ R×R. Then and only then, there exists a continuous functionα : R→ C and constants a, b, c such thatu(x, y) = α(x+ y) + a(x− y)2 + b(x− y) + c (x, y ∈ R)holds.On a discrete version of the wave equation 7In [1] J. A. Baker investigated a partial difference equation rather simi-lar to equation (W ). Namely, in the aforementioned paper he presented thecontinuous solutions of equationu(x+ h, y)− 2u(x+ h, y) = u(x, y + h)− 2u(x, y + h) (x, y, h ∈ R) ,where u : R×R→ C is the unknown function. We would like to remark thatin view of Theorem 3.2 we can determine not only the continuous but alsothe general solutions of this equation. To see this, it is enough to substituteh = 1 and apply Theorem 3.2.Corollary 3.4. Assume that the function u : R × R → C fulfills functionalequationu(x+ h, y)− 2u(x+ h, y) = u(x, y + h)− 2u(x, y + h)for all x, y, h ∈ R. Then and only then, there exists functions α, β : R → Csuch that β is a generalized second order polynomial, further, there is anantisymmetric and biadditive function B : R× R→ C such thatu(x, y) = α(x+ y) + β(x− y) +B(x, y) (x, y ∈ R)is fulfilled.References[1] J. A. Baker, An analogue of the wave equation and certain related functionalequations, Canad. Math. Bull. 12 (1969), 837–846. MR 0254455 (40 #7663)[2] M. Hosszu´, On the functional equation F (x + y, z) + F (x, y) = F (x, y + z) +F (y, z), Period. Math. Hungar. 1 (1971), no. 3, 213–216. MR 0289991 (44#7176)[3] M. Lefranc, Analyse spectrale sur Zn, C. R. Acad. Sci. Paris 246 (1958), 1951–1953. MR 0098951 (20 #5396)[4] P. K. Sahoo and L. Sze´kelyhidi, A functional equation on Zn ⊕Zn, Acta Math.Hungar. 94 (2002), no. 1-2, 93–98. MR 1905789 (2003f:39086)[5] , On the general solution of a functional equation on Z⊕Z, Arch. Math.(Basel) 81 (2003), no. 2, 233–239. MR 2009566 (2004h:39063)[6] L. Schwartz, Sur une proprie´te´ de synthe`se spectrale dans les groupes non com-pacts, C. R. Acad. Sci. Paris 227 (1948), 424–426. MR 0027096 (10,249e)[7] L. Sze´kelyhidi, On discrete spectral synthesis, Functional equations—results andadvances, Adv. Math. (Dordr.), vol. 3, Kluwer Acad. Publ., Dordrecht, 2002,pp. 263–274. MR 1912720 (2003g:43006)[8] , Difference equations via spectral synthesis, Ann. Univ. Sci. Budapest.Sect. Comput. 24 (2004), 3–14. MR 2168033 (2007j:43006)[9] , Discrete spectral synthesis and its applications, Springer Monographsin Mathematics, Springer, Dordrecht, 2006. MR 2279454 (2009d:43001)8 E. GselmannEszter GselmannDepartment of AnalysisInstitute of MathematicsUniversity of DebrecenP. O. Box: 12.DebrecenH–4010Hungarye-mail: gselmann@science.unideb.huSubmitted: 28 February, 2014.

On a discrete version of the wave equation

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On a discrete version of the wave equation

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