Discrete Bessel functions and partial difference equations

Transcription

1 Disrete Bessel funtions and partial differene equations Antonín Slavík Charles University, Faulty of Mathematis and Physis, Sokolovská 83, Praha 8, Czeh Republi Abstrat We introdue a new lass of disrete Bessel funtions and disrete modified Bessel funtions of integer order. After obtaining some of their basi properties, we show that these funtions lead to fundamental solutions of the disrete wave equation and disrete diffusion equation. Keywords: Bessel funtion; modified Bessel funtion; Bessel differene equation; disrete wave equation; disrete diffusion equation; fundamental solution MSC 010 subjet lassifiation: 39A1, 39A14, 33C10, 33C05, 39A06, 39A10, 34A33 1 Introdution In their reent paper [], M. Bohner and T. Cuhta have proposed a new definition of the disrete Bessel funtion J n t = 1n t n n t n F n!, n t + 1 ; n + 1; 1, t N 0, 1.1 where F is the hypergeometri funtion and x k denotes the Pohhammer symbol also known as the rising fatorial given by xx + 1 x + k 1 for k N, x k = 1 for k = 0. The disrete Bessel funtion given by 1.1 is different from the one studied in earlier papers [4, 5], and its advantage is that it shares many properties with the lassial Bessel funtion. For example, it satisfies the differene equation tt 1 yt + t yt 1 + tt 1yt n yt = 0 where ft = ft + 1 ft is the forward differene, whih is a disrete analogue of the Bessel differential equation t y t + ty t + t n yt = 0. The goal of this paper is to introdue a new lass of disrete Bessel funtions denoted by J n, where n N 0 is the order and is a parameter, and to show that these disrete Bessel funtions provide fundamental solutions to the disrete wave equation ux, t = ux + 1, t ux, t + ux 1, t, x Z, t N 0 1. with ux, t being the seond-order forward differene of u with respet to t. The disrete Bessel funtion J n given by 1.1 is a speial ase of J n orresponding to = 1. 1

2 We also introdue a new lass of disrete modified Bessel funtions denoted by I n, whih an be used to onstrut fundamental solutions of the disrete diffusion equation ux, t = ux + 1, t ux, t + ux 1, t, x Z, t N where ux, t is the forward differene of u with respet to t. Our motivation omes from the theory of lattie differential equations, i.e., equations with disrete spae and ontinuous time. In this ontext, it is known that the fundamental solutions of the lattie wave equation u t x, t = ux + 1, t ux, t + ux 1, t, x Z, t R + 0, 1.4 whih is a semidisrete analogue of 1., have the form u 1 x, t = J x t and u x, t = t 0 J xs ds, where J x is the lassial Bessel funtion see [8, Example 3.3]. Similarly, the fundamental solution of the lattie diffusion equation u t x, t = ux + 1, t ux, t + ux 1, t, x Z, t R + 0, 1.5 whih is a semidisrete analogue of 1.3, has the form ux, t = e t I x t, where I x is the lassial modified Bessel funtion see [10, Example 3.1]. The orresponding formulas for fundamental solutions of the partial differene equations 1. and 1.3 will be obtained in Setions 3 and 4, respetively. To ahieve this goal, we need the disrete analogues of the funtions t J n t and t I n t for an arbitrary > 0, whih are preisely the funtions J n and I n mentioned earlier. While the funtions t J n t and t I n t satisfy the differential equations t y t + ty t + ± t n yt = 0, we will show that their disrete ounterparts J n and I n are solutions of the differene equations tt 1 yt + t yt 1 ± tt 1yt n yt = By expanding the differenes, we obtain the equivalent form t n yt tt 1yt ± tt 1yt = We remark that the fundamental solutions of the partial differene equations 1. and 1.3 are already available in the existing literature [8, 10], but they are expressed in a different form than we obtain in Setions 3 and 4. Expressing them in terms of the disrete Bessel funtions an simplify the study of their properties. For example, following the method from [], we prove that the funtion J n is osillatory. This fat implies that for eah fixed x, the first fundamental solution to 1. is osillatory as a funtion of t; this result is new and would be diffiult to obtain by different methods. Disrete Bessel funtions Both types of the Bessel funtions, J n and I n, will be defined in terms of the hypergeometri series F α, β; γ; z = k=0 α k β k z k. γ k k! Definition.1. For eah C, we define the disrete Bessel funtion Jnt = /n t n F n!, n t + 1 ; n + 1;, t N 0, n N 0,.1

3 and the disrete modified Bessel funtion Int = /n t n F n!, n t + 1 ; n + 1;, t N 0, n N 0.. Note that if n > t, then t n = 0 and therefore Jnt = Int = 0. Otherwise, if n t, then one of the frations n t and n t + 1 is a nonpositive integer, whih means that the hypergeometri series ourring in.1 and. have only finitely many nonzero terms. As in [], the definition of Jnt an be extended to all t Z, but the same extension is not always possible for Int. Similarly, it would be possible to onsider Bessel funtions of non-integer orders n. However, for simpliity, we restrit ourselves to nonnegative integer values of t and n; this ase is the most interesting one for appliations to partial differene equations. For = 1, the funtion J n oinides with the disrete Bessel funtion 1.1 introdued in []. For appliations in partial differene equations, the most useful ase is when is a positive real number. One advantage of allowing to be omplex is the onnetion formula I nt = i n J i n t, whih is a straightforward onsequene of the definitions. Our first goal is to prove that J n and I n satisfy the differene equations The next result generalizes [, Theorem 1]. Theorem.. If C and n N 0, then the funtion satisfies the differene equation or equivalently B n t = /n t n F n!, n t + 1 ; n + 1; ±, t N 0, tt 1 B n t + t B n t 1 tt 1B n t n B n t = 0, t, t n B n t tt 1B n t tt 1B n t = 0, t. Proof. We use the ontiguous relation see [7, formula ] γ α βf α, β; γ; z γ αf α 1, β; γ; z + β1 zf α, β + 1; γ; z = 0 with α = n t + 1, β = n t + 1, γ = n + 1, z = ± to get t 1 F + 1, n t + 1 ; n + 1; ± + n t F n + t + 1, n t F, n t + 1 ; n + 1; ± + 3 ; n + 1; ± = 0. By multiplying the equation with /n n! t n+1, using the definition of B n and the symmetry of F in the first two arguments, we obtain tt 1B n t 1 n + t t + nb n t + 1 tt 1B n t = 0. Corollary.3. For eah C and n N 0, the funtion J n is a solution of the differene equation tt 1 yt + t yt 1 + tt 1yt n yt = 0, t, and the funtion I n is a solution of the differene equation tt 1 yt + t yt 1 tt 1yt n yt = 0, t. 3

4 The next task is to obtain expressions for differenes of the disrete Bessel funtions. The following result generalizes Theorems 5, 6 and Corollary 7 from []. Our proof is simpler than in [] and relies on the ontiguous relations for the hypergeometri funtion. Theorem.4. Assume that C. For eah n N 0, onsider the funtion B n t = /n t n F n! Then we have the following identities:, n t + 1 ; n + 1; ±, t N 0. t B n t 1 = nb n t ± tb n+1 t 1, n 0, t 1,.3 t B n t 1 = nb n t + tb n 1 t 1, n 1, t 1,.4 nb n t = t Bn 1 t 1 B n+1 t 1, n 1, t 1,.5 Proof. To prove.3, we need to show that ± n + 1 tf B n t = Bn 1 t ± B n+1 t, n 1, t = tb n t 1 + n tb n t ± tb n+1 t 1. Using the definition of B n and dividing by / n t n+1 /n + 1!, we see it is enough to show that 0 = n + 1F + 1, n t + 1; n + 1; ± + n + 1F + 1, n t ; n + 1; ± + 1; n + ; ±. + 3, n t To prove this, we use the ontiguous relations see [7, formulas and ] αγ1 zf α + 1, β + 1; γ; z = γ γ β 1F α, β; γ; z γ α β 1F α, β + 1; γ; z, αγ1 zf α + 1, β + 1; γ; z = αγf α, β + 1; γ; z αγ β 1zF α + 1, β + 1; γ + 1; z. By equating the right-hand sides and dividing by γ β 1, we get γf α, β; γ; z γf α, β + 1; γ; z = αzf α + 1, β + 1; γ + 1; z. The desired relation now follows by letting α = n t + 1, β = n t, γ = n + 1, z = ±. To prove.4, we have to show that 0 = tb n t 1 n + tb n t + tb n 1 t 1. Using the definition of B n and dividing by / n t n /n!, we see it is enough to show that 0 = t + nf + 1, n t + 1; n + 1; ± n + tf, n t + 1 ; n + 1; ± + nf, n t + 1 ; n; ±. To prove this, we use the ontiguous relation see [7, formula ] γ αf α, β; γ + 1; z + αf α + 1, β; γ + 1; z γf α, β; γ; z = 0. The desired relation now follows by letting α = n t, β = n t + 1, γ = n, z = ±. Identity.5 is obtained by subtrating.3 from.4. To get the identity.6, add.3 and.4, divide by t, and replae t by t

5 Corollary.5. For eah C, the following relations hold: t J nt 1 = nj nt tj n+1t 1, n 0, t 1,.7 t Jnt 1 = njnt + tjn 1t 1, n 1, t 1,.8 njnt = t J n 1 t 1 + Jn+1t 1, n 1, t 1,.9 J nt = J n 1 t J n+1t, n 1, t 0,.10 t I nt 1 = ni nt + ti n+1t 1, n 0, t 1,.11 t Int 1 = nint + tin 1t 1, n 1, t 1,.1 nint = t I n 1 t 1 In+1t 1, n 1, t 1,.13 I nt = I n 1 t + I n+1t, n 1, t The next theorem provides additional information about the values and differenes of J n and I n. Theorem.6. For eah C, the funtions J n and I n have the following properties: J 00 = I 00 = 1. J nt = I nt = 0 for all t N 0 and n N suh that n > t. J n0 = I n0 = 0 for all n N 0 \ 1}, and J 10 = I 10 = /. Proof. The first two statements follow from the definitions of Jn and In; note that t n = 0 for all t N 0 and n N suh that n > t. Using.10 and.14, we get Jn0 = J n 1 0 Jn+10 and In0 = I n In+10 for all n N. Both expressions are equal to / if n = 1, and zero for all n N \ 1}. For n = 0, the relations.7 and.11 with t = 1 imply J00 = I00 = 0. The remaining results in this setion are onerned with the sign of J n and I n if is a real number. The first statement generalizes [, Theorem 1]. Theorem.7. For eah R \ 0} and n N 0, the funtion J n is osillatory i.e., J nt hanges sign or vanishes for infinitely many values of t N 0. Proof. To simplify notation, we denote yt = J nt. Aording to Corollary.3 with t replaed by t +, we see that y satisfies the differene equation t + t + 1 yt + t + yt t + t + 1yt n yt + = 0, t N 0. Using the formulas yt = yt + 1 yt yt and yt + = yt yt + 1, we obtain yt t + t yt + 1 t + t + t + 1 n + yt + 1 t + 1t + n = 0, and therefore yt = t + t n t t + t yt + 1 t + 1t + n t + t yt + 1, t N 0. Let vt = t 1+ t/ t t n t Γ t n + 1Γ t+n + 1 =, t n, n + 1, n +,...}. 1 + t/ Γt + 1 5

6 One an verify using a omputer system suh as Mathematia or by a hand alulation similar to [, Lemma 11] that Let vt vt + vt t + 1t + + n t t + t = 0, t n, n + 1, n +,...},.15 vt lim t vt + 1 = Using the produt rule twie, we get ut = vtyt, t n, n + 1, n +,...}. ut = yt + 1 vt + vt yt, ut = yt + 1 vt + vt vt yt vt yt ut + 1 = vt + 1 vt + vt vt yt + 1 t + t n t +vt t + t yt + 1 t + 1t + n t + t vt = ut + 1 vt + 1 t + 1t + n vt t + t vt yt + 1 vt vt + vt t + t n t t + t The last term vanishes thanks to.15, and therefore t + 1t + n vt ut + ut + 1 t + t vt + 1 vt = 0. vt + 1 This is a seond-order differene equation of the form ut + qtut + 1 = 0, where qt = t + 1t + n vt t + t vt + 1 ut + 1 vt + 1. vt + vt vt. vt + 1 By Wintner s theorem see [3, Theorem 4.45], suh equation is osillatory if t=n qt =. To verify this fat, it is enough to show that lim t qt > 0. Using.16, we alulate lim qt = t + 1 lim vt vt + t vt + 1 lim t vt vt vt + 1 = = + 1 > 0. This shows that u is osillatory. Sine v is positive, y is osillatory. Theorem.8. For eah 0 and n N 0, the funtion I n is nonnegative. For eah < 0, the funtion I n is nonnegative if n is even and nonpositive if n is odd. Proof. The first statement where 0 is easily proved by indution with respet to t. For t = 0, it follows from the first and seond part of Theorem.6 that I n0 0 for all n N 0. Suppose that I nt 0 for all n N 0. By the relation.14, we have I nt 0 for all n N. If n = 0, then the relation.11 with t replaed by t + 1 implies I 0t = I 1t 0. Consequently, we have I nt + 1 = I nt + I nt 0 for all n N 0. The seond statement where < 0 is a onsequene of the first part and the identity whih follows immediately from the definition. I nt = 1 n I n t, 6

7 3 Disrete wave equation In this setion, we explore the relation between the disrete Bessel funtion J n and the disrete wave equation ux, t = ux + 1, t ux, t + ux 1, t, x Z, t N 0 the forward differene operator always applies to the time variable t; differenes with respet to the spae variable x are never onsidered in this paper. Suppose that u 1 : Z N 0 R is the solution orresponding to the initial onditions 1 if x = 0, u 1 x, 0 = 0 if x 0, u 1 x, 0 = 0, x Z. Then it is not diffiult to hek that the funtion u : Z N 0 R given by t 1 u x, t = u 1 x, s s=0 where the sum is understood as empty if t = 0 is the solution of the disrete wave equation satisfying the onditions u x, 0 = 0, x Z, 1 if x = 0, u x, 0 = 0 if x 0. In [8, Theorem 3.], it is shown that for arbitrary bounded real sequenes u 0 x} x Z, v 0 x} x Z, the funtion ux, t = k Zu 0 k u 1 x k, t + v 0 k u x k, t, x Z, t N 0, 3.1 is the solution of the disrete wave equation satisfying ux, 0 = u 0 x, ux, 0 = v 0 x, x Z. In fat, it is not diffiult to see use indution with respet to t that we have u 1 x, t = u x, t = 0 whenever x > t. Hene, on the right-hand side of the formula 3.1, the terms orresponding to k Z suh that x k > t do not ontribute to ux, t, and we an write ux, t = x+t k=x t u 0 k u 1 x k, t + v 0 k u x k, t, x Z, t N 0. The solutions u 1, u are referred to as the fundamental solutions of the disrete wave equation. The next theorem shows that u 1 and onsequently also u an be expressed in terms of the disrete Bessel funtion J n. Theorem 3.1. For eah > 0, the solution of the initial-value problem ux, t = ux + 1, t ux, t + ux 1, t, x Z, t N 0, 3. 1 if x = 0, ux, 0 = if x 0, ux, 0 = 0, x Z, 3.4 7

8 is given by Moreover, for eah x Z, the funtion t ux, t osillatory. ux, t = J x t, x Z, t N Proof. Let u be defined by 3.5. The relations 3.3 and 3.4 follow from Theorem.6. If x 1, we use the identity.10 to alulate ux, t = Jxt = Jx 1t Jx+1t, ux, t = Jx t Jxt + Jx+t = ux 1, t ux, t + ux + 1, t. Similarly, if x 1, we obtain ux, t = J xt = J x 1t J x+1t, ux, t = J x t J xt + J x+t = ux + 1, t ux, t + ux 1, t. Finally, for x = 0, we use the identity.7 with n = 0 and t replaed by t + 1 to get and onsequently by identity.10 u0, t = J 0 t = J 1 t, u0, t = J 0 t + J t = J t J 0 t + J t = u1, t u0, t + u 1, t. Thus, the relation 3. holds for all x Z, t N 0. The fat that t ux, t is osillatory follows from Theorem.7. Remark 3.. The first fundamental solution of the disrete wave equation an be alternatively expressed using the multinomial oeffiients as follows see [8, Example 3.5]: ux, t = t t 1 j j+x j, t j x, j + x j=0 4 Disrete diffusion equation We now turn our attention to the disrete diffusion equation wx, t = d wx + 1, t wx, t + wx 1, t, x Z, t N 0. The solution w : Z N 0 R orresponding to the initial onditions 1 if x = 0, wx, 0 = 0 if x 0, is alled the fundamental solution. In [9, Corollary 3.8], it is shown that for an arbitrary bounded real sequene u 0 x} x Z, the funtion ux, t = k Z u 0 k wx k, t, x Z, t N 0, 4.1 is the solution of the disrete diffusion equation satisfying ux, 0 = u 0 x, x Z. 8

9 Observing that wx, t = 0 whenever x > t use indution with respet to t, we an simplify the formula 4.1 to ux, t = x+t k=x t u 0 k wx k, t, x Z, t N 0. For d 1/, the next theorem shows that the fundamental solution w an be onstruted using the disrete modified Bessel funtion I n. Theorem 4.1. For eah d 1/, the solution of the initial-value problem is given by wx, t = d wx + 1, t wx, t + wx 1, t, x Z, t N 0, 4. 1 if x = 0, wx, 0 = if x 0. wx, t = 1 d t I d/1 d x t, x Z, t N Moreover, for eah x Z, the funtion t wx, t is nonnegative if d 0, 1/, and osillatory if d > 1/. Proof. Let w be defined by 4.4. To simplify notation, let = d/1 d. The relation 4.3 follows from Theorem.6. Let zx, t = I x x Z, t N 0. If x 1, we use the identity.14 to alulate Similarly, if x 1, we obtain zx, t = I x t = I x+1t + I x 1t = zx + 1, t + zx 1, t. zx, t = I xt = I x+1t + I x 1t = zx 1, t + zx + 1, t. Finally, if x = 0, identity.11 with n = 0 and t replaed by t + 1 implies that Note that Hene, by the produt rule, we have z0, t = I 0 t = I 1 t = z1, t + z 1, t. 1 d t = 1 d t+1 1 d t = d1 d t. wx, t = zx, t1 d t = zx, t1 d t+1 + zx, t 1 d t = d zx + 1, t + zx 1, t 1 d t+1 zx, td1 d t 1 d = d zx + 1, t zx, t + zx 1, t 1 d t = d wx + 1, t wx, t + wx 1, t. The fat that t wx, t is nonnegative if d 0, 1/ and osillatory if d > 1/ follows from the definition of w and Theorem.8. Remark 4.. An alternative form of the fundamental solution to the disrete diffusion equation is see [10, Example 3.3] t t wx, t = d j+x 1 d t j x. j, t j x, j + x j=0 This formula is valid also for d = 1/, when it redues to t 1 t t+x wx, t = if t + x is even, 0 if t + x is odd. 9

10 5 Conlusion We onlude the paper by pointing out two possible diretions for further researh: The lassial Bessel funtions have their multivariable ounterparts [1], whih found appliations in various areas of physis see, e.g., [6] and the referenes there. Is there a reasonable extension of the disrete Bessel funtions to several variables? If yes, is it related to the higher-dimensional disrete diffusion/wave equations? Note that an expliit formula for the fundamental solution of the n-dimensional disrete diffusion equation, whih does not rely on Bessel funtions, an be found in [10]. In Setions 3 and 4, we were dealing with the disrete diffusion/wave equations whose left-hand sides involve forward differenes of first and seond order with respet to time. In some situations, it might be more appropriate to onsider the bakward first-order differene for the diffusion equation, and the entral or bakward seond-order differene for the wave equation. Is it possible to express their solutions with the help of some Bessel-type funtions? Aknowledgement This paper was supported by the Czeh Siene Foundation, grant no. GA S. I am grateful to the anonymous referees for their suggestions, whih helped to improve the exposition of the paper. Referenes [1] P. Appell, Sur l inversion approhée de ertaines intégrales réelles et sur l extension de l équation de Kepler et des fontions de Bessel, C. R. Aad. Si , [] M. Bohner, T. Cuhta, The Bessel differene equation, Pro. Amer. Math. So , [3] M. Bohner, A. Peterson, Dynami Equations on Time Sales: An Introdution with Appliations, Birkhäuser, Boston, 001. [4] R. H. Boyer, Disrete Bessel funtions, J. Math. Anal. Appl. 1961, [5] J. J. Gergen, Bessel differene systems of zero order, J. Math. Anal. Appl , [6] H. J. Korsh, A. Klumpp, D. Witthaut, On two-dimensional Bessel funtions, J. Phys. A, Math. Gen , [7] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, eds., NIST Handbook of Mathematial Funtions, Cambridge University Press, New York, 010. Online version at [8] A. Slavík, Disrete-spae systems of partial dynami equations and disrete-spae wave equation, Qual. Theory Dyn. Syst , [9] A. Slavík, P. Stehlík, Dynami diffusion-type equations on disrete-spae domains, J. Math. Anal. Appl , [10] A. Slavík, P. Stehlík, Expliit solutions to dynami diffusion-type equations and their time integrals. Appl. Math. Comput ,