This paper deals with the fractional generalization of the integro-differential diffusion-wave equation for the Heisenberg sub-Laplacian, with homogeneous Bitsadze-Samarskii type time-nonlocal conditions. For the considered problem, we show the existence, uniqueness and the explicit representation formulae for the solution

Avalishvili G

B. T. Torebek

Bitsadze AV

Fujita Y.

Il'in VA

Kilbas AA

M. Ruzhansky

Martin-Vaquero J

N. Tokmagambetov

Tanabe H.

Integral Transforms and Special Functions

English

Ruzhansky, M

Tokmagambetov, N

Torebek, BT

Queen Mary Research Online

BITSADZE-SAMARSKII TYPE PROBLEM FOR THEINTEGRO-DIFFERENTIAL DIFFUSION-WAVE EQUATION ONTHE HEISENBERG GROUPMICHAEL RUZHANSKY, NIYAZ TOKMAGAMBETOV, AND BERIKBOL T. TOREBEKAbstract. This paper deals with the fractional generalization of the integro-differential diffusion-wave equation for the Heisenberg sub-Laplacian, with homoge-neous Bitsadze-Samarskii type time-nonlocal conditions. For the considered prob-lem, we show the existence, uniqueness, and the explicit representation formulaefor the solution.1. IntroductionThe purpose of this paper is to study Bitsadze-Samarskii type nonlocal problem forthe time-fractional diffusion-wave equation with the Heisenberg sub-Laplacian ∆Hnin the space variables.In [1] Bitsadze and Samarskii established the solvability of the new class of non-local problems for the elliptic equations, which relate the values of the solution onparts of the boundary with its values inside the domain. Such problems are called theBitsadze-Samarskii problems. For the motivation of studying the Bitsadze-Samarskiitype nonlocal problems we refer to [2]–[8] and references therein.Certain types of physical problems can be modeled by heat and wave equationswith Bitsadze-Samarskii type initial conditions. The time multi-point heat and waveproblems can arise from studying the atomic reactors [9] –[10] and of some inverse heatconduction problems for determining the unknown physical parameters [11]. Well-posedness and numerical simulations of time multi-point heat and wave problemswere studied in [9]–[15].The version of such equations on the Heisenberg group serves as a basic model forthe analysis of the sub-elliptic diffusion and wave propagation models, providing newinsights and techniques for the whole problem.Thus, we consider the fractional integro-differential diffusion-wave equation(1.1) Dα+0,tu(t, x)− Iβ+0,t∆Hnu(t, x) = f(t, x), t > 0, x ∈ Hn,where f(t, x) is a sufficiently smooth function, Iβ+0,t is the Riemann-Liouville fractionalintegral of order β > 0 (see e.g. [16])Iβ+0,tu(t, x) =1Γ(β)t∫0(t− s)β−1u(s, x)ds,Date: July 20, 2019.2010 Mathematics Subject Classification. 35R11, 34B10, 35R03.Key words and phrases. time-fractional diffusion-wave equation, Riemann-Liouville derivative,Bitsadze-Samarskii type problem, Mittag-Leffler function, Heisenberg sub-Laplacian.12 MICHAEL RUZHANSKY, NIYAZ TOKMAGAMBETOV, AND BERIKBOL T. TOREBEKand ∂α+0,t is the Riemann-Liouville fractional derivative of order 0 < α ≤ 2 ([16])defined asDα+0,tu(t, x) =∂[α]+1∂t[α]+1I1+[α]−α+0,t u(t, x),where [α] is the integer part of α.When 1 < α < 2, β = 0, equation (1.1) is the time-fractional wave equation andwhen 0 < α < 1, β = 0, equation (1.1) is the time-fractional diffusion equation.When α = 2, β = 0, it represents the classical wave equation; while if α = 1, β = 0,it represents the classical diffusion equation.Many mathematical formulations of physical phenomena contain integro-differential equations, these equations arise in many fields such as fluid dynamics,biological models and chemical kinetics. If α = 1, the equation (1.1) describes theheat conduction with memory [17]–[18], and many authors studied the analoguesproblems (see [19]–[25]).1.1. Heisenberg group. Let Hn be the Heisenberg group, that is, the space R2n+1endowed with the group lawξ ◦ ξ′ =(x+ x′, y + y′, s+ s′ + 2n∑i=1(xiy′i − x′iyi)),where ξ = (z, s) = (x, y, s) = (x1, ..., xn, y1, ..., yn, s), z = (x, s), x ∈ Rn, y ∈ Rn,s ∈ R, n > 1; ξ′ = (x′, y′, s′) ∈ R2n+1. This group multiplication endows Hn with astructure of a nilpotent Lie group. A family of dilations is defined asδτ (x, y, s) = (τx, τy, τ2s), τ > 0.The homogeneous dimension with respect to these dilations is Q = 2n + 2. The leftinvariant vector fields on the Heisenberg group areXi =∂∂xi+ 2yi∂∂s, Yi =∂∂yi− 2xi∂∂s, i = 1, 2, ..., n.The horizontal gradient is∇Hn = (X1, ..., Xn, Y1, ..., Yn).Hence, the sub-Laplacian ∆Hn is denoted by∆Hn =n∑i=1(X2i + Y2i)= ∇Hn · ∇Hn .The (Kaplan) distance function on Hn is given bydist(ξ, ξ′) ={((x− x′)2 + (y − y′)2)2 + (s− s′ − 2(x · y′ − x′ · y)2)}1/4, ξ, ξ′ ∈ Hn.If ξ′ = 0, then the distance function isdist(ξ) ={(|x|2 + |y|2)2 + s2}1/4={|z|4 + s2}1/4, |z| =√|x|2 + |y|2.BITSADZE-SAMARSKII TYPE PROBLEM 31.2. Group Fourier transform. We begin with a reminder of the definition of thegroup Fourier transform on the Heisenberg group (see many sources, but e.g. [26]–[27] for its use in similar contexts). For f ∈ S(Hn) the group Fourier transform isdefined asf̂(λ) :=∫Hnf(x)πλ(x)∗dx,with the Schrödinger representationsπλ : L2(Rn)→ L2(Rn)for all λ ∈ R∗ := R\{0}. The inverse group Fourier transform formula can be writtenasf(x) =∫λ∈R∗Tr[f̂(λ)πλ(x)]|λ|ndλ,where Tr is the trace operator. The Plancherel formula becomes‖f‖2L2(Hn) =∫λ∈R∗‖f̂(λ)‖2HS[L2(Rn)]|λ|ndλ,where ‖ · ‖2HS[L2(Rn)] is the Hilbert-Schmidt norm on L2 (Rn) . For more details onPlanchrel formula and the Hilbert Schmidt norm, we refer to [28, Chapter 6, Propo-sition 6.2.7].2. Main resultsIn this paper we will work in the functional space C([0, T ];L2(Hn)) with the norm‖u‖C([0,T ];L2(Hn)) := supt∈[0,T ]‖u(t, ·)‖L2(Hn)for all u ∈ C([0, T ];L2(Hn)).Problem 2.1. Assume that f ∈ C([0, T ];L2(Hn)). In a domain Ω = {(t, x) :(0, T )×Hn} consider the equation (1.1) with Bitsadze-Samarskii type time-nonlocalconditions(2.1) I1+[α]−α+0,t u(0, x) +m∑i=1µiI2−α+0,tu(Ti, x) = 0, [α]Dα−1+0,tu(0, x) = 0, x ∈ Hn,where m ∈ N, µi ∈ R, 0 < T1 ≤ T2 ≤ ... ≤ Tm = T.We seek a solution t1+[α]−αu ∈ C([0, T ];L2(Hn)) of the problem (1.1), (2.1) suchthat Dα+0,tu ∈ C([0, T ];L2(Hn)) and ∆Hnu ∈ C([0, T ];L2(Hn)).The condition (2.2) below can be interpreted as a multi–point non-resonance con-dition. Note that a similar problem for the time-fractional multi-term diffusion-waveequation was investigated by the authors in [29].4 MICHAEL RUZHANSKY, NIYAZ TOKMAGAMBETOV, AND BERIKBOL T. TOREBEKTheorem 2.2. Let f ∈ C([0, T ];L2(Hn)), Dα+0,tf ∈ C([0, T ];L2(Hn)), and assumethat the conditions(2.2)∣∣∣∣∣1 +m∑i=1µiTα−[α]−1i Eα+β,α−1(−|λ|νlTα+βi)∣∣∣∣∣ ≥M > 0hold for all l ∈ Nn (where M is a constant), whereνl =n∑j=1(2lj + 1), l = (l1, ..., ln) ∈ Nn.Then there exists a unique solution of Problem 2.1, and it can be written as(2.3) u(t, x) =∫R∗Tr[K̂(t, λ)πλ(x)]dλ,whereK̂(t, λ)l,k = F̂ (t, λ)l,k −m∑i=1µiF̂ (Ti, λ)l,ktα−[α]−1Eα+β,α−1(−|λ|νltα+β)1 +m∑i=1µiTα−[α]−1i Eα+β,α−1(−|λ|νlTα+βi) ,for all λ ∈ R∗ and k ∈ N. HereF̂ (t, λ)l,k =t∫0sα−1Eα+β,α(−|λ|νlsα+β)f̂(t− s, λ)l,kds,andEα,β(z) =∞∑k=0zkΓ(αk + β)is the Mittag-Leffler function ([16]).2.1. Proof of Theorem 2.2.2.1.1. Proof of the existence result. Let us take the group Fourier transform fromSubsection 1.2 with respect to x ∈ Hn, that is,(2.4) Dα+0,tû(t, λ) + σ∆Hn (λ)Iβ+0,tû(t, λ) = f̂(t, λ),where σ∆Hn (λ) is the symbol of the Heisenberg sub-Laplacian. It has the followingform(2.5) σ∆Hn (λ) = |λ|Hτ ≡ |λ|(−∆τ + |τ |2),where Hτ is a harmonic oscillator operator for the variable τ ∈ Rn. For more infor-mation about the operator Hτ , we refer to [28]–[30].It is known that the operator Hτ is essentially self-adjoint in L2(Rn) with a discretespectrum νl, l = (l1, ..., ln) ∈ Nn.Corresponding to µl, the harmonic oscillator operator has the complete system oforthonormal eigenfunctions {el}l∈N on L2(Rn). They take the formel(τ) =n∏j=1Plj(τj)e−−|τ |22 ,BITSADZE-SAMARSKII TYPE PROBLEM 5where Pm(·) is the Hermite polynomial of order m. That is,Pm(t) = cme|t|22(t− ddt)me−|t|22 , t > 0, cm = 2−m/2(m!)−1/2π−1/4.For more details, see [30].Consequently, the equation (2.4) can be rewritten as(2.6) Dα+0,tû(t, λ)l,k + |λ|νlIβ+0,tû(t, λ)l,k = f̂(t, λ)l,k,for all λ ∈ R∗, and any l, k ∈ N. Hereû(t, λ)l,k = (û(t, λ)el, ek)L2(Rn)andf̂(t, λ)l,k = (f̂(t, λ)el, ek)L2(Rn).According to [31], the solution for equation (2.6) satisfying initial conditionsI2−α+0,t û(0, λ)l,k = C, [α]Dα−1+0,tû(0, λ)l,k = 0,can be represented in the form(2.7)û(t, λ)l,k =t∫0sα−1Eα+β,α(−|λ|νlsα+β)f̂(t− s, λ)l,kds+ CEα+β,α−1(−|λ|νlTα+βi).Then, it is not difficult to show, that the solutions of the equation (2.6) satisfyingthe following conditions(2.8) I2−α+0,t û(0, λ)l,k +m∑i=1µiI2−α+0,t û(Ti, λ)l,k = 0, [α]Dα−1+0,tû(0, λ)l,k = 0,can be represented in the form(2.9) û(t, λ)l,k = F̂ (t, λ)l,k −m∑i=1µiF̂ (Ti, λ)l,ktα−[α]−1Eα+β,α−1(−|λ|νltα+β)1 +m∑i=1µiTα−[α]−1i Eα+β,α−1(−|λ|νlTα+βi) ,whereF̂ (t, λ)l,k =t∫0sα−1Eα+β,α(−|λ|νlsα+β)f̂(t− s, λ)l,kds.Indeed, the formula (2.9) can be checked by the direct calculation from (2.7) underthe conditions (2.8).Now, applying the inverse group Fourier transform, we obtain the solution of Prob-lem 2.1 in the form (2.3).We note-that the above expression is well-defined in view of the non-resonanceconditions (2.2). Finally, based on (2.9), we rewrite our formal solution as (2.3).6 MICHAEL RUZHANSKY, NIYAZ TOKMAGAMBETOV, AND BERIKBOL T. TOREBEK2.1.2. Convergence of the formal solution. Here, we prove convergence of the obtainedintegrals corresponding to functions t1+[α]−αu(x, t), Dα+0,tu(x, t), and ∆Hnu(x, t). Toprove the convergence, we use the estimate for the Mittag-Leffler function(2.10) |Eα,β (z)| ≤C1 + |z|.Let us first prove the convergence of (2.3). From the estimate (2.10), we have thefollowing inequalities ∣∣∣F̂ (t, λ)l,k∣∣∣ ≤ C1∣∣∣f̂(t, λ)l,k∣∣∣1 + |λ|νl,∣∣Eα+β,α−1 (−|λ|νltα+β)∣∣ ≤ C21 + |λ|νltα+β,for some constants C1, C2 > 0. Hence, from these estimates it follows that|t1+[α]−αû(t, λ)l,k|2 ≤∣∣∣t1+[α]−αF̂ (t, λ)l,k∣∣∣2+1Mn∑i=1|µi|2|F̂ (Ti, λ)l,k|2∣∣Eα+β,α−1 (−|λ|νltα+β)∣∣2≤ C∣∣∣f̂(t, λ)l,k∣∣∣2(1 + |λ|νl)2+ C1Mn∑i=1|µi|2∣∣∣f̂(Ti, λ)l,k∣∣∣2(1 + |λ|νl)21(1 + |λ|νltα+β)2.Thus, since for any Hilbert-Schmidt operator A one has‖A‖2HS =∑l,k|(Aφl, φk)|2for any orthonormal basis {φ1, φ2, . . .}, then we can consider the infinite sum over l, kof the inequalities provided by (2.9). This gives(2.11) ‖t1+[α]−αû(t, λ)‖2HS ≤ C‖(1 + σ∆Hn (λ))−1)f̂(t, λ)‖2HS,since supt∈[0,T ]11+|λ|νltα+β= 1. Thus, integrating both sides of (2.11) against thePlancherel measure on R∗ and using the Plancherel identity (see e. g. [28]), weobtain‖t1+[α]−αu‖C([0,T ];L2(Hn)) ≤ C‖(I + ∆Hn)−1f‖C([0,T ];L2(Hn))and‖∆Hnu‖C([0,T ];L2(Hn)) ≤ C‖f‖C([0,T ];L2(Hn)).Since f ∈ C([0, T ];L2(Hn)), we get‖t1+[α]−αu(t, x)‖C([0,T ];L2(Hn)) <∞and‖∆Hnu(t, x)‖C([0,T ];L2(Hn)) <∞.The convergence of the integral corresponding to Dα+0,tu(x, t) can be shown in asimilar way.BITSADZE-SAMARSKII TYPE PROBLEM 7To show the uniqueness of the solution, let us assume that there are two differentfunctions u and v satisfying Problem 2.1. Now we introduce a new function w as thedifference of the solutions u and v, that is, w := u− v.Indeed, w satisfies the homogeneous equation(2.12) Dα+0,tw(t, x)− Iβ+0,t∆Hw(t, x) = 0, t > 0, x ∈ Hn,with boundary conditions(2.13) I2−α+0,tw(0, x) +m∑i=1µiI2−α+0,tw(Ti, x) = 0, [α]Dα−1+0,tw(0, x) = 0, x ∈ Hn,where µi ∈ R, 0 < T1 ≤ T2 ≤ ... ≤ Tm = T.Again, repeating the same arguments, for the solution w of the problem (2.12)–(2.13) we obtain the estimate‖t1+[α]−αw‖C([0,T ];L2(Hn)) ≤ 0.Thus, 0 = w = u− v. The proof is complete.3. Disclosure statementNo potential conflict of interest was reported by the authors.4. FundingThe authors were supported in parts by the FWO Odysseus Project. The firstauthor was supported in parts by the EPSRC grant EP/R003025/1 and by the Lev-erhulme Grant RPG-2017-151. The second author was supported in parts by theMinistry of Education and Science of the Republic of Kazakhstan (MESRK) GrantAP05130994. The third author was supported by MESRK Grant AP05131756. Nonew data was collected or generated during the course of research.References[1] Bitsadze AV, Samarskii AA. On some simplest generalizations of linear elliptic problems. Dokl.Akad. Nauk SSSR. 1969;185:739–740.[2] Avalishvili G, Avalishvili M, Miara B. Nonclassical problems with nonlocal initial conditionsfor second-order evolution equations. Asymptotic Anal. 2012;76:171–192.[3] Il’in VA, Moiseev EI. 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Integrodifferential equation which interpolates the heat equation and the wave equa-tion II. Osaka J. Math. 1990;27:797–804.[24] Borikhanov M, Kirane M, Torebek BT. Maximum principle and its application for the non-linear time-fractional diffusion equations with Cauchy-Dirichlet conditions. Appl. Math. Lett.2018;81:14–20.[25] Kirane M, Torebek BT. Extremum principle for the Hadamard derivatives and its application tononlinear fractional partial differential equations. Fract. Calc. Appl. Anal. 2019;22(2):358–378.[26] Ruzhansky M, Tokmagambetov N. Wave equation for operators with discrete spectrum andirregular propagation speed. Arch. Ration. Mech. Anal. 2017;226:1161–1207.[27] Ruzhansky M, Tokmagambetov N. Nonlinear damped wave equations for the sub-Laplacian onthe Heisenberg group and for Rockland operators on graded Lie groups. J. Differ. Equations.2018;265(10):5212–5236.[28] Fischer V, Ruzhansky M. Quantization on nilpotent Lie groups. Volume 314 of Progress inMathematics. Birkhäuser/Springer, [Open access book], 2016.[29] Ruzhansky M, Tokmagambetov N, Torebek BT. On a non-local problem for a multi-termfractional diffusion-wave equation. arXiv:1812.01336.[30] Nicola F, Rodino L. Global pseudo-differential calculus on Euclidean spaces. Volume 4 ofPseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel, 2010.[31] Al Saqabi BN, Tuan VK. Solution of a fractional differintegral equation. Integral Transform.Spec. Funct. 1996;4:321–326.Michael Ruzhansky:Department of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, BelgiumandSchool of Mathematical SciencesQueen Mary University of LondonUnited KingdomBITSADZE-SAMARSKII TYPE PROBLEM 9E-mail address michael.ruzhansky@ugent.beNiyaz Tokmagambetov:Al–Farabi Kazakh National University71 Al–Farabi ave., Almaty, 050040andDepartment of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, BelgiumandInstitute of Mathematics and Mathematical Modeling125 Pushkin str., Almaty, 050010Kazakhstan,E-mail address tokmagambetov@math.kzBerikbol T. Torebek:Al–Farabi Kazakh National University71 Al–Farabi ave., Almaty, 050040andDepartment of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, BelgiumandInstitute of Mathematics and Mathematical Modeling125 Pushkin str., Almaty, 050010Kazakhstan,E-mail address torebek@math.kz

Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group

Ruzhansky, Michael

Tokmagambetov, Niyaz

Torebek, Berikbol

Ghent University Academic Bibliography

Full Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=gitr20Integral Transforms and Special FunctionsISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: https://www.tandfonline.com/loi/gitr20Bitsadze–Samarskii type problem for the integro-differential diffusion–wave equation on theHeisenberg groupM. Ruzhansky, N. Tokmagambetov & B. T. TorebekTo cite this article: M. Ruzhansky, N. Tokmagambetov & B. T. Torebek (2020) Bitsadze–Samarskiitype problem for the integro-differential diffusion–wave equation on the Heisenberg group, IntegralTransforms and Special Functions, 31:1, 1-9, DOI: 10.1080/10652469.2019.1652823To link to this article:  https://doi.org/10.1080/10652469.2019.1652823© 2019 The Author(s). Published by InformaUK Limited, trading as Taylor & FrancisGroupPublished online: 18 Aug 2019.Submit your article to this journal Article views: 261View related articles View Crossmark dataINTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS2020, VOL. 31, NO. 1, 1–9https://doi.org/10.1080/10652469.2019.1652823Research ArticleBitsadze–Samarskii type problem for the integro-differentialdiffusion–wave equation on the Heisenberg groupM. Ruzhanskya,b, N. Tokmagambetova,c,d and B. T. Torebeka,c,daDepartment of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium;bSchool of Mathematical Sciences, Queen Mary University of London, London, UK; cAl-Farabi Kazakh NationalUniversity, Almaty, Kazakhstan; dInstitute of Mathematics and Mathematical Modeling, Almaty, KazakhstanABSTRACTThis paper deals with the fractional generalization of the integro-differential diffusion–waveequation for theHeisenberg sub-Laplacian,with homogeneous Bitsadze–Samarskii type time-nonlocal condi-tions. For the considered problem, we show the existence, unique-ness and the explicit representation formulae for the solution.ARTICLE HISTORYReceived 13 February 2019Accepted 3 August 2019KEYWORDSTime-fractionaldiﬀusion–wave equation;Riemann–Liouvillederivative;Bitsadze–Samarskii typeproblem; Mittag–Leﬄerfunction; Heisenbergsub-LaplacianAMS 2010MATHEMATICSSUBJECTCLASSIFICATIONS35R11; 34B10; 35R031. IntroductionThe purpose of this paper is to study Bitsadze–Samarskii type nonlocal problem for thetime-fractional diffusion–wave equation with the Heisenberg sub-Laplacian Hn in thespace variables.In [1], Bitsadze and Samarskii established the solvability of the new class of non-local problems for the elliptic equations, which relate the values of the solution on partsof the boundary with its values inside the domain. Such problems are called the Bit-sadze–Samarskii problems. For the motivation of studying the Bitsadze–Samarskii typenonlocal problems, we refer to [2–8] and references therein.Certain types of physical problems can be modelled by heat and wave equations withBitsadze–Samarskii type initial conditions. The time multi-point heat and wave problemscan arise from studying the atomic reactors [9,10] and of some inverse heat conduc-tion problems for determining the unknown physical parameters [11]. Well-posednessand numerical simulations of time multi-point heat and wave problems were studied in[9,10,12–15].CONTACT N. Tokmagambetov tokmagambetov@math.kz Al-Farabi Kazakh National University, 71 Al-Farabiave., Almaty, Kazakhstan Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty, Kazakhstan© 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis GroupThis is anOpenAccess article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.2 M. RUZHANSKY ET AL.The version of such equations on the Heisenberg group serves as a basic model for theanalysis of the sub-elliptic diffusion and wave propagation models, providing new insightsand techniques for the whole problem.Thus we consider the fractional integro-differential diffusion–wave equationDα+0,tu(t, x) − Iβ+0,tHnu(t, x) = f (t, x), t > 0, x ∈ Hn, (1.1)where f (t, x) is a sufficiently smooth function, Iβ+0,t is the Riemann–Liouville fractionalintegral of order β > 0 [16]Iβ+0,tu(t, x) =1(β)∫ t0(t − s)β−1u(s, x) ds,and ∂α+0,t is the Riemann–Liouville fractional derivative of order 0 < α ≤ 2 ([16]) definedasDα+0,tu(t, x) =∂ [α]+1∂t[α]+1I1+[α]−α+0,t u(t, x),where [α] is the integer part of α.When 1 < α < 2,β = 0, Equation (1.1) is the time-fractional wave equation and when0 < α < 1,β = 0, Equation (1.1) is the time-fractional diffusion equation. When α =2,β = 0, it represents the classical wave equation, while if α = 1,β = 0, it represents theclassical diffusion equation.Many mathematical formulations of physical phenomena contain integro-differentialequations, these equations arise in many fields such as fluid dynamics, biological modelsand chemical kinetics. Ifα = 1, Equation (1.1) describes the heat conductionwithmemory[17,18], and many authors studied the analogue problems [19–25].1.1. Heisenberg groupLetHn be the Heisenberg group, that is, the space R2n+1 endowed with the group lawξ ◦ ξ ′ =(x + x′, y + y′, s + s′ + 2n∑i=1(xiy′i − x′iyi)),where ξ = (z, s) = (x, y, s) = (x1, . . . , xn, y1, . . . , yn, s), z = (x, s), x ∈ Rn, y ∈ Rn, s ∈R, n > 1; ξ ′ = (x′, y′, s′) ∈ R2n+1. This group multiplication endowsHn with a structureof a nilpotent Lie group. A family of dilations is defined asδτ (x, y, s) = (τx, τy, τ 2s), τ > 0.The homogeneous dimension with respect to these dilations is Q = 2n + 2. The leftinvariant vector fields on the Heisenberg group areXi = ∂∂xi+ 2yi ∂∂s, Yi = ∂∂yi− 2xi ∂∂s, i = 1, 2, . . . , n.The horizontal gradient is∇Hn = (X1, . . . ,Xn,Y1, . . . ,Yn).INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 3Hence, the sub-Laplacian Hn is denoted byHn =n∑i=1(X2i + Y2i) = ∇Hn · ∇Hn .The (Kaplan) distance function onHn is given bydist(ξ , ξ ′) = {((x − x′)2 + (y − y′)2)2 + (s − s′ − 2(x · y′ − x′ · y)2)}1/4 , ξ , ξ ′ ∈ Hn.If ξ ′ = 0, then the distance function isdist(ξ) = {(|x|2 + |y|2)2 + s2}1/4 = {|z|4 + s2}1/4 , |z| = √|x|2 + |y|2.1.2. Group Fourier transformWe begin with a reminder of the definition of the group Fourier transform on the Heisen-berg group (seemany sources, but e.g. [26,27] for its use in similar contexts). For f ∈ S(Hn)the group Fourier transform is defined asf̂ (λ) :=∫Hnf (x)πλ(x)∗ dx,with the Schrödinger representationsπλ : L2(Rn) → L2(Rn)for all λ ∈ R∗ := R \ {0}. The inverse group Fourier transform formula can be written asf (x) =∫λ∈R∗Tr[̂f (λ)πλ(x)] |λ|n dλ,where Tr is the trace operator. The Plancherel formula becomes‖f ‖2L2(Hn) =∫λ∈R∗‖̂f (λ)‖2HS[L2(Rn)]|λ|n dλ,where ‖ · ‖2HS[L2(Rn)] is the Hilbert–Schmidt norm on L2(Rn). For more details onPlancherel formula and the Hilbert Schmidt norm, we refer to [28, Chapter 6, Proposition6.2.7].2. Main resultsIn this paper, we will work in the functional space C([0,T]; L2(Hn)) with the norm‖u‖C([0,T];L2(Hn)) := supt∈[0,T]‖u(t, ·)‖L2(Hn)for all u ∈ C([0,T]; L2(Hn)).4 M. RUZHANSKY ET AL.Problem 2.1: Assume that f ∈ C([0,T]; L2(Hn)). In a domain  = {(t, x) : (0,T) ×Hn}consider Equation (1.1) with Bitsadze–Samarskii type time-nonlocal conditionsI1+[α]−α+0,t u(0, x) +m∑i=1μiI2−α+0,t u(Ti, x) = 0, [α]Dα−1+0,tu(0, x) = 0, x ∈ Hn, (2.1)wherem ∈ N,μi ∈ R, 0 < T1 ≤ T2 ≤ · · · ≤ Tm = T.We seek a solution t1+[α]−αu ∈ C([0,T]; L2(Hn)) of the problems (1.1) and (2.1) suchthat Dα+0,tu ∈ C([0,T]; L2(Hn)) and Hnu ∈ C([0,T]; L2(Hn)).The condition (2.2) can be interpreted as a multi-point non-resonance condition. Notethat a similar problem for the time-fractional multi-term diffusion–wave equation wasinvestigated by the authors in [29].Theorem 2.2: Let f ∈ C([0,T]; L2(Hn)), Dα+0,tf ∈ C([0,T]; L2(Hn)), and assume that theconditions ∣∣∣∣∣1 +m∑i=1μiTα−[α]−1i Eα+β ,α−1(−|λ|νlTα+βi)∣∣∣∣∣ ≥ M > 0 (2.2)hold for all l ∈ Nn (where M is a constant), whereνl =n∑j=1(2lj + 1), l = (l1, . . . , ln) ∈ Nn.Then there exists a unique solution of Problem 2.1, and it can be written asu(t, x) =∫R∗Tr[K̂(t, λ)πλ(x)] dλ, (2.3)whereK̂(t, λ)l,k = F̂(t, λ)l,k −∑mi=1 μîF(Ti, λ)l,ktα−[α]−1Eα+β ,α−1(−|λ|νltα+β)1 +∑mi=1 μiTα−[α]−1i Eα+β ,α−1 (−|λ|νlTα+βi ) ,for all λ ∈ R∗ and k ∈ N. HereF̂(t, λ)l,k =∫ t0sα−1Eα+β ,α(−|λ|νlsα+β) f̂ (t − s, λ)l,k ds,andEα,β(z) =∞∑k=0zk(αk + β)is the Mittag–Leffler function [16].INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 52.1. Proof of Theorem 2.22.1.1. Proof of the existence resultLet us take the group Fourier transform from Section 1.2 with respect to x ∈ Hn, that is,Dα+0,tû(t, λ) + σHn (λ)Iβ+0,tû(t, λ) = f̂ (t, λ), (2.4)where σHn (λ) is the symbol of the Heisenberg sub-Laplacian. It has the following form:σHn (λ) = |λ|Hτ ≡ |λ|(−τ + |τ |2) , (2.5)whereHτ is a harmonic oscillator operator for the variable τ ∈ Rn. For more informationabout the operatorHτ , we refer to [28,30].It is known that the operator Hτ is essentially self-adjoint in L2(Rn) with a discretespectrum νl, l = (l1, . . . , ln) ∈ Nn.Corresponding to μl, the harmonic oscillator operator has the complete system oforthonormal eigenfunctions {el}l∈N on L2(Rn). They take the formel(τ ) =n∏j=1Plj(τj)e−(−|τ |2/2),where Pm(·) is the Hermite polynomial of orderm. That is,Pm(t) = cme|t|2/2(t − ddt)me−(|t|2/2), t > 0, cm = 2−m/2(m!)−1/2π−1/4.For more details, see [30].Consequently, Equation (2.4) can be rewritten asDα+0,tû(t, λ)l,k + |λ|νlIβ+0,tû(t, λ)l,k = f̂ (t, λ)l,k, (2.6)for all λ ∈ R∗, and any l, k ∈ N. Hereû(t, λ)l,k = (̂u(t, λ)el, ek)L2(Rn)andf̂ (t, λ)l,k = (̂f (t, λ)el, ek)L2(Rn).According to [31], the solution for Equation (2.6) satisfying initial conditionsI2−α+0,t û(0, λ)l,k = C, [α]Dα−1+0,t û(0, λ)l,k = 0,can be represented in the formû(t, λ)l,k =∫ t0sα−1Eα+β ,α(−|λ|νlsα+β) f̂ (t − s, λ)l,k ds + CEα+β ,α−1 (−|λ|νlTα+βi ) .(2.7)Then, it is not difficult to show that the solutions of Equation (2.6) satisfying the followingconditions:I2−α+0,t û(0, λ)l,k +m∑i=1μiI2−α+0,t û(Ti, λ)l,k = 0, [α]Dα−1+0,t û(0, λ)l,k = 0, (2.8)6 M. RUZHANSKY ET AL.can be represented in the formû(t, λ)l,k = F̂(t, λ)l,k −∑mi=1 μîF(Ti, λ)l,ktα−[α]−1Eα+β ,α−1(−|λ|νltα+β)1 +∑mi=1 μiTα−[α]−1i Eα+β ,α−1 (−|λ|νlTα+βi ) , (2.9)whereF̂(t, λ)l,k =∫ t0sα−1Eα+β ,α(−|λ|νlsα+β) f̂ (t − s, λ)l,k ds.Indeed, the formula (2.9) can be checked by the direct calculation from (2.7) under theconditions (2.8).Now, applying the inverse group Fourier transform, we obtain the solution of Prob-lem 2.1 in the form (2.3).We note that the above expression is well defined in view of the non-resonanceconditions (2.2). Finally, based on (2.9), we rewrite our formal solution as (2.3).2.1.2. Convergence of the formal solutionHere, we prove convergence of the obtained integrals corresponding to functionst1+[α]−αu(x, t),Dα+0,tu(x, t) andHnu(x, t). To prove the convergence, we use the estimatefor the Mittag–Leffler function∣∣Eα,β (z)∣∣ ≤ C1 + |z| . (2.10)Let us first prove the convergence of (2.3). From the estimate (2.10), we have the followinginequalities:∣∣̂F(t, λ)l,k∣∣ ≤ C1 ∣∣̂f (t, λ)l,k∣∣1 + |λ|νl ,∣∣Eα+β ,α−1 (−|λ|νltα+β)∣∣ ≤ C21 + |λ|νltα+βfor some constants C1,C2 > 0. Hence, from these estimates it follows that|t1+[α]−αû(t, λ)l,k|2 ≤∣∣∣t1+[α]−αF̂(t, λ)l,k∣∣∣2+ 1Mn∑i=1|μi|2 |̂F(Ti, λ)l,k|2∣∣Eα+β ,α−1 (−|λ|νltα+β)∣∣2≤ C∣∣̂f (t, λ)l,k∣∣2(1 + |λ|νl)2+ C 1Mn∑i=1|μi|2∣∣̂f (Ti, λ)l,k∣∣2(1 + |λ|νl)21(1 + |λ|νltα+β)2.INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 7Thus, since for any Hilbert–Schmidt operator A one has‖A‖2HS =∑l,k|(Aφl,φk)|2for any orthonormal basis {φ1,φ2, . . .}, then we can consider the infinite sum over l, k ofthe inequalities provided by (2.9). This gives‖t1+[α]−αû(t, λ)‖2HS ≤ C‖(1 + σHn (λ))−1)̂f (t, λ)‖2HS, (2.11)since supt∈[0,T] 1/(1 + |λ|νltα+β) = 1. Thus integrating both sides of (2.11) against thePlancherel measure onR∗ and using the Plancherel identity [28], we obtain‖t1+[α]−αu‖C([0,T];L2(Hn)) ≤ C‖(I + Hn)−1f ‖C([0,T];L2(Hn))and‖Hnu‖C([0,T];L2(Hn)) ≤ C‖f ‖C([0,T];L2(Hn)).Since f ∈ C([0,T]; L2(Hn)), we get‖t1+[α]−αu(t, x)‖C([0,T];L2(Hn)) < ∞and‖Hnu(t, x)‖C([0,T];L2(Hn)) < ∞.The convergence of the integral corresponding to Dα+0,tu(x, t) can be shown in a similarway.To show the uniqueness of the solution, let us assume that there are two different func-tions u and v satisfying Problem 2.1. Now we introduce a new function w as the differenceof the solutions u and v, that is, w:=u−v.Indeed, w satisfies the homogeneous equationDα+0,tw(t, x) − Iβ+0,tHw(t, x) = 0, t > 0, x ∈ Hn, (2.12)with boundary conditionsI2−α+0,t w(0, x) +m∑i=1μiI2−α+0,t w(Ti, x) = 0, [α]Dα−1+0,tw(0, x) = 0, x ∈ Hn, (2.13)where μi ∈ R, 0 < T1 ≤ T2 ≤ · · · ≤ Tm = T.Again, repeating the same arguments, for the solution w of the problems (2.12)and (2.13) we obtain the estimate‖t1+[α]−αw‖C([0,T];L2(Hn)) ≤ 0.Thus 0 = w = u − v. The proof is complete.Disclosure statementNo potential conflict of interest was reported by the authors.8 M. RUZHANSKY ET AL.FundingThe authors were supported in parts by the FWO Odysseus Project. The first author was supportedin parts by the EPSRC grant EP/R003025/1 and by the Leverhulme Trust Grant RPG-2017-151. Thesecond author was supported in parts by the Ministry of Education and Science of the Republicof Kazakhstan (MESRK) Grant AP05130994. 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Bitsadze–Samarskii type problem for the integro-differential diffusion–wave equation on the Heisenberg group

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Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group

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