February  2020, 25(2): 569-597. doi: 10.3934/dcdsb.2019255

Well-posedness results for fractional semi-linear wave equations

1. 

Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

2. 

Department of Mathematics, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin-10, Turkey

3. 

Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, 32004-Ourense, Spain

* Corresponding author: Arran Fernandez

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received  April 2019 Revised  May 2019 Published  November 2019

This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some $ L^{p} $-estimates are derived, by considering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly elliptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro-differential equations.

Citation: Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
References:
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E. AlvarezC. G. GalV. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Analysis, 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.  Google Scholar

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I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.  doi: 10.1016/j.aim.2009.11.010.  Google Scholar

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A. BernardisF. J. Martín-ReyesP. R. Stinga and J. L. Torrea, Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.  doi: 10.1016/j.jde.2015.12.042.  Google Scholar

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C. Bucur and F. Ferrari, An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887.  doi: 10.1515/fca-2016-0047.  Google Scholar

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X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 2007, 1245–1260. doi: 10.1080/03605300600987306.  Google Scholar

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A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954.  Google Scholar

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A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.  Google Scholar

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A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.  Google Scholar

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S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

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P. FelmerA. Quaas and J. G. Tan, Positive solutions of nonlinear schrödinger equation with the fractional laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[21]

A. Fernandez, An elliptic regularity theorem for fractional partial differential operators, Comp. Appl. Math., 37 (2018), 5542-5553.  doi: 10.1007/s40314-018-0618-2.  Google Scholar

[22]

Y. Fujita, Energy inequalities for integro-partial differential equations with Riemann-Liouville integrals, SIAM J. Math. Anal., 23 (1992), 1182-118.  doi: 10.1137/0523066.  Google Scholar

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Y. V. Gorbatenko, Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space, Methods Funct. Anal. Topology, 17 (2011), 1-9.   Google Scholar

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R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

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[26]

H. Hirata and C. X. Miao, Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative., Adv. Differential Equations, 7 (2002), 217-236.   Google Scholar

[27]

Y. H. Huang, Explicit Barenblatt profiles for fractional porous medium equations, Bulletin of the London Mathematical Society, 46 (2014), 857-869.  doi: 10.1112/blms/bdu045.  Google Scholar

[28]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behaviour for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[29]

V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional wave equations, Electron. J. Differential Equations, 2017 (2017), 42 pp.  Google Scholar

[30]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.  Google Scholar

[31]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[32]

K.-H. Kim and S. Lim, Asymptotic behaviours of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[33]

A. N. Kochubeǐ, Diffusion of fractional order, Differ. Equ., 26 (1990), 485-492.   Google Scholar

[34]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[35]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[36]

L. Maligranda, On interpolation of nonlinear operators, Comment. Math. Prace Mat., 28 (1989), 253-275.   Google Scholar

[37]

L. Maligranda, Marcinkiewicz interpolation theorem and marcinkiewicz spaces, Wiad. Mat., 48 (2012), 157-171.  doi: 10.14708/wm.v48i2.328.  Google Scholar

[38] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[39]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.1090/S0002-9947-1972-0293384-6.  Google Scholar

[40]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[41]

A. Niang, Boundary regularity for a degenerate elliptic equation with mixed boundary conditions, Commun. Pure Appl. Anal., 18 (2019), 107-128.  doi: 10.3934/cpaa.2019007.  Google Scholar

[42]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[43]

E. Otárola and A. J. Salgado, Regularity of solutions to space-time fractional wave equations: A PDE approach, Fract. Calc. Appl. Anal., 21 (2018), 1262-1293.  doi: 10.1515/fca-2018-0067.  Google Scholar

[44]

A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II, Math. Nach., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144.  Google Scholar

[45]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[46]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[47]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[48]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Part. Diff. Eqs., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[49]

L. Tartar, An introduction to Sobolev spaces and interpolation spaces., Lecture Notes of the Unione Matematica Italiana, volume 3, Springer, Berlin, 2007.  Google Scholar

[50]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103908.  Google Scholar

[51]

E. Valdinoci, From the lung jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.   Google Scholar

[52]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.  doi: 10.1016/j.matpur.2013.07.001.  Google Scholar

[53] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.   Google Scholar

show all references

References:
[1]

N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, Contemporary Research in Elliptic PDEs and Related Topics, 2019, 1–105. doi: 10.1007/978-3-030-18921-1_1.  Google Scholar

[2]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.  Google Scholar

[3]

M. AllenL. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[4]

E. AlvarezC. G. GalV. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Analysis, 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.  Google Scholar

[5]

I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.  doi: 10.1016/j.aim.2009.11.010.  Google Scholar

[6]

D. Baleanu and A. Fernandez, A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-12.  doi: 10.14232/ejqtde.2017.1.15.  Google Scholar

[7]

A. BernardisF. J. Martín-ReyesP. R. Stinga and J. L. Torrea, Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.  doi: 10.1016/j.jde.2015.12.042.  Google Scholar

[8]

M. Š. Birman and M. Z. Solomjak, Spektralnaya Teoriya Samosopryazhennykh Operatorov v Gilbertovom Prostranstve, Leningrad. Univ., Leningrad, 1980. Google Scholar

[9]

C. Bucur and F. Ferrari, An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887.  doi: 10.1515/fca-2016-0047.  Google Scholar

[10]

X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[11]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 2007, 1245–1260. doi: 10.1080/03605300600987306.  Google Scholar

[12]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954.  Google Scholar

[13]

M. D'Abbicco, M. R. Ebert and T. Picon, Global existence of small data solutions to the semilinear fractional wave equation, New Trends in Analysis and Interdisciplinary Applications, Trends Math. Res. Perspect., Birkhäuser/Springer, Cham, (2017), 465–471. doi: 10.1007/978-3-319-48812-7_59.  Google Scholar

[14]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.  Google Scholar

[15]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.  Google Scholar

[16]

S. Dipierro and E. Valdinoci, A simple mathematical model inspired by the Purkinje cells: From delayed travelling waves to fractional diffusion, Bull. Math. Biol., 80 (2018), 1849-1870.  doi: 10.1007/s11538-018-0437-z.  Google Scholar

[17]

J.-D. Djida and A. Fernandez, Interior regularity estimates for a degenerate elliptic equation with mixed boundary conditions, Axioms, 7 (2018), 1-16.  doi: 10.3390/axioms7030065.  Google Scholar

[18]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[19]

L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[20]

P. FelmerA. Quaas and J. G. Tan, Positive solutions of nonlinear schrödinger equation with the fractional laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[21]

A. Fernandez, An elliptic regularity theorem for fractional partial differential operators, Comp. Appl. Math., 37 (2018), 5542-5553.  doi: 10.1007/s40314-018-0618-2.  Google Scholar

[22]

Y. Fujita, Energy inequalities for integro-partial differential equations with Riemann-Liouville integrals, SIAM J. Math. Anal., 23 (1992), 1182-118.  doi: 10.1137/0523066.  Google Scholar

[23]

Y. V. Gorbatenko, Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space, Methods Funct. Anal. Topology, 17 (2011), 1-9.   Google Scholar

[24]

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

[25]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[26]

H. Hirata and C. X. Miao, Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative., Adv. Differential Equations, 7 (2002), 217-236.   Google Scholar

[27]

Y. H. Huang, Explicit Barenblatt profiles for fractional porous medium equations, Bulletin of the London Mathematical Society, 46 (2014), 857-869.  doi: 10.1112/blms/bdu045.  Google Scholar

[28]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behaviour for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[29]

V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional wave equations, Electron. J. Differential Equations, 2017 (2017), 42 pp.  Google Scholar

[30]

Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.  doi: 10.1515/fca-2017-0006.  Google Scholar

[31]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[32]

K.-H. Kim and S. Lim, Asymptotic behaviours of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[33]

A. N. Kochubeǐ, Diffusion of fractional order, Differ. Equ., 26 (1990), 485-492.   Google Scholar

[34]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[35]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973.  Google Scholar

[36]

L. Maligranda, On interpolation of nonlinear operators, Comment. Math. Prace Mat., 28 (1989), 253-275.   Google Scholar

[37]

L. Maligranda, Marcinkiewicz interpolation theorem and marcinkiewicz spaces, Wiad. Mat., 48 (2012), 157-171.  doi: 10.14708/wm.v48i2.328.  Google Scholar

[38] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[39]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.1090/S0002-9947-1972-0293384-6.  Google Scholar

[40]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[41]

A. Niang, Boundary regularity for a degenerate elliptic equation with mixed boundary conditions, Commun. Pure Appl. Anal., 18 (2019), 107-128.  doi: 10.3934/cpaa.2019007.  Google Scholar

[42]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[43]

E. Otárola and A. J. Salgado, Regularity of solutions to space-time fractional wave equations: A PDE approach, Fract. Calc. Appl. Anal., 21 (2018), 1262-1293.  doi: 10.1515/fca-2018-0067.  Google Scholar

[44]

A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II, Math. Nach., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144.  Google Scholar

[45]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[46]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[47]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[48]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Part. Diff. Eqs., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[49]

L. Tartar, An introduction to Sobolev spaces and interpolation spaces., Lecture Notes of the Unione Matematica Italiana, volume 3, Springer, Berlin, 2007.  Google Scholar

[50]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103908.  Google Scholar

[51]

E. Valdinoci, From the lung jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.   Google Scholar

[52]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.  doi: 10.1016/j.matpur.2013.07.001.  Google Scholar

[53] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.   Google Scholar
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