×

Fractional Klein-Gordon equation with singular mass. (English) Zbl 1498.35364

Summary: We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of \(\delta^2\) is observed.

MSC:

35L81 Singular hyperbolic equations
35A35 Theoretical approximation in context of PDEs
35D30 Weak solutions to PDEs
35L05 Wave equation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bagley, R.; Torvik, P., On the appearance of the fractional derivative in the behavior of real materials, J Appl Mech, 51, 2, 294-298 (1984) · Zbl 1203.74022
[2] URL: https://tel.archives-ouvertes.fr/tel-01127419
[3] 384 pages · Zbl 0789.26002
[4] 322 pages · Zbl 0292.26011
[5] Podlubny, I., Fractional differential equations, 340pages (1998), Elsevier Science
[6] Abuteen, E.; Freihat, A.; Al-Smadi, M.; Khalil, H.; Khan, R., Approximate series solution of nonlinear, fractional Klein-Gordon equations using fractional reduced differential transform method, J Math Stat, 12, 1, 23-33 (2016)
[7] Çulha; Daşcı̇oğlu, A., Analytic solutions of the space-time conformable fractional Klein-Gordon equation in general form, Waves Random Complex Media, 29, 4, 775-790 (2019) · Zbl 1505.35348
[8] Ege, S.; Misirli, E., Solutions of the space-time fractional foam drainage equation and the fractional Klein-Gordon equation by use of modified Kudryashov method, Int J Res Advent Technol, 2, 3, 384-388 (2014)
[9] Gepreel, K.; Mohamed, M., Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chin Phys B, 22, 1, 010201 (2013)
[10] Garra, R.; Orsingher, E.; Polito, F., Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications, ICFDA’14 International conference on fractional differentiation and its applications, 1-6 (2014)
[11] Khader, M.; Adel, M., Analytical and numerical validation for solving the fractional Klein-Gordon equation using the fractional complex transform and variational iteration methods, Nonlinear Eng., 5, 3, 141-145 (2016)
[12] Kurulay, M., Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method, Adv Differ Equ, 187 (2012) · Zbl 1377.35270
[13] Ran, M.; Zhang, C., Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions, Comput Math Appl, 71, 5, 1151-1162 (2016) · Zbl 1443.65138
[14] Sweilam, N.; Khader, M.; Mahdy, A., On the numerical solution for the linear fractional Klein-Gordon equation using legendre pseudospectral method, Int J Pure Appl Math, 84, 4, 307-319 (2013)
[15] Singh, H.; Kumar, D.; Singh, J.; Singh, C., A reliable numerical algorithm for the fractional Klein-Gordon equation, Eng Trans, 67, 1, 21-34 (2019)
[16] Topsakal, M.; Taşcan, F., Exact travelling wave solutions for space-time fractional Klein-Gordon equation and (2+1)-dimensional time-fractional Zoomeron equation via Auxiliary equation method, Appl Math Nonlinear Sci, 5, 1, 437-446 (2020) · Zbl 1506.35274
[17] Ziane, D.; Cherif, M., A new analytical solution of Klein-Gordon equation with local fractional derivative, Asian-Eur J Math, 2150029 (2020)
[18] Zhang, J.; J. Wang, Y. Z., Numerical analysis for Klein-Gordon equation with time-space fractional derivatives, Asian-Eur J Math, 43, 6, 3689-3700 (2020) · Zbl 1447.65040
[19] Arda, A.; Sever, R.; Tezcan, C., Analytical solutions to the Klein-Gordon equation with position-dependent mass for \(q\)-parameter Pöschl-Teller potential, Chin Phys Lett, 27, 010306 (2010)
[20] de Souza Dutra, A.; Jia, C., Classes of exact Klein-Gordon equations with spatially dependent masses: regularizing the one-dimensional inversely linear potential, Phys Lett A, 352, 6, 484-487 (2006) · Zbl 1187.81116
[21] Wang, Z.; Long, Z.; Long, C.; Wang, L., Analytical solutions of position-dependent mass Klein-Gordon equation for unequal scalar and vector Yukawa potentials, Indian J Phys, 89, 1059-1064 (2015)
[22] Wang, B.; Long, Z.; Long, C.; Wu, S., Klein-Gordon oscillator with position-dependent mass in the rotating cosmic string spacetime, Mod Phys Lett A, 33, 4, 1850025 (2018) · Zbl 1381.81043
[23] Ghosh, U.; Banerjee, J.; Sarkar, S.; Das, S., Fractional Klein-Gordon equation composed of Jumarie fractional derivative and its interpretation by a smoothness parameter, Pramana J Phys, 90, 74, 1-10 (2018)
[24] Golmankhaneh, A.; Golmankhaneh, A.; Baleanu, D., On nonlinear fractional Klein-Gordon equation, Pramana J Phys, 91, 3, 446-451 (2011) · Zbl 1203.94031
[25] Schwartz, L., Sur l’impossibilité de la multiplication des distributions, C R Acad Sci Paris, 239, 847-848 (1954) · Zbl 0056.10602
[26] Garetto, C.; Ruzhansky, M., Hyperbolic second order equations with non-regular time dependent coefficients, Arch Ration Mech Anal, 217, 1, 113-154 (2015) · Zbl 1320.35181
[27] Garetto C.. On the wave equation with multiplicities and space-dependent irregular coefficients. arXiv preprint arXiv:2004.096572020
[28] Ruzhansky, M.; Tokmagambetov, N., Wave equation for operators with discrete spectrum and irregular propagation speed, Arch Ration Mech Anal, 226, 1161-1207 (2017) · Zbl 1386.35263
[29] Munoz, J.; Ruzhansky, M.; Tokmagambetov, N., Wave propagation with irregular dissipation and applications to acoustic problems and shallow water, J Math Pures Appl, 123, 127-147 (1993) · Zbl 1418.35277
[30] Bakhti B.. Interacting fluids in an arbitrary external field. arXiv preprint arXiv:1702.049052017.
[31] Nezza, E. D.; Palatucci, G.; Valdinoci, E. E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math, 136, 5, 521-573 (2012) · Zbl 1252.46023
[32] Garofalo N.. Fractional thoughts. arXiv preprint arXiv:1712.03347v42018.
[33] Evans, L., Partial differential equations (1998), American Mathematical Society · Zbl 0902.35002
[34] 302 pages · Zbl 0818.46036
[35] Altybay, A.; Ruzhansky, M.; Tokmagambetov, N., Wave equation with distributional propagation speed and mass term: numerical simulations, Appl Math E-Notes, 24, 552-562 (2019) · Zbl 1432.35145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.