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Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

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Abstract

Generalizations of fractional derivatives of noninteger orders for N-dimensional Euclidean space are proposed. These fractional derivatives of the Riesz type can be considered as partial derivatives of noninteger orders. In contrast to the usual Riesz derivatives, the suggested derivatives give the usual partial derivatives for integer values of orders. For integer values of orders, the partial fractional derivatives of the Riesz type are equal to the standard partial derivatives of integer orders with respect to coordinate. Fractional generalizations of the nonlinear equations such as sine-Gordon, Boussinesq, Burgers, Korteweg–de Vries and Monge–Ampere equations for nonlocal continuum are considered.

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References

  1. Letnikov, A.V.: Historical development of the theory of differentiation of fractional order. Mat. Sb. 3, 85–119 (1868). (in Russian)

    Google Scholar 

  2. Debnath, L.: A brief historical introduction to fractional calculus. Int. J. Math. Educ. Sci. Technol. 35(4), 487–501 (2004)

    Article  MathSciNet  Google Scholar 

  3. Tenreiro Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Tenreiro Machado, J.A., Galhano, A.M., Trujillo, J.J.: Science metrics on fractional calculus development since 1966. Fract. Calc. Appl. Anal. 16(2), 479–500 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tenreiro Machado, J.A., Galhano, A.M.S.F., Trujillo, J.J.: On development of fractional calculus during the last fifty years. Scientometrics 98(1), 577–582 (2014)

    Article  Google Scholar 

  6. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  7. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  8. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  9. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Springer, Netherlands (2011)

    Book  MATH  Google Scholar 

  10. Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers, Vol. I. Background and Theory. Springer, Higher Education Press (2012)

  11. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  12. Valerio, D., Trujillo, J.J., Rivero, M., Tenreiro Machado, J.A., Baleanu, D.: Fractional calculus: a survey of useful formulas. Eur. Phys. J. Spec. Top. 222(8), 1827–1846 (2013)

    Article  Google Scholar 

  13. Ortigueira, M.D., Tenreiro Machado, J.A.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    Article  MathSciNet  Google Scholar 

  14. Liu, Cheng-shi: Counterexamples on Jumarie’s two basic fractional calculus formulae. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 92–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tarasov, V.E.: On chain rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 30(1–3), 1–4 (2016)

    Article  MathSciNet  Google Scholar 

  16. Tarasov, V.E.: Local fractional derivatives of differentiable functions are integer-order derivatives or zero. Int. J. Appl. Comput. Math. 2(2), 195–201 (2016)

    Article  MathSciNet  Google Scholar 

  17. Tarasov, V.E.: Comments on Riemann–Christoffel tensor in differential geometry of fractional order application to fractal space-time, [Fractals 21 (2013) 1350004]. Fractals 23(2), 1575001 (2015)

    Article  MathSciNet  Google Scholar 

  18. Tarasov, V.E.: Comments on the Minkowski’s space-time is consistent with differential geometry of fractional order, [Physics Letters A 363 (2007) 5–11]. Phys. Lett. A 379(14–15), 1071–1072 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tarasov, V.E.: Leibniz rule and fractional derivatives of power functions. J. Comput. Nonlinear Dyn. 11(3), 031014 (2016)

    Article  MathSciNet  Google Scholar 

  20. Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18(11), 2945–2948 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Carpinteri, A., Mainardi, F. (eds.): Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)

    MATH  Google Scholar 

  22. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    MATH  Google Scholar 

  23. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  24. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2011)

    Google Scholar 

  25. Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers. Vol. II. Applications. Springer, Higher Education Press (2012)

  26. Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics. Wiley-ISTE, London (2014)

    Book  MATH  Google Scholar 

  27. Tarasov, V.E.: Review of some promising fractional physical models. Int. J. Mod. Phys. B 27(9), 1330005 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tarasov, V.E.: Toward lattice fractional vector calculus. J. Phys. A 47(35), 355204 (2014). (51 pages)

  29. Tarasov, V.E.: Lattice fractional calculus. Appl. Math. Comput. 257, 12–33 (2015)

  30. Riesz, M.: L’integrale de Riemann–Liouville et le probleme de Cauchy pour l’equation des ondes. Bull. de la Soc. Math. de France. Supplement. 67, 153–170 (1939). https://eudml.org/doc/86724

  31. Riesz, M.: L’intégrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81(1), 1–222 (1949). doi:10.1007/BF02395016. (in French)

    Article  MathSciNet  MATH  Google Scholar 

  32. Prado, H., Rivero, M., Trujillo, J.J., Velasco, M.P.: New results from old investigation: a note on fractional m-dimensional differential operators. The fractional Laplacian. Fract. Calc. Appl. Anal. 18(2), 290–306 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lizorkin, P.I.: Characterization of the spaces \(L^r_p(\mathbb{R}^n)\) in terms of difference singular integrals. Mat. Sb. 81(1), 79–91 (1970). (in Russian)

    MathSciNet  Google Scholar 

  34. Samko, S.: Convolution and potential type operators in \(L^{p(x)}\). Integral Transforms Spec. Funct. 7(3–4), 261–284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Samko, S.: Convolution type operators in \(L^{p(x)}\). Integral Transforms Spec. Funct. 7(1–2), 123–144 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. Spec. Funct. 16(5–6), 461–482 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Samko, S.: A new approach to the inversion of the Riesz potential operator. Fract. Calc. Appl. Anal. 1(3), 225–245 (1998)

    MathSciNet  MATH  Google Scholar 

  38. Rafeiro, H., Samko, S.: Approximative method for the inversion of the Riesz potential operator in variable Lebesgue spaces. Fract. Calc. Appl. Anal. 11(3), 269–280 (2008)

    MathSciNet  MATH  Google Scholar 

  39. Rafeiro, H., Samko, S.: On multidimensional analogue of Marchaud formula for fractional Riesz-type derivatives in domains in \(R^n\). Fract. Calc. Appl. Anal. 8(4), 393–401 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Almeida, A., Samko, S.: Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Funct. Spaces Appl. 4(2), 113–144 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Samko, S.G.: On spaces of Riesz potentials. Math. USSR-Izv. 10(5), 1089–1117 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ortigueira, M.D., Laleg-Kirati, T.-M., Tenreiro Machado, J.A.: Riesz potential versus fractional Laplacian. J. Stat. Mech. Theory Exp. 2014(9), P09032 (2014)

    Article  MathSciNet  Google Scholar 

  43. Cerutti, R.A., Trione, S.E.: The inversion of Marcel Riesz ultrahyperbolic causal operators. Appl. Math. Lett. 12(6), 25–30 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cerutti, R.A., Trione, S.E.: Some properties of the generalized causal and anticausal Riesz potentials. Appl. Math. Lett. 13(4), 129–136 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tarasov, V.E.: Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grunwald–Letnikov–Riesz type. Mech. Mater. 70(1), 106–114 (2014). arXiv:1502.06268

  46. Tarasov, V.E.: Fractional-order difference equations for physical lattices and some applications. J. Math. Phys. 56(10), 103506 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tarasov, V.E.: Three-dimensional lattice models with long-range interactions of Grunwald–Letnikov type for fractional generalization of gradient elasticity. Meccanica 51(1), 125–138 (2016). doi:10.1007/s11012-015-0190-4

    Article  MathSciNet  MATH  Google Scholar 

  48. Ortigueira, M.D., Magin, R.L., Trujillo, J.J., Velasco, M.P.: A real regularised fractional derivative. Signal Image Video Process. 6(3), 351–358 (2012)

    Article  Google Scholar 

  49. Ortigueira, M.D., Trujillo, J.J.: A unified approach to fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5151–5157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tarasov, V.E.: Exact discretization by Fourier transforms. Commun. Nonlinear Sci. Numer. Simul. 37, 3161 (2016)

    Article  MathSciNet  Google Scholar 

  51. Tarasov, V.E.: United lattice fractional integro-differentiation. Fract. Calc. Appl. Anal. 19(3) (2016) accepted for publication

  52. Fichtenholz, G.M.: Differential and Integral Calculus, vol. 1, 7th Ed. Nauka, Moscow, 1969. (in Russian)

  53. Tarasov, V.E.: Non-linear fractional field equations: weak non-linearity at power-law non-locality. Nonlinear Dyn. 80(4), 1665–1672 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Tarasov, V.E.: Large lattice fractional Fokker–Planck equation. J. Stat. Mech. Theory Exp. 2014(9), P09036 (2014). arXiv:1503.03636

  55. Tarasov, V.E.: Fractional Liouville equation on lattice phase-space. Phys. A Stat. Mech. Appl. 421, 330–342 (2015). arXiv:1503.04351

  56. Tarasov, V.E.: Fractional quantum field theory: from lattice to continuum. Adv. High Energy Phys. 2014, 957863 (2014). 14 pages

    Article  MathSciNet  Google Scholar 

  57. Tarasov, V.E., Aifantis, E.C.: Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality. Commun. Nonlinear Sci. Numer. Simul. 22(13), 197–227 (2015). (arXiv:1404.5241)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tarasov, V.E.: Discretely and continuously distributed dynamical systems with fractional nonlocality. In: Cattani, C., Srivastava, H.M., Yang, X.-J. (eds.) Fractional Dynamics (De Gruyter Open, Warsaw, Berlin, 2015) Chapter 3. pp. 31–49. doi:10.1515/9783110472097-003

  59. Davydov, A.S.: Theory of Molecular Excitons. Plenum, New York (1971)

    Book  Google Scholar 

  60. Scott, A.C.: Davydov’s soliton. Phys. Rep. 217, 1–67 (1992)

    Article  MATH  Google Scholar 

  61. Gaididei, YuB, Mingaleev, S.F., Christiansen, P.L., Rasmussen, K.O.: Effects of nonlocal dispersive interactions on self-trapping excitations. Phys. Rev. E 55, 6141–6150 (1997)

    Article  Google Scholar 

  62. Rasmussen, K.O., Christiansen, P.L., Johansson, M., Gaididei, YuB, Mingaleev, S.F.: Localized excitations in discrete nonlinear Schröedinger systems: effects of nonlocal dispersive interactions and noise. Phys. D 113, 134–151 (1998)

    Article  MATH  Google Scholar 

  63. Gaididei, Yu., Flytzanis, N., Neuper, A., Mertens, F.G.: Effect of nonlocal interactions on soliton dynamics in anharmonic lattices. Phys. Rev. Lett. 75, 2240–2243 (1995)

    Article  MATH  Google Scholar 

  64. Mingaleev, S.F., Gaididei, Y.B., Mertens, F.G.: Solitons in anharmonic chains with power-law long-range interactions. Phys. Rev. E 58, 3833–3842 (1998)

    Article  Google Scholar 

  65. Mingaleev, S.F., Gaididei, Y.B., Mertens, F.G.: Solitons in anharmonic chains with ultra-long-range interatomic interactions. Phys. Rev. E 61, R1044–R1047 (2000). arxiv:patt-sol/9910005

  66. Korabel, N., Zaslavsky, G.M.: Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction. Phys. A 378(2), 223–237 (2007)

    Article  Google Scholar 

  67. Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91–107 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  68. Dyson, F.J.: Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 212–215 (1969)

    Article  MathSciNet  Google Scholar 

  69. Dyson, F.J.: An Ising ferromagnet with discontinuous long-range order. Commun. Math. Phys. 21, 269–283 (1971)

    Article  MathSciNet  Google Scholar 

  70. Nakano, H., Takahashi, M.: Quantum Heisenberg chain with long-range ferromagnetic interactions at low temperatures. J. Phys. Soc. Jpn. 63, 926–933 (1994). arxiv:cond-mat/9311034

  71. Nakano, H., Takahashi, M.: Quantum Heisenberg model with long-range ferromagnetic interactions. Phys. Rev. B 50, 10331–10334 (1994)

    Article  Google Scholar 

  72. Nakano, H., Takahashi, M.: Quantum Heisenberg ferromagnets with long-range interactions. J. Phys. Soc. Jpn. 63, 4256–4257 (1994)

    Article  Google Scholar 

  73. Nakano, H., Takahashi, M.: Magnetic properties of quantum Heisenberg ferromagnets with long-range interactions. Phys. Rev. B 52, 6606–6610 (1995)

    Article  Google Scholar 

  74. Joyce, G.S.: Absence of ferromagnetism or antiferromagnetism in the isotropic Heisenberg model with long-range interactions. J. Phys. C 2, 1531–1533 (1969)

    Article  Google Scholar 

  75. Sousa, J.R.: Phase diagram in the quantum XY model with long-range interactions. Eur. Phys. J. B 43, 93–96 (2005)

    Article  Google Scholar 

  76. Braun, O.M., Kivshar, Y.S., Zelenskaya, I.I.: Kinks in the Frenkel–Kontorova model with long-range interparticle interactions. Phys. Rev. B 41, 7118–7138 (1990)

    Article  Google Scholar 

  77. Flach, S.: Breathers on lattices with long-range interaction. Phys. Rev. E 58, R4116–R4119 (1998)

    Article  Google Scholar 

  78. Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295, 181–264 (1998)

    Article  MathSciNet  Google Scholar 

  79. Gorbach, A.V., Flach, S.: Compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms. Phys. Rev. E 72, 056607 (2005)

    Article  MathSciNet  Google Scholar 

  80. Laskin, N., Zaslavsky, G.M.: Nonlinear fractional dynamics on a lattice with long-range interactions. Phys. A 368, 38–54 (2006). arxiv:nlin.SI/0512010

  81. Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of coupled oscillators with long-range interaction. Chaos 16(2), 023110 (2006). arxiv:nlin.PS/0512013

  82. Tarasov, V.E., Zaslavsky, G.M.: Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11(8), 885–898 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  83. Zaslavsky, G.M., Edelman, M., Tarasov, V.E.: Dynamics of the chain of oscillators with long-range interaction: from synchronization to chaos. Chaos 17(4), 043124 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  84. Korabel, N., Zaslavsky, G.M., Tarasov, V.E.: Coupled oscillators with power-law interaction and their fractional dynamics analogues. Commun. Nonlinear Sci. Numer. Simul. 12(8), 1405–1417 (2007). arxiv:math-ph/0603074

  85. Tarasov, V.E.: Continuous limit of discrete systems with long-range interaction. J. Phys. A 39(48), 14895–14910 (2006). arXiv:0711.0826

  86. Tarasov, V.E.: Map of discrete system into continuous. J. Math. Phys. 47(9), 092901 (2006). arXiv:0711.2612

  87. Biler, P., Funaki, T., Woyczynski, W.A.: Fractal Burger equation. J. Differ. Equ. 14, 9–46 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  88. Burgers, J.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (2008)

    Google Scholar 

  89. Momani, S.: An explicit and numerical solutions of the fractional KdV equation. Math. Comput. Simul. 70, 110–1118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  90. Miskinis, P.: Weakly nonlocal supersymmetric KdV hierarchy. Nonlinear Anal. Model. Control 10, 343–348 (2005)

    MathSciNet  MATH  Google Scholar 

  91. de Bouard, A., Saut, J.-C.: Solitary waves of generalized Kadomtsev–Petviashvili equations, Annales de l’Institut Henri Poincare. Anal. Non Lineaire 14(2), 211–236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  92. Jones, K.L.: Three-dimensional Korteweg–de Vries equation and traveling wave solutions. Int. J. Math. Math. Sci. 24(6), 379–384 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  93. Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  94. Shen, S., Liu, F., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J. Appl. Math. 73(6), 850–872 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  95. Li, ChP, Zeng, F.H.: Finite difference methods for fractional differential equations. Int. J. Bifurc. Chaos 22(4), 1230014 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  96. Li, ChP, Zeng, F.H.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34(2), 149–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  97. Huang, Y.H., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014). arXiv:1311.7691

  98. Tarasov, V.E.: General lattice model of gradient elasticity. Mod. Phys. Lett. B. 28(7), 1450054 (2014) (17 pages). arXiv:1501.01435

  99. Tarasov, V.E.: Lattice model with nearest-neighbor and next-nearest-neighbor interactions for gradient elasticity. Discontinuity Nonlinearity Complex. 4(1), 11–23 (2015). arXiv:1503.03633

  100. Tarasov, V.E.: Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323(11), 2756–2778 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  101. Samko, S.: On local summability of Riesz potentials in the case \(Re \alpha >0\). Anal. Math. 25, 205–210 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  102. Webb, G.M., Zank, G.P.: Painleve analysis of the three-dimensional Burgers equation. Phys. Lett. A 150(1), 14–22 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  103. Shandarin, S.F.: Three-dimensional Burgers equation as a model for the large-scale structure formation in the Universe, Chapter In: The IMA Volumes in Mathematics and its Applications (1996) pp. 401–413. arXiv:astro-ph/9507082

  104. Dai, Ch-Q, Yu, F.-B.: Special solitonic localized structures for the (3+1)-dimensional Burgers equation in water waves. Wave Motion 51(1), 52–59 (2014)

    Article  MathSciNet  Google Scholar 

  105. Korpusov, M.O.: Blowup of solutions of the three-dimensional Rosenau–Burgers equation. Theor. Math. Phys. 170(3), 280–286 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  106. Wazwaz, A.-M.: A variety of (3+1)-dimensional Burgers equations derived by using the Burgers recursion operator. Math. Methods Appl. Sci. (2014). doi:10.1002/mma.3255 published online: 18 AUG

  107. Johnson, R.S.: A two-dimensional Boussinesq equation for water waves and some of its solutions. J. Fluid Mech. 323(1), 65–78 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  108. El-Sabbagh, M.F., Ali, A.T.: New exact solutions for (3+1)-dimensional Kadomtsev–Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation. Int. J. Nonlinear Sci. Numer. Simul. 6(2), 151–162 (2005)

    Article  MathSciNet  Google Scholar 

  109. Yong-Qi, Wu: Periodic wave solution to the (3+1)-dimensional Boussinesq equation. Chin. Phys. Lett. 25(8), 2739–2742 (2008)

    Article  Google Scholar 

  110. Huan, Zhang, Bo, Tian, Hai-Qiang, Zhang, Tao, Geng, Xiang-Hua, Meng, Wen-Jun, Liu, Ke-Jie, Cai: Periodic wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional Kadomtsev–Petviashvili equation. Commun. Theor. Phys. 50(5), 1169–1176 (2008)

    Article  MathSciNet  Google Scholar 

  111. Moleleki, L.D., Khalique, C.M.: Symmetries, traveling wave solutions, and conservation laws of a (3+1)-dimensional Boussinesq equation. Adv. Math. Phys. 2014. (2014) Article ID 672679

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    Vasily E. Tarasov

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Tarasov, V.E. Partial fractional derivatives of Riesz type and nonlinear fractional differential equations. Nonlinear Dyn 86, 1745–1759 (2016). https://doi.org/10.1007/s11071-016-2991-y

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