Symmetry analysis of space-time fractional Poisson equation with a delay

References

• Y.A. Ahmad, On the Conservation Laws for a Certain Class of Nonlinear Wave Equation via a New Conservation Theorem, Communications in Nonlinear Science and Numerical Simulation 17(4), (2012), 1566–1575.
   Web of Science ®Google Scholar
• M.M. Alshamrani, H.A. Zedan, and M. Abu-Nawas, Lie group method and fractional differential equations, JNSA 10(8) (2017), 4175–4180.
   Google Scholar
• G. Bluman, A. Cheviakov, and S. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, Vol. 168, Springer-Verlag, New York, 2010.
   Google Scholar
• Cheng Chen and Yao-Lin Jiang, Lie Group Analysis and Invariant Solutions for Nonlinear Time-Fractional Diffusion-Convection Equations, Commun. Theor. Phys. 68 (2017), Article ID 295.
   Web of Science ®Google Scholar
• E.H. El Kinani and A. Ouhadan, Lie symmetry analysis of some time fractional partial differential equations, Int. J. Mod. Phys. Conf. Ser. 38 (2015), 15600751–15600758.
   Google Scholar
• Jan Čermák, Jan Horníček, and Tomáš Kisela, Stability regions for fractional differential systems with a time delay, Communications in Nonlinear Science and Numerical Simulation 31(1–3) (2016), 108–123.
   Web of Science ®Google Scholar
• E. Fredericks and F.M. Mahomed, Formal Approach for Handling Lie Point Symmetries of Scalar First-Order Ito Stochastic Ordinary Differential Equations, Journal of Nonlinear Mathematical Physics 15 (2008), 44–59.
   Web of Science ®Google Scholar
• Manoj Gaur and Karanjeet Singh, Symmetry analysis of time-fractional potential Burger’s equation, Math. Commun. 22 (2017), 1–11.
   Web of Science ®Google Scholar
• R.K. Gazizov, A.A. Kasatkin and S.Yu Lukashchuk, Continuous transformation groups of fractional differential equations, Vestn. USATU 9 (2007), 125–35.
   Google Scholar
• R.K. Gazizov, A.A. Kasatkin and S.Yu Lukashchuk, Symmetry properties of fractional diffusion equations, Physica Scripta T136 (2009), Article ID 014016.
   Google Scholar
• D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.
   Google Scholar
• D.J. Griffiths and C. Reed, Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, NJ, USA, 1999.
   Google Scholar
• Y.N. Grigoriev, N.H. Ibragimov, V.F. Kovalev, and S.V. Meleshko, Symmetries of Integro-Differential Equation, Springer, New York, 2010.
   Google Scholar
• J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
   Google Scholar
• J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology & Society 15(2) (1999), 86–90.
   Google Scholar
• J. Hu, Y.J. Ye, S.F. Shen, and J. Zhang, Lie symmetry analysis of the time fractional KdV-type equation, Appl. Math. Comput. 233 (2014), 439–444.
   Web of Science ®Google Scholar
• N.H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, John Wiley and Sons, Chichester, 1999.
   Google Scholar
• Ram Jiwari, Vikas Kumar, Ram Karan, and Ali Saleh Alshomrani, Haar wavelet quasilinearization approach for MHD Falkner-Skan flow over permeable wall via Lie Group method, International Journal of Numerical Methods for Heat and Fluid Flow 27 (2017), 1332–1350.
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, In: North-Holland Mathematics Studies, Jan V. Mill, (ed.), Vol. 204, Elsevier, Amsterdam, 2006.
   Google Scholar
• V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Longma Group, Harlow, 1994.
   Google Scholar
• Vikas Kumar and Aisha Alqahtani, Lie symmetry analysis, soliton and numerical solutions of boundary value problem for variable coefficients coupled KdV-Burgers equation, Nonlinear Dynamics 90 (2017), 2903–2915.
   Web of Science ®Google Scholar
• Vikas Kumar, R.K. Gupta, and Ram Jiwari, Exact and Numerical Solutions of Coupled Short Pulse Equation with Time Dependent Coefficients, Nonlinear Dynamics 79 (2015), 455–464.
   Web of Science ®Google Scholar
• Vikas Kumar and Lakhveer Kaur, On the solutions of field equations due to rotating bodies in General Relativity, St. Petersburg Polytechnical University Journal: Physics and Mathematics 03 (2017), 352–358.
   Google Scholar
• Vikas Kumar, Lakhveer Kaur, Ajay Kumar, and Mehmet Emir Koksal, Lie symmetry based-analytical and numerical approach for modified Burgers-KdV equation, Results in Physics 08 (2017), 1136–1142.
   Google Scholar
• Y.Y. Li, Y. Zhao, G.N. Xie, D. Baleanu, X.J. Yang, and K. Zhao, Local fractional Poisson and Laplace equations with applications to electrostatics in fractal domain, Advances in Mathematical Physics 2014 (2014), Article ID 590574.
   Web of Science ®Google Scholar
• F.-S. Long and S.V. Meleshko, On the complete group classification of the one-dimensional nonlinear Klein-Gordon equation with a delay, Mathematical Methods in the Applied Sciences 39(12) (2016), 3255–3270.
   Web of Science ®Google Scholar
• F.-S. Long and S.V. Meleshko, Symmetry analysis of the nonlinear two dimensional Klein-Gordon equation with a time-varying delay, Mathematical Methods in the Applied Sciences 40 (2017), 4658–4673.
   Web of Science ®Google Scholar
• S.Yu. Lukashchuk and A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. 257 (2015), 335–343.
   Web of Science ®Google Scholar
• F.M. Mahomed, Symmetry Group Classification of Ordinary Differential Equations: Survey of Some Results, Mathematical Methods in the Applied Sciences 30(16) (2007), 1995–2012.
   Web of Science ®Google Scholar
• S.V. Meleshko and S. Moyo, On the complete group classification of the reaction-diffusion equation with a delay, Journal of Mathematical Analysis and Applications 338 (2008), 448–466.
   Web of Science ®Google Scholar
• L. Morgado, N.J. Ford, and P.M. Lima, Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics 252 (2013), 159–168.
   Web of Science ®Google Scholar
• Aminu M. Nass, Symmetry Analysis and Invariants Solutions of Laplace Equation on Surfaces of Revolution, Advances in Mathematics: Scientific Journal 3(1) (2014), 23–31.
   Google Scholar
• Aminu M. Nass and E. Fredericks, Lie Symmetry of Itô Stochastic Differential Equation Driven by Poisson Process, American Review of Mathematics and Statistics 4(1) (2016), 17–30.
   Google Scholar
• Aminu M. Nass and E. Fredericks, Symmetry of Jump-Diffusion Stochastic Differential Equations, Global and Stochastic Analysis 3(1) (2016), 11–23.
   Google Scholar
• Aminu M. Nass and E. Fredericks, N-Symmetry of Itô Stochastic Differential Equation Driven by Poisson Process, International Journal of Pure and Applied Mathematics 110(1) (2016), 165–182.
   Google Scholar
• Aminu M. Nass and E. Fredericks, W-symmetries of jump-diffusion Itô stochastic differential equations, Nonlinear Dyn. 90(4) (2017), 2869–2877.
   Web of Science ®Google Scholar
• Peter O’Neil, Beginning Partial Differential Equations, 3rd Edition, John Wiley and Sons, Hoboken, New Jersey, 2014.
   Google Scholar
• L.V. Ovsiannikov, Group Analysis of Differential Equations, Nauka, Moscow, 1978.
   Google Scholar
• S. Persides, The Laplace and Poisson equations in Schwarzschild’s space-time, Journal of Mathematical Analysis and Applications 43 (1973), 571–578.
   Web of Science ®Google Scholar
• V.G. Pimenov and A.S. Hendy, Numerical studies for fractional functional differential equations with delay based on BDF-Type shifted Chebyshev approximations, Hindawi Publishing Corporation Abstract and Applied Analysis 2015 (2015), Article ID 510875, http://dx.doi.org/10.1155/2015/510875.
   Google Scholar
• E.M. Poliscuk, Equations of the Laplace and Poisson type in a function space, Matematicheskii Sbornik 114(2) (1967), 261–292.
   Google Scholar
• P. Pue-on and S.V. Meleshko Group classification of second-order delay ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation 15 (2010), 1444–1453.
   Web of Science ®Google Scholar
• M.N. Sadiku, Numerical Techniques in Electromagnetic, CRC Press, Boca Raton, 2000.
   Google Scholar
• A.P.S. Selvadurai, Partial Differential Equations in Mechanics 2, The Biharmonic, Poisson’s Equation, Springer-Verlag, Berlin/Heidelberg, 2000.
   Google Scholar
• K. Singla and R.K. Gupta, Space-time fractional nonlinear partial differential equations: symmetry analysis and conservation laws, Nonlinear Dyn. 89(1) (2017), 321–331.
   Web of Science ®Google Scholar
• K. Singla and R.K. Gupta, Generalized Lie symmetry approach for fractional order systems of differential equations, III, Journal of Mathematical Physics 58 (2017), Article ID 061501.
   Web of Science ®Google Scholar
• J. Tanthanuch, Symmetry analysis of the nonhomogeneous inviscid burgers equation with delay, Communications in Nonlinear Science and Numerical Simulation 17 (2012), 4978–4987.
   Web of Science ®Google Scholar
• J. Tanthanuch and S.V. Meleshko, On definition of an admitted Lie group for functional differential equations, Communications in Nonlinear Science and Numerical Simulation 9 (2004), 117–125.
   Google Scholar
• G. Unal, Symmetries of Itô and Stratonovich Dynamical Systems and their Conserved Quantities, Nonlinear Dynamics 32 (2003), 417–426.
   Web of Science ®Google Scholar
• S.C. Wafo and F.M. Mahomed, Integration of stochastic ordinary differential equations from a symmetry standpoint, Journal of Physics A: Mathematical and General 34 (2001), 177–194.
   Google Scholar
• Gw Wang, Tz Xu, and T. Feng, Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation, PLoS ONE 9(2) (2014), Article ID e88336, https://doi.org/10.1371/journal.pone.0088336.
   Web of Science ®Google Scholar
• Qifeng Zhang, Maohua Ran, and Dinghua Xu, Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay, Applicable Analysis 96(11) (2017), 1867–1884.
   Web of Science ®Google Scholar