On Complete Group Classification of Time Fractional Systems Evolution Differential Equation with a Constant Delay
Yıl 2023,
Cilt: 6 Sayı: 1, 12 - 23, 29.03.2023
Kassimu Mpungu
,
Aminu Ma'aruf Nass
Öz
A fractional order system of evolution partial differential equations with a constant delay is considered. By exploiting the Lie symmetry method, we give a complete group classification of the system. Furthermore, we establish the corresponding symmetry reductions and construct some analytical solutions to the system.
Kaynakça
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- [2] S. Djennadi, N. Shawagfeh, O.A. Arqub, A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time–space fractional diffusion equation, Partial Differ. Equ. Appl. Math., 4, (2021), 100164.
- [3] O. A. Arqub, Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method, Int. J. Numer. Methods Heat Fluid Flow, 30(11), (2020), 4711-4733.
- [4] M. D. Johansyah, A. K. Supriatna, E. Rusyaman, J. Saputra, Application of fractional differential equation in economic growth model: a systematic review approach, AIMS Math., 6(9), (2021), 10266-10280.
- [5] H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real-world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64, (2018), 213-231.
- [6] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic press, 1982.
- [7] R. Al-Deiakeh, O. A. Arqub, M. Al-Smadi, S. Momani, Lie symmetry analysis, explicit solutions, and conservation laws of the time-fractional Fisher equation in two-dimensional space, J. Ocean Eng. Sci., 7(4), (2021), 345-352.
- [8] S. Dillen, M. Oberlack, Y. Wang, Analytical investigation of rotationally symmetrical oscillating flows of viscoelastic fluids, J. Non-Newton. Fluid Mech., 272, (2019), 104168.
- [9] M. I. El Bahi, K. Hilal, Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-like equation, J. Funct. Spaces, 2021, (2021), Article ID 6628130, 10 pages.
- [10] M. S. Hashemi, D. Baleanu, Lie Symmetry Analysis of Fractional Differential Equations, Chapman and Hall/CRC, 2020.
- [11] B. C. Kaibe, J. G. O’Hara, Symmetry Analysis of an Interest Rate Derivatives PDE Model in Financial Mathematics, Symmetry, 11(8), (2019), 1056.
- [12] D. Klingenberg, M. Oberlack, D. Pluemacher, Symmetries and turbulence modeling, Phys. Fluids, 32(2), (2020), 025108.
- [13] N. Maarouf, H. Maadan, K. Hilal, Lie symmetry analysis and explicit solutions for the time-fractional regularized long-wave equation, Int. J. Differ. Equ., 2021, (2021), Article ID 6614231, 11 pages.
- [14] A. M. Nass, K. Mpungu, Lie symmetry reductions and integrability of approximated small delay stochastic differential equations, Afr. Mat., 32(1–2), (2021), 199-209.
- [15] A. M. Nass, Lie point symmetries of autonomous scalar first-order Itˆo Stochastic delay ordinary differential equations, J. Theor. Probab., 35(3), (2022), 1939-1951.
- [16] H. Wang, A. M. Nass, Z. Zou, Lie symmetry analysis of C1(a;b;c) partial differential equations, Adv. Math. Phys., 2021, (2021), Article. ID 9113423,
7 pages.
- [17] Z. Y. Zhang, G. F. Li, Lie symmetry analysis and exact solutions of the time-fractional biological population model, Phys. A, 540, (2020), 123134.
- [18] K. Singla, R. K. Gupta, Comment on ‘Lie symmetries and group classification of a class of time fractional evolution systems’ [J. Math. Phys. 56, 123504 (2015)], J. Math. Phys., 58(5), (2017), 054101.
- [19] C. Abdallah, P. Dorato, J. Benites-Read, R. Byrne, Delayed positive feedback can stabilize oscillatory systems, Proceedings of the 1993 American Control Conference (ACC 1993), (1993), 3106-3107.
- [20] S. Sen, P. Ghosh, S. S. Riaz, D. S. Ray, Time-delay-induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80(4), (2009), 046212.
- [21] C. Dong, Q. Xiao, M. Wang, T. Morstyn, M. D. McCulloch, H. Jia, Distorted stability space and instability triggering mechanism of EV aggregation delays in the secondary frequency regulation of electrical grid-electric vehicle system, IEEE Trans. Smart Grid, 11(6), (2020) 5084-5098.
- [22] O. A. Arqub, T. Hayat, M. Alhodaly, Analysis of Lie symmetry, explicit series solutions, and conservation laws for the nonlinear time-fractional phi-four equation in two-dimensional space, Int. J. Appl. Comput. Math, 8(3), (2022), 145.
- [23] Y. Feng, J. Yu, Lie symmetry analysis of fractional ordinary differential equation with neutral delay, AIMS Math., 6(4), (2021), 3592–3605.
- [24] A. M. Nass, Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay, Appl. Math. Comput., 347, (2019), 370-380.
- [25] A. M. Nass, K. Mpungu, R. I. Nuruddeen, Group classification of space-time fractional nonlinear Poisson equation, Math. Commun., 24(2), (2019), 221-233.
- [26] K. Mpungu, A. M. Nass, Symmetry analysis of time fractional convection-reaction-diffusion equation with a delay, Results Nonlinear Anal., 2(3), (2019), 113-124.
- [27] A. M. Nass, Symmetry analysis of space-time fractional Poisson equation with a delay, Quaest. Math., 42(9), (2019), 1221-1235.
- [28] M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 2004(3), (2004), 197–211.
- [29] A. A. Kilbas, T. Pierantozzi, J. J. Trujillo, L. V´azquez, On the solution of fractional evolution equations, J. Phys. A: Math. Gen., 37(9), (2004), 3271.
- [30] P. Loreti, D. Sforza, Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces, Fractal Fract., 5(4), (2021), 138.
- [31] A. Suechoei, P. S. Ngiamsunthorn, Extremal solutions of jCaputo fractional evolution equations involving integral kernels, AIMS Math., 6(5), (2021), 4734-4757.
- [32] F. S. Long, S. V. Meleshko, On the complete group classification of the one-dimensional nonlinear Klein-Gordon equation with a delay, Math. Methods Appl. Sci., 39(12), (2016), 3255-3270.
- [33] J. Tanthanuch, Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Commun. Nonlinear Sci. Numer. Simul., 17(12), (2012), 4978-4987.
Yıl 2023,
Cilt: 6 Sayı: 1, 12 - 23, 29.03.2023
Kassimu Mpungu
,
Aminu Ma'aruf Nass
Kaynakça
- [1] W. Beghami, B. Maayah, S. Bushnaq, O. A. Arqub, The Laplace optimized decomposition method for solving systems of partial differential equations of fractional order, Int. J. Appl. Comput. Math, 8(2), (2022), 52.
- [2] S. Djennadi, N. Shawagfeh, O.A. Arqub, A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time–space fractional diffusion equation, Partial Differ. Equ. Appl. Math., 4, (2021), 100164.
- [3] O. A. Arqub, Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method, Int. J. Numer. Methods Heat Fluid Flow, 30(11), (2020), 4711-4733.
- [4] M. D. Johansyah, A. K. Supriatna, E. Rusyaman, J. Saputra, Application of fractional differential equation in economic growth model: a systematic review approach, AIMS Math., 6(9), (2021), 10266-10280.
- [5] H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real-world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64, (2018), 213-231.
- [6] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic press, 1982.
- [7] R. Al-Deiakeh, O. A. Arqub, M. Al-Smadi, S. Momani, Lie symmetry analysis, explicit solutions, and conservation laws of the time-fractional Fisher equation in two-dimensional space, J. Ocean Eng. Sci., 7(4), (2021), 345-352.
- [8] S. Dillen, M. Oberlack, Y. Wang, Analytical investigation of rotationally symmetrical oscillating flows of viscoelastic fluids, J. Non-Newton. Fluid Mech., 272, (2019), 104168.
- [9] M. I. El Bahi, K. Hilal, Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-like equation, J. Funct. Spaces, 2021, (2021), Article ID 6628130, 10 pages.
- [10] M. S. Hashemi, D. Baleanu, Lie Symmetry Analysis of Fractional Differential Equations, Chapman and Hall/CRC, 2020.
- [11] B. C. Kaibe, J. G. O’Hara, Symmetry Analysis of an Interest Rate Derivatives PDE Model in Financial Mathematics, Symmetry, 11(8), (2019), 1056.
- [12] D. Klingenberg, M. Oberlack, D. Pluemacher, Symmetries and turbulence modeling, Phys. Fluids, 32(2), (2020), 025108.
- [13] N. Maarouf, H. Maadan, K. Hilal, Lie symmetry analysis and explicit solutions for the time-fractional regularized long-wave equation, Int. J. Differ. Equ., 2021, (2021), Article ID 6614231, 11 pages.
- [14] A. M. Nass, K. Mpungu, Lie symmetry reductions and integrability of approximated small delay stochastic differential equations, Afr. Mat., 32(1–2), (2021), 199-209.
- [15] A. M. Nass, Lie point symmetries of autonomous scalar first-order Itˆo Stochastic delay ordinary differential equations, J. Theor. Probab., 35(3), (2022), 1939-1951.
- [16] H. Wang, A. M. Nass, Z. Zou, Lie symmetry analysis of C1(a;b;c) partial differential equations, Adv. Math. Phys., 2021, (2021), Article. ID 9113423,
7 pages.
- [17] Z. Y. Zhang, G. F. Li, Lie symmetry analysis and exact solutions of the time-fractional biological population model, Phys. A, 540, (2020), 123134.
- [18] K. Singla, R. K. Gupta, Comment on ‘Lie symmetries and group classification of a class of time fractional evolution systems’ [J. Math. Phys. 56, 123504 (2015)], J. Math. Phys., 58(5), (2017), 054101.
- [19] C. Abdallah, P. Dorato, J. Benites-Read, R. Byrne, Delayed positive feedback can stabilize oscillatory systems, Proceedings of the 1993 American Control Conference (ACC 1993), (1993), 3106-3107.
- [20] S. Sen, P. Ghosh, S. S. Riaz, D. S. Ray, Time-delay-induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80(4), (2009), 046212.
- [21] C. Dong, Q. Xiao, M. Wang, T. Morstyn, M. D. McCulloch, H. Jia, Distorted stability space and instability triggering mechanism of EV aggregation delays in the secondary frequency regulation of electrical grid-electric vehicle system, IEEE Trans. Smart Grid, 11(6), (2020) 5084-5098.
- [22] O. A. Arqub, T. Hayat, M. Alhodaly, Analysis of Lie symmetry, explicit series solutions, and conservation laws for the nonlinear time-fractional phi-four equation in two-dimensional space, Int. J. Appl. Comput. Math, 8(3), (2022), 145.
- [23] Y. Feng, J. Yu, Lie symmetry analysis of fractional ordinary differential equation with neutral delay, AIMS Math., 6(4), (2021), 3592–3605.
- [24] A. M. Nass, Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay, Appl. Math. Comput., 347, (2019), 370-380.
- [25] A. M. Nass, K. Mpungu, R. I. Nuruddeen, Group classification of space-time fractional nonlinear Poisson equation, Math. Commun., 24(2), (2019), 221-233.
- [26] K. Mpungu, A. M. Nass, Symmetry analysis of time fractional convection-reaction-diffusion equation with a delay, Results Nonlinear Anal., 2(3), (2019), 113-124.
- [27] A. M. Nass, Symmetry analysis of space-time fractional Poisson equation with a delay, Quaest. Math., 42(9), (2019), 1221-1235.
- [28] M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 2004(3), (2004), 197–211.
- [29] A. A. Kilbas, T. Pierantozzi, J. J. Trujillo, L. V´azquez, On the solution of fractional evolution equations, J. Phys. A: Math. Gen., 37(9), (2004), 3271.
- [30] P. Loreti, D. Sforza, Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces, Fractal Fract., 5(4), (2021), 138.
- [31] A. Suechoei, P. S. Ngiamsunthorn, Extremal solutions of jCaputo fractional evolution equations involving integral kernels, AIMS Math., 6(5), (2021), 4734-4757.
- [32] F. S. Long, S. V. Meleshko, On the complete group classification of the one-dimensional nonlinear Klein-Gordon equation with a delay, Math. Methods Appl. Sci., 39(12), (2016), 3255-3270.
- [33] J. Tanthanuch, Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Commun. Nonlinear Sci. Numer. Simul., 17(12), (2012), 4978-4987.