Some newly explored exact solitary wave solutions to nonlinear inhomogeneous Murnaghan’s rod equation of fractional order

References

• K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type. Berlin: Springer-Verlag, 2010.
   Google Scholar
• Iqbal M, seadawy AR, Lu D. Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods. Mod Phys Lett A. 2018;33(32):1850183.
   Web of Science ®Google Scholar
• Lu D, Seadawy AR, Iqbal M. Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications. Results Phys. 2018;11:1161–1171.
   Web of Science ®Google Scholar
• Iqbal M, Seadawy AR, Lu D, et al. Construction of a weakly nonlinear dispersion solitary wave solution for the Zakharov–Kuznetsov-modified equal width dynamical equation. Indian J Phys. 2020;94:1465–1474.
   Web of Science ®Google Scholar
• Seadawy AR, Iqbal M, Lu D. Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A. 2020;544:123560.
   Web of Science ®Google Scholar
• Iqbal M, Seadawy AR, Lu D. Dispersive solitary wave solutions of nonlinear further modified Korteweg–de Vries dynamical equation in an unmagnetized dusty plasma. Mod Phys Lett A. 2018;33(37):1850217.
   Web of Science ®Google Scholar
• Iqbal M, Seadawy AR, Lu D. Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions. Mod Phys Lett B. 2019;33(18):1950210.
   Web of Science ®Google Scholar
• Seadawy AR, Iqbal M, Lu D. Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal width dynamical equation. Comput Math Appl. 2019;78(11):3620–3632.
   Web of Science ®Google Scholar
• Seadawy AR, Iqbal M, Lu D. Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive–Petviashvili modified equal width-Burgers equation. Indian J Phys. 2020;94(5):675–687.
   Web of Science ®Google Scholar
• Iqbal M, Seadawy AR, Lu D, et al. Construction of bright–dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation. Mod Phys Lett A. 2019;34(37):1950309.
   Web of Science ®Google Scholar
• Seadawy AR, Iqbal M, Lu D. Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics. Indian J Phys. 2020;94:823–832.
   Web of Science ®Google Scholar
• Seadawy AR, Iqbal M, Lu D. Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J Taibah Univ Sci. 2019;13(1):1060–1072.
   Web of Science ®Google Scholar
• Iqbal M, Seadawy AR, Khalil OH, et al. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear Nizhnik–Novikov–Vesselov dynamical equation. Results hys. 2020;16:102838.
   Google Scholar
• Mohyud-Din ST, Noor MA. Homotopy perturbation method for solving partial differential equations. Z Naturforsch A. 2009;64(3–4):157–170.
   Google Scholar
• Shang Y. A lie algebra approach to susceptible-infected-susceptible epidemics. Electron J Differ Equ. 2012;2012:1–7.
   Google Scholar
• Shang Y. Lie algebraic discussion for affinity based information diffusion in social networks. Open Phys. 2017;15(1):705–711.
   Web of Science ®Google Scholar
• Sakar MG, Ergören H. Alternative variational iteration method for solving the time-fractional Fornberg–whitham equation. Appl Math Model. 2015;39(14):3972–3979.
   Web of Science ®Google Scholar
• Abdou M, Soliman A. New applications of variational iteration method. Physica D. 2005;211(1):1–8.
   Web of Science ®Google Scholar
• Zhang S, Xia T. An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations. Commun Nonlinear Sci Numer Simul. 2008;13(7):1294–1301.
   Web of Science ®Google Scholar
• Navickas Z, Ragulskis M, Listopadskis N, et al. Comments on ‘soliton solutions to fractional-order nonlinear differential equations based on the exp-function method’. Optik. 2017;132:223–231.
   Web of Science ®Google Scholar
• Navickas Z, Telksnys T, Ragulskis M. Comments on ‘The exp-function method and generalized solitary solutions’. Comput Math Appl. 2015;69(8):798–803.
   Web of Science ®Google Scholar
• Zhang S, Xia TC. A further improved extended Fan sub-equation method and its application to the (3+1)-dimensional Kadomstev–Petviashvili equation. Phys Lett A. 2006;356(2):119–123.
   Web of Science ®Google Scholar
• Guner O, Atik H, Kayyrzhanovich AA. New exact solution for space–time fractional differential equations via (G′/G)-expansion method. Optik. 2017;130:696–701.
   Web of Science ®Google Scholar
• Manafian J, Lakestani M. Application of tan(ϕ/2)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik. 2016;127(4):2040–2054.
   Web of Science ®Google Scholar
• Akbulut A, Kaplan M, Tascan F. The investigation of exact solutions of nonlinear partial differential equations by using exp(−Φ(ξ)) method. Optik. 2017;132:382–387.
   Web of Science ®Google Scholar
• Kudryashov NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul. 2012;17(6):2248–2253.
   Web of Science ®Google Scholar
• Zhu S. The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2+1)-dimensional Boiti–Leon–Pempinelle equation. Chaos Solitons Fractals. 2008;37(5):1335–1342.
   Web of Science ®Google Scholar
• Bekir A, Cevikel AC. The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics. J King Saud Univ - Sci. 2011;23(2):127–132.
   Google Scholar
• Li B, Chen Y, Xuan H, et al. Generalized Riccati equation expansion method and its application to the (3+1)-dimensional Jumbo–Miwa equation. Appl Math Comput. 2004;152(2):581–595.
   Web of Science ®Google Scholar
• Tala-Tebue E, Tsobgni-Fozap DC, Kenfack-Jiotsa A, et al. Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G′/G)-expansion method including the generalized Riccati equation. Eur Phys J Plus. 2014;129(6):136.
   Web of Science ®Google Scholar
• Salathiel Y, Amadou Y, Betchewe G, et al. Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method. Nonlinear Dyn. 2017;87(4):2435–2443.
   Web of Science ®Google Scholar
• Koonprasert S, Sirisubtawee S, Ampun S. More explicit solitary solutions of the space-time fractional fifth order nonlinear Sawada–Kotera equation via the improved generalized Riccati equation mapping method. 2017.
   Google Scholar
• Bibi S, Ahmed N, Faisal I, et al. Some new solutions of the Caudrey–Dodd–Gibbon (CDG) equation using the conformable derivative. Adv Differ Equ. 2019;2019(1):89.
   Google Scholar
• Samsonov A. Strain solitons in solids and how to construct them. Boca Raton (FL): Chapman and Hall/CRC; Dec 18, 2020.
   Google Scholar
• Erbay HA, Erbay S, Erkip A. Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations. Nonlinear Anal: Theory Methods Appl. 2014;95:313–322.
   Web of Science ®Google Scholar
• Cattani C, Sulaiman TA, Baskonus HM, et al. Solitons in an inhomogeneous Murnaghan’s rod. Eur Phys J Plus. 2018;133(6):228.
   Web of Science ®Google Scholar
• Silambarasan R, Baskonus HM, Bulut H. Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan’s rod. Eur Phys J Plus. 2019;134(3):125.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Slovak Republic: Academic Press, 1999.
   Google Scholar
• He J-H, Elagan SK, Li ZB. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys Lett A. 2012;376(4):257–259.
   Web of Science ®Google Scholar
• Guner O, Bekir A. On the concept of exact solution for nonlinear differential equations of fractional-order. Math Methods Appl Sci. 2016;39(14):4035–4043.
   Web of Science ®Google Scholar
• Abbott S, Weisstein EW. CRC concise encyclopedia of mathematics. Math Gazette. 2007;84(501):549.
   Google Scholar