In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: β , and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark-bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

Department of Computer Science and Mathematics Lebanese American University Beirut 11022801 Lebanon

Department of Mathematics Art and Science Faculty Siirt University 56100 Siirt Turkey

Department of Mathematics Ghazni University Ghazni Afghanistan

Department of Mathematics University of Engineering and Technology Lahore Pakistan

IT4Innovations VSB Technical University of Ostrava Ostrava Czech Republic

Zobrazit více v PubMed

Miah, M. M. et al. Abundant closed form wave solutions to some nonlinear evolution equations in mathematical physics. J. Ocean Eng. Sci.5(3), 269–278 (2020). PubMed PMC

El-Shiekh, R. M. et al. Solitary wave solutions for the variable-coefficient coupled nonlinear Schrodinger equations and Davey–Stewartson system using modified sine-Gordon equation method. J. Ocean Eng. Sci.5(2), 180–185 (2020).

Shallal, M. A. et al. Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method. J. Ocean Eng. Sci.5(3), 223–229 (2020).

Parto-Haghighi, M. & Manafian, J. Solving a class of boundary value problems and fractional Boussinesq-like equations with [Image: see text]-derivatives by fractional-order exponential trial functions. J. Ocean Eng. Sci.5(3), 197–204 (2020).

Miah, M. M. et al. Further investigations to extract abundant new exact traveling wave solutions of some NLEEs. J. Ocean Eng. Sci.4(4), 387–394 (2019).

Vithya, A. & Rajan, M. M. Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients. J. Ocean Eng. Sci.5(3), 205–213 (2020).

Hosseini, K. et al. Invariant subspaces, exact solutions and stability analysis of nonlinear water wave equations. J. Ocean Eng. Sci.5(1), 35–40 (2020).

Bulut, H. et al. New solitary wave structures to the (3+ 1) dimensional Kadomtsev–Petviashvili and Schrodinger equation. J. Ocean Eng. Sci.4(4), 373–378 (2019).

Raza, N. & Javid, A. Optical dark and singular solitons to the Biswas–Milovic equation in nonlinear optics with spatio-temporal dispersion. Optik158, 1049–57 (2018).

Jhangeer, A. et al. Construction of traveling waves patterns of (1+ n)-dimensional modified Zakharov Kuznetsov equation in plasma physics. Results Phys.19, 103330 (2020).

Raza, N., Jhangeer, A., Rezazadeh, H. & Bekir, A. Explicit solutions of the (2+1)-dimensional Hirota–Maccari system arising in nonlinear optics. Int. J. Mod. Phys. B33, 1950360 (2019).

Gunerhan, H. et al. Exact optical solutions of the (2+ 1) dimensions Kundu–Mukherjee–Naskar model via the new extended direct algebraic method. Mod. Phys. Lett. B34(22), 2050225 (2020).

Park, C. et al. On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering. Alex. Eng. J.59(3), 1099–1105 (2020).

Bekir, A. & Guner, O. Topological (dark) soliton solutions for the Camassa–Holm type equations. Ocean Eng.74, 276–279 (2013).

Lu, D. et al. Applications of extended simple equation method on unstable nonlinear Schrodinger equations. Optik140, 136–144 (2017).

Biswas, A. et al. Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method. Optik157, 1235–1240 (2018).

Guner, O. & Bekir, A. Solving nonlinear space-time fractional differential equations via ansatz method. Comput. Method. Diff. Equ.6(1), 1–11 (2018).

Kumar, D. et al. The sine–Gordon expansion method to look for the traveling wave solutions of the Tzitzeica type equations in nonlinear optics. Optik149, 439–446 (2017).

Milici, C. et al.Introduction to Fractional Differential Equations (Springer, Cham, 2018).

Machado, J. T. et al. Recent history of the fractional calculus: Data and statistics. Handb. Fract. Calc. Appl.1, 1–21 (2019).

Khalil, R. et al. A new definition of fractional derivative. J. Comput. Appl. Math.264, 65–70 (2014).

Scott, A. C. Encyclopedia of Nonlinear Science (Routledge, Beijing, 2005).

Sousa, J. V. et al. A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties. Int. J. Anal. Appl.16, 83–96 (2018).

Bibi, S. et al. Some new exact solitary wave solutions of the van der Waals model arising in nature. Results Phys.9, 648–655 (2018).

Lu, D. et al. Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods. Results Phys.7, 2028–2035 (2017).

Seadawy, A. R. Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev–Petviashvili dynamical equation for dispersive shallow-water waves. Eur. Phys. J. Plus132(1), 1–13 (2017).

Hussain, A. et al. Optical solitons of fractional complex Ginzburg-Landau equation with conformable, beta, and M-truncated derivatives: A comparative study. Adv. Diff. Equ.2020, 1–19 (2020).

Rezazadeh, H. New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik167, 218–227 (2018).

Ghanbari, B., Osman, M. S. & Baleanu, D. Generalized exponential rational function method for extended Zakharov Kuzetsov equation with conformable derivative. Mod. Phys. Lett. A34, 1950155 (2019).

AkgAl, A. et al. Approximate solutions to the conformable Rosenau–Hyman equation using the two-step Adomian decomposition. Math. Methods Appl. Sci.43(13), 7632–7639 (2020).

Huy Tuan, N. et al. On well-posedness of the sub-diffusion equation with conformable derivative model. Commun. Nonlinear Sci. Numer. Simul.89, 105332 (2020).

Atangana, A., Baleanu, D. & Alsaedi, A. New properties of conformable derivative. Open Math.13, 889–898 (2015).

Abdeljawad, T. On conformable fractional calculus. J. Comput. Appl. Math.279, 57–66 (2015).

Jhangeer, A. et al. New complex waves of perturbed Schrdinger equation with Kerr law nonlinearity and Kundu–Mukherjee–Naskar equation. Results Phys.16, 102816 (2020).

Argentina, M., Clerc, M. G. & Soto, R. Van der Waals-like transition in fluidized granular matter. Phys. Rev. Lett.89(4), 044301 (2002). PubMed

Clerc, M. G. & Escaff, D. Solitary waves in van der Waals-like transition in fluidized granular matter. Phys. A371(1), 33–36 (2006). PubMed