Home Technology The expa function method and the conformable time-fractional KdV equations
Article Open Access

The expa function method and the conformable time-fractional KdV equations

  • Asim Zafar EMAIL logo
Published/Copyright: July 12, 2019
Download Article (PDF)
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Nonlinear Engineering
From the journal Nonlinear Engineering Volume 8 Issue 1

Abstract

The nonlinear fractional differential equations (FDEs) are produced by mathematical modelling of some nonlinear physical systems. The study of such nonlinear physical models through wave solutions analysis corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, we are going to explore the conformable time-fractional KdV equations using the expa function method. The way to reach explicit exact wave solutions is to transform the fractional order PDE into a nonlinear ODE of discrete order through travelling wave transforms. The subsequent equation has been explored by utilizing the exp a function approach. Consequently, some new explicit exact wave solutions of the said equations are effectively formulated and graphically conveyed with the help of numerical simulation.

Keywords: ime-fractional KdV and time-fractional modified KdV equations; Conformable derivatives; exp a function method; Explicit exact solutions

1 Introduction

The Korteweg– de Vries (KdV) equations have been observed in a broad variety of material science phenomena as a model for the development and communication of nonlinear waves. The KdV equation was introduced by Korteweg and de Vries to describe shallow water waves of long wavelength with small amplitude. Subsequently the KdV equation has been merged into a number of other physical contexts as collision-free hydromagnetic waves, stratified internal waves, particle acoustic waves, plasma physics, and so on [1, 2, 3, 4, 5, 6, 7].

Various fields of science and engineering are influenced by Leibnitz’s work on fractional calculus and having substantially increasing impact on these sciences during last twenty years ostensibly [8, 9, 10]. From different minds for different applications, many definitions of fractional derivatives, Like Hilfer, Riemann-Liouville, Caputo form and so on, have been introduced in the literature [11, 12], but the most fascinating definition and the geometrical explanation for complex fractional transform and fractional derivative are given in [13, 14].

The explicit exact wave solutions have always been a particular importance among the researchers in many fields of nonlinear sciences. The availability of symbolic computation softwares is a direct help to minimize the manual labor for finding problematic solutions to nonlinear evolution equations. Various significant methods have been presented to explore exact solutions in different research articles, for example, the ansatz method, modified simple equation method, the extended trail equation, the first integral, the improved tan(ϕ(η)/2)-expansion, the exp-function and the exp(–ϕ(ϵ)) methods [15, 16, 17, 18, 19, 20, 21]. Also some other reliable techniques, like homotopy method [22, 23, 24, 25], generalized Kudryashov method [26, 27, 28, 29], modified Kudryashov and the sine-Gordon expansion approach [30, 31, 32, 33] have been done by different researchers. Furthermore, the auxiliary equation method, the extended tanh-function method, generalized projective Riccati equation method, functional variable method, and Jacobi elliptic function expansion method have been explored in [34, 35, 36, 37, 38, 39] for integral and fractional order as well. The expa function method is a new and an efficient technique which has been acknowledged by the scholars rapidly. In particular, Ali and Hassan, Hosseini et al., Zayed and Al-Nowehy all have utilized the expa function method in [40, 41, 42, 43] respectively.

The paper’s aim is to apply the expa function approach to nonlinear conformable time-fractional KdV equations. The investigation for exact solutions is as follow. The description of conformable derivative and expa function approach is provided in section 2. The section 3 represents the method utilization for extracting new exact solutions with their graphical representation. The conclusion is provided in the last section.

2 A new approach for fractional derivatives

A simple but a fascinating definition of the fractional derivative called conformable fractional derivative has been introduced by Khalil et al. [13]. The Liebniz rule and the chain rule both are obeyed by this derivative. Recall the definition of the conformable derivative and with some properties.

Definition 1

Suppose h : ℝ+ → ℝ be a function. Then, for all t > 0,

Dty(h(t))=limε→0h(t+εt1−y)−h(t)ε

is known as the conformable fractional derivative of h of order y, 0 < y ≤ 1. Some useful properties are being listed as follows:

Dty(a h+b g)=aDty(h)+bDty(g), for all a, b ∈ ℝ

Dty(h g)=h Dty(g)+g Dty(h)

Let h : ℝ+ → ℝ be a differentiable and y-differentiable function, g be a differentiable function defined in the range of h.

Dty(h∘g(t))=t1−y g′(t) h′(g(t)).

On the top of that, the following rules hold.

Dty(tp) = ptp–y, for all p ∈ ℝ

Dty(λ) = 0, where λ is constant.

Dty(h/g)=gDty(h)−hDty(g)g2.

Conjointly, if h is differentiable, then Dty(h(t))=t1−ydh(t)dt.

2.1 A transitory explanation of expa function method

The present subsection provides a concise explanation for the expa [40, 42, 44] in generating new explicit exact solutions of nonlinear conformable time-fractional DE. For this purpose, suppose that we have a nonlinear conformable time-fractional DE that can be presented in the form

F(u,Dty u,Dxu,Dt2y u,Dxxu,...)=0(1)

The FDE (1) can be changed into the following nonlinear ODE of integer order

P(U,U′,U″,...,)=0(2)

with the use of following transformation

u(x,t)=U(ξ),ξ=kx−ltyy,(3)

where k, l are nonzero arbitrary constants.

Let us try to search a non-trivial solution for the Eq. (2) in the following form

U(ξ)=a0+a1aξ+...+aNaNξb0+b1aξ+...+bNaNξ(4)

where ai and bi for (0 ≤ i ≤ N) are found later and N is a free positive integer.

Replacing the Eq.(4) in the nonlinear Eq.(2), yields

P(aξ)=q0+q1aξ+...+qτaτξ=0(5)

Setting li(0 ≤ i ≤ τ) in Eq.(4) to be zero, results in a set of nonlinear equations as below

qi=0,  i=0,...,τ(6)

which by solving the generated set (6), we approach to non-trivial solutions of the nonlinear FDE (1).

3 Explicit exact wave solutions for time fractional KdV equations

In this section, the following time fractional KdV and modified KdV equations [7], are going to be considered for solution via expa method.

Dtyu+pu2ux+quxxx=0,  0 <y≤ 1,(7)

Using the transformation (3), integrating w.r.t. ξ and taking constant of integration is zero, we get

−lu+13pku3+qk3u″=0.(8)

Through balancing, we select N = 1, the nontrivial solution (4) reduces to:

U(ξ)=α1αξ+α0β1αξ+β0,   α≠1(9)

By setting the above solution in reduced equation Eq. (8) and equating factors of each power of aξ in the resulting equation, we reach a nonlinear algebraic set as

α03kp−3α0β02l=03α1β02k3qlog2⁡(α)−3α0β0β1k3qlog2⁡(α)+3α02α1kp−3α1β02l−6α0β0β1l=03α0β12k3qlog2⁡(α)−3α1β0β1k3qlog2⁡(α)+3α0α12kp−3α0β12l−6α1β0β1l=0α13kp−3α1β12l=0

which its solution yields

α0=∓−3q2pβ0klog⁡(α), α1=±−3q2pβ1klog⁡(α),l=−12k3qlog2⁡(α)

Thus, the following new explicit exact solutions to the conformable time fractional third order modified KdV equation can be written as

U1,2(ξ)=ıklog⁡(α)3q2p(∓β0±β1αξ)β0+β1αξ(10)

where ξ=kx+12k3qlog2⁡(α)tyy.

Fig. 1 Solution profile of U1,2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
Fig. 1

Solution profile of U1,2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3

We now consider the time fractional conformable KdV Equation as follows.

Dtyu+puux+quxxx=0,  0 <y≤ 1,(11)

Using the transformation (3), integrating w.r.t. ξ and taking constant of integration is zero, we get

−lu+12pku2+qk3u″=0.(12)

Through balancing the terms we select N = 2, the non-trivial solution (4) becomes

U(ξ)=α2α2ξ+α1αξ+α0β2α2ξ+β1αξ+β0,   α≠1(13)

By setting the above non-trivial solution in reduced equation Eq. (12) and equating factors of each power of αξ in the resulting equation, we reach a nonlinear algebraic set as

α02β0kp−2α0β02l=0,2α1β02k3qlog2⁡(α)−2α0β0β1k3qlog2⁡(α)+2α0α1β0kp+α02β1kp−2α1β02l−4α0β0β1l=0,8α2β02k3qlog2⁡(α)+2α0β12k3qlog2⁡(α)−2α1β0β1k3qlog2⁡(α)−8α0β0β2k3qlog2⁡(α)+α12β0kp+2α0α2β0kp+2α0α1β1kp+α02β2kp−2α2β02l−2α0β12l−4α1β0β1l−4α0β0β2l=0,6α2β0β1k3qlog2⁡(α)−12α1β0β2k3qlog2⁡(α)+6α0β1β2k3qlog2⁡(α)+2α1α2β0kp+α12β1kp+2α0α2β1kp+2α0α1β2kp−2α1β12l−4α2β0β1l−4α1β0β2l−4α0β1β2l=0,2α2β12k3qlog2⁡(α)+8α0β22k3qlog2⁡(α)−8α2β0β2k3qlog2⁡(α)−2α1β1β2k3qlog2⁡(α)+α22β0kp+2α1α2β1kp+α12β2kp+2α0α2β2kp−2α2β12l−2α0β22l−4α2β0β2l−4α1β1β2l=0,2α1β22k3qlog2⁡(α)−2α2β1β2k3qlog2⁡(α)+α22β1kp+2α1α2β2kp−2α1β22l−4α2β1β2l=0,α22β2kp−2α2β22l=0.

which its solution yields

α0=−2β0k2qlog2⁡(α)p, α1=4β1k2qlog2⁡(α)p,α2=−β12k2qlog2⁡(α)2β0p, β2=β124β0, l=−k3qlog2⁡(α),α0=0, α1=6β1k2qlog2⁡(α)p, α2=0,β2=β124β0, l=k3qlog2⁡(α)

Thus, the following new explicit exact solution to the conformable time fractional third order KdV equation can be written as

U1(ξ)=−β12k2qlog2⁡(α)2β0pα2ξ+4β1k2qlog2⁡(α)pαξ−2β0k2qlog2⁡(α)pβ124β0α2ξ+β1αξ+β0(14)

where ξ=kx+k3qlog2⁡(α)tyy.

U2(ξ)=6β1k2qlog2⁡(α)pαξβ124β0α2ξ+β1αξ+β0(15)

where ξ=kx−k3qlog2⁡(α)tyy.

Fig. 2 Solution profile of U1 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
Fig. 2

Solution profile of U1 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3

Fig. 3 Solution profile of U2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
Fig. 3

Solution profile of U2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3

4 Conclusion

We have succeeded in exploring the explicit exact wave solutions of conformable time fractional KdV equations via expa function method. These solutions are given explicitly and verified by setting back in the reduced equations with the aid of Mathematica. Furthermore, the numerical simulation of some but different kind of solutions through MATLAB has been given for the reader to visualize the basic phenomena.

References

[1] Dr. D. J. Korteweg and Dr. G. de Vries. Xli. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39(240):422–443, 1895.10.1080/14786449508620739Search in Google Scholar

[2] Hans Schamel. A modified korteweg-de vries equation for ion acoustic wavess due to resonant electrons. Journal of Plasma Physics, 9(3):377–387, 1973.10.1017/S002237780000756XSearch in Google Scholar

[3] Zuntao Fu, Shida Liu, and Shikuo Liu. New solutions to mkdv equation. Physics Letters A, 326(5):364 – 374, 2004.10.1016/j.physleta.2004.04.059Search in Google Scholar

[4] Abdul-Majid Wazwaz. New sets of solitary wave solutions to the kdv, mkdv, and the generalized kdv equations. Communications in Nonlinear Science and Numerical Simulation, 13(2):331 – 339, 2008.10.1016/j.cnsns.2006.03.013Search in Google Scholar

[5] SR Mousavian, H Jafari, CM Khalique, and SA Karimi. New exact-analytical solutions for the mkdv equation. TJMCS, 2(3):413–416, 2011.10.22436/jmcs.02.03.02Search in Google Scholar

[6] S. Saha Ray. Soliton solutions for time fractional coupled modified kdv equations using new coupled fractional reduced differential transform method. Journal of Mathematical Chemistry, 51(8):2214–2229, 2013.10.1007/s10910-013-0210-3Search in Google Scholar

[7] S. Sahoo and S. Saha Ray. Solitary wave solutions for time fractional third order modified kdv equation using two reliable techniques (G′G)-expansion method and improved (G′G)-expansion method. PhysicaA, 448:265–282, 2016.10.1016/j.physa.2015.12.072Search in Google Scholar

[8] G. Samko, A.A. Kilbas, and Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, 1993.Search in Google Scholar

[9] A. Kilbas, M. H. Srivastava, and J. J. Trujillo. Theory and application of fractional differential equations. North Holland Mathematics Studies, 204, 2006.Search in Google Scholar

[10] Kenneth S Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. 1993.Search in Google Scholar

[11] Guy Jumarie. Modified riemann-liouville derivative and fractional taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9-10):1367–1376, 2006.10.1016/j.camwa.2006.02.001Search in Google Scholar

[12] Michele Caputo and Mauro Fabrizio. A new definition of fractional derivative without singular kernel. Progress in fractional differentiation and applications, 1(2):73 – 85, 2015.Search in Google Scholar

[13] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh. A new deflation of fractional derivative. J.Comput.Appl.Math., 264:65–70, 2014.10.1016/j.cam.2014.01.002Search in Google Scholar

[14] Ji-Huan He, S.K. Elagan, and Z.B. Li. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A, 376(4):257 – 259, 2012.10.1016/j.physleta.2011.11.030Search in Google Scholar

[15] Kamyar Hosseini, Peyman Mayeli, and Reza Ansari. Bright and singular soliton solutions of the conformable time-fractional klein–gordon equations with different nonlinearities. Waves in Random and Complex Media, 28(3):426–434, 2018.10.1080/17455030.2017.1362133Search in Google Scholar

[16] M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, and A. Biswas. Optical solitons perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Optik, 127(22):10659–10669, 2016.10.1016/j.ijleo.2016.08.076Search in Google Scholar

[17] Yücel Çenesiz, Dumitru Baleanu, Ali Kurt, and Orkun Tasbozan. New exact solutions of burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27(1):103–116, 2017.10.1080/17455030.2016.1205237Search in Google Scholar

[18] Mostafa Eslami and Hadi Rezazadeh. The first integral method for wu–zhang system with conformable time-fractional derivative. Calcolo, 53(3):475–485, 2016.10.1007/s10092-015-0158-8Search in Google Scholar

[19] Kamyar Hosseini, Jalil Manafian, Farzan Samadani, Mohammadreza Foroutan, Mohammad Mirzazadeh, and Qin Zhou. Resonant optical solitons with perturbation terms and fractional temporal evolution using improved tan(ϕ(η)/2)-expansion method and exp function approach. Optik, 158:933 – 939, 2018.10.1016/j.ijleo.2017.12.139Search in Google Scholar

[20] Orkun Tasbozan, Yücel Çenesiz, Ali Kurt, and Dumitru Baleanu. New analytical solutions for conformable fractional pdes arising in mathematical physics by exp-function method. Open Physics, 15(1):647–651.10.1515/phys-2017-0075Search in Google Scholar

[21] K. Hosseini, D. Kumar, M. Kaplan, and E.Y. Bejarbaneh. New exact traveling wave solutions of the unstable nonlinear schrödinger equations. Commun. Theor. Phys., 68:761–767, 2017.10.1088/0253-6102/68/6/761Search in Google Scholar

[22] Ali Kurt, Orkun Tasbozan, and Yucel Cenesiz. Homotopy analysis method for conformable burgers–korteweg-de vries equation. Bull. Math. Sci. Appl, 17:17–23, 2016.10.18052/www.scipress.com/BMSA.17.17Search in Google Scholar

[23] A Kurt, O Tasbozan, and D Baleanu. Newsolutions for conformable fractional nizhnik-novikov-veselov system via (G′G)-expansion method and homotopy analysis method. Optical and Quantum Electronics, 49(10):333, 2017.10.1007/s11082-017-1163-8Search in Google Scholar

[24] OS lyiola, O Tasbozan, A Kurt, and Y Çenesiz. On the analytical solutions of the system of conformable time-fractional robertson equations with 1 - d diffusion. Chaos, Solitons & Fractals, 94:1–7, 2017.10.1016/j.chaos.2016.11.003Search in Google Scholar

[25] Ali Kurt, Yücel Çenesiz, and Orkun Tasbozan. On the solution of burgers’ equation with the new fractional derivative. Open Physics, 13(1), 2015.10.1515/phys-2015-0045Search in Google Scholar

[26] Y. Pandir, Y. Gurefe, and E. Misirli. A new approach to kudryashov’s method for solving some nonlinear physical models. Int.J. Phys. Sci., 7:2860–2866, 2012.10.5897/IJPS12.071Search in Google Scholar

[27] M. Kaplan, A. Bekir, and A. Akbulut. A generalized kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dyn., 85:2843–2850, 2016.10.1007/s11071-016-2867-1Search in Google Scholar

[28] K. Hosseini, P. Mayeli, and D. Kumar. New exact solutions of the coupled sine-gordon equations in nonlinear optics using the modified kudryashov method. Journal of Modern Optics, 65(3):361–364, 2018.10.1080/09500340.2017.1380857Search in Google Scholar

[29] K. Hosseini, E. Yazdani Bejarbaneh, A. Bekir, and M. Kaplan. New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Optical and Quantum Electronics, 49(7):241, 2017.10.1007/s11082-017-1070-zSearch in Google Scholar

[30] Alper Korkmaz and Kamyar Hosseini. Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods. Optical and Quantum Electronics, 49(8):278, 2017.10.1007/s11082-017-1116-2Search in Google Scholar

[31] D. Kumar, K. Hosseini, and F. Samadani. The sine-gordon expansion method to look for the traveling wave solutions of the tzitzéica type equations in nonlinear optics. Optik - International Journal for Light and Electron Optics, 149:439 – 446, 2017.10.1016/j.ijleo.2017.09.066Search in Google Scholar

[32] K Hosseini, A Bekir, M Kaplan, and Ö Güner. On a new technique for solving the nonlinear conformable time-fractional differential equations. Optical and Quantum Electronics, 49(11):343, 2017.10.1007/s11082-017-1178-1Search in Google Scholar

[33] Orkun Tasbozan, Mehmet Şenol, Ali Kurt, and Ozan Özkan. Newsolutions of fractional drinfeld-sokolov-wilson system in shallow water waves. Ocean Engineering, 161:62–68, 2018.10.1016/j.oceaneng.2018.04.075Search in Google Scholar

[34] A. Akbulut and K. Kaplan. Auxiliary equation method for time fractional differential equation with conformable derivatives. Computers and Mathematics with Applications, 75:876–882, 2018.10.1016/j.camwa.2017.10.016Search in Google Scholar

[35] A. Bekir. Application of the extended tanh method for coupled nonlinear evolution equation. Commun. Nonlinear Sci., 13:1742–1751, 2008.10.1016/j.cnsns.2007.05.001Search in Google Scholar

[36] Hadi Rezazadeh, Alper Korkmaz, Mostafa Eslami, Javad Vahidi, and Rahim Asghari. Traveling wave solution of conformable fractional generalized reaction duffing model by generalized projective riccati equation method. Optical and Quantum Electronics, 50(3):150, 2018.10.1007/s11082-018-1416-1Search in Google Scholar

[37] Yücel Çenesiz, Orkun Tasbozan, and Ali Kurt. Functional variable method for conformable fractional modified kdv-zk equation and maccari system. Tbilisi Mathematical Journal, 10(1):118–126, 2017.10.1515/tmj-2017-0010Search in Google Scholar

[38] Orkun Tasbozan, Yücel Çenesiz, and Ali Kurt. New solutions for conformable fractional boussinesq and combined kdv-mkdv equations using jacobi elliptic function expansion method. The European Physical Journal Plus, 131(7):244, 2016.10.1140/epjp/i2016-16244-xSearch in Google Scholar

[39] Ali Kurt, Yücel Cenesiz, and Orkun Tasbozan. Exact solution for the conformable burgers’ equation by the hopf-cole transform. Cankaya University Journal of Science and Engineering, 13(2).Search in Google Scholar

[40] A. T. Ali and E. R. Hassan. General expa function method for nonlinear evolution equations. Appl. Math. Comput., 217:451–459, 2010.10.1016/j.amc.2010.06.025Search in Google Scholar

[41] E. M. E. Zayed and A. G. Al-Nowehy. Generalized kudryashov method and general exp a function method for solving a high order nonlinear schrödinger equation. J. Space Explor., 6:1–26, 2017.Search in Google Scholar

[42] K. Hosseini, Z. Ayati, and R. Ansari. New exact solution of the tzitzéica type equations in nonlinear optics using the expa function method. J. Mod. Opt., 65(7):847–851, 2018.10.1080/09500340.2017.1407002Search in Google Scholar

[43] K. Hosseini, A. Zabihi, F. Samadani, and R. Ansari. New explicit exact solutions of the unstable nonlinear schrödinger’s equation using the expa and hyperbolic function methods. Optical and Quantum Electronics, 50(2):82, 2018.10.1007/s11082-018-1350-2Search in Google Scholar

[44] Asim Zafar. Rational exponential solutions of conformable space-time fractional equal-width equations. Nonlinear Engineering, 2018.10.1515/nleng-2018-0076Search in Google Scholar

Received: 2018-06-26
Revised: 2018-09-01
Accepted: 2018-11-06
Published Online: 2019-07-12
Published in Print: 2019-01-28

© 2019 Asim Zafar, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
https://doi.org/10.1515/nleng-2018-0094
Keywords for this article
ime-fractional KdV and time-fractional modified KdV equations; Conformable derivatives; exp a function method; Explicit exact solutions
Creative Commons
BY 4.0

Articles in the same Issue

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Winner of the OpenAthens UX Award 2026
Winner of the OpenAthens UX Award 2026
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 10.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/nleng-2018-0094/html
Scroll to top button