THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

Authors of this Paper

External Links

online first only

A modified exp-function method for fractional partial differential equations

ABSTRACT
This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu's method. The results show that the present method is effective, and can be applied to other fractional differential equations.
KEYWORDS
PAPER SUBMITTED: 2020-04-28
PAPER REVISED: 2020-06-18
PAPER ACCEPTED: 2020-06-18
PUBLISHED ONLINE: 2021-01-31
DOI REFERENCE: https://doi.org/10.2298/TSCI200428017T
REFERENCES
  1. Tian, A.H., et al. Fractional prediction of ground temperature based on soil field spectrum, Thermal Science, 24(2020), 4, pp.2301-2309
  2. Wang, K.L., Yao, S.W. He's fractional derivative for the evolution equation, Thermal Science, 24(2020), 4, pp. 2507-2513
  3. Shen Y, El-Dib, Y.O. A periodic solution of the fractional sine-Gordon equation arising in architectural engineering, Journal of Low Frequency Noise Vibration and Active Control, 2020 DOI: 10.1177/1461348420917565
  4. He, J.H., The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise Vibration and Active Control, (38) 2019: pp. 1252 - 1260
  5. He, J.H. Exp - function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 14(2013), 6, pp. 363 - 366
  6. Ji, F.Y., et al. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Applied Mathematical Modelling , 82(2020) , June, pp. 437 - 448
  7. He, J.H., et al. Difference equation vs differential equation on different scales, International Journal of Numerical Methods for Heat and Fluid Flow. 2020, DOI: 10.1108/HFF - 03 - 2020 - 0178
  8. Zhang , S., et al. Simplest exp-function method for exact solutions of Mikhauilov-Novikov - Wang equation, Therm al Science , 23(2019), 4, pp.2381 - 2388
  9. Yang, Y.-J. The fractional residual method for solving the local fractional differential equations, Thermal Science, 2020 24(4):2535-2542
  10. He, J.H., Homotopy perturbation method: a new nonlinear analytical technique , Applied Mathematics and Computation , 1 35(2003),1,pp.73 - 79
  11. Yu , D.N., et a l. Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators , Journal of Low Frequency Noise Vibration and Active Control, (38 ) 2019 , 3 - 4, pp. 1540 - 15 54
  12. Kuang, W.X., et al. Homotopy perturbation method with an auxiliary term for the optimal design of a tangent nonlinear packaging system , Journal of Low Frequency Noise Vibration and Active Control, (38 ) 2019 , 3 - 4, pp. 1075 - 108 0
  13. He, J . H ., Latifizadeh , H. A general numerical algorithm for nonlinear differential equations by the variational iteration method, International Journal of Numerical Methods for Heat and Fluid Flow , 2020, DOI :10.1108/HFF - 01 - 2020 - 0029
  14. Anjum, N., He, J.H. Laplace transform: Making the variational iteration method easier , Applied Mathematics Letters, 92(2019), Jun. , pp. 134 - 138
  15. Yang Yong-Ju. The local fractional variational iteration method a promising technology for fractional calculus, Thermal Science, 2020 24(4):2605-2614
  16. He, J.H., Notes on the optimal variational iteration method , Applied Mathematics Letters , 25(2012), 10, pp.1579 - 158 1
  17. He, J. H., J in, X., A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube, Mathematical Methods in the Applied Sciences, 2020 , Article DOI: 10.1002/mma.6321 , http: //dx.doi.org/10.1002/mma.6321
  18. He, J. H., A short re view on analytica l methods for to a fully fourth - order nonlinear integral boundary value problem with fractal derivatives , International Journal of Numerical Methods for H eat and Fluid Flow, 2020, DOI (10.1108/HFF - 01 - 2020 - 0060)
  19. Tariq, H., Akram, G ., New approach for exact solutions of time fractional Cahn-Allen equation and time fractional Phi - 4 equation, Physics A , 473 (201 7 ) ,pp.3 52 - 362
  20. Hosseini , K ., et al. , New exact travelling wave solutions of the Tzitzeica type equations using a novel exponential rational function method , Optic , 148 (201 7 ) ,pp. 85 - 89
  21. Wu , W . T ., Mathematics Mechanization , Science Press , Beijing , 2000
  22. He, J.H., et al. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus , Physics Letters A , 376(2012), 4, pp. 257 - 259
  23. H e, J.H., Li, Z.B. Converting fractional differential equations into partial differential equation, Thermal Science, 16(2012), 2, pp. 331 - 33 4
  24. L i, Z.B., et al. Exact solution of time - fractional heat conduction equation by the fractional complex transform, Thermal Science , 16 (2012), 2 , pp. 335 - 338 出版年 : 2012
  25. He , J . H . , Ain , Q . T . New promises and future challenges of fractal calculus: from two - scale Thermodynamics to fractal variational principle, Thermal Science, 24 (2020) , 2A, pp. 659 - 681
  26. He , J.H. , Ji, F.Y. Two - scale mathematics and fractional calculus for thermodynamics, Therm. Sci., 23 (2019) , 4, pp. 2131 - 2133
  27. Ain, Q.T, He, J.H. On two - scale dimension and its applications, Thermal Science, 23(2019), 3B, pp. 1707 - 1712
  28. Wang , K.L. & Yao, S.W. Numerical method for fractional Zakharov-Kuznetsov equations with He's fractional derivative, Thermal Science, 23(2019), 4, pp. 2170 - 2163
  29. He , J.H., et al. A new fractional derivative and its application to explanation of polar bear hairs , Journal of King Saud University Science, 28(201 6), 2, pp. 192 - 190
  30. H e , J.H. & Li, Z.B. A fractional model for dye removal , Journal of Kin g Saud University Science, 28(2016), 1, pp. 16 - 14
  31. H e, J.H. A Tutorial Review on Fractal Spacetime and Fractional Calculus , International Journal of Theoretical Physics, 53(2014), 11, pp. 3718 - 3698
  32. Wang , Q.L., et al. Fractal calculus and its application to explanation of biomechanism of polar hairs (vol 26, 1850086, 2018) , Fractals , 27( 2019), 5, Article Number : 1992001
  33. Wang , Q.L., et al. Fractal calculus and its application to explanation of biomechanism of polar hairs (vol 26, 1850086, 2018) , Fractals , 2 6 (201 8 ), 6 , Article Number : 1850086
  34. He, J. H., Fractal calculus and its geometrical explanation . Results in Physics , 2018, 10: 272 - 276 .
  35. Tian, Y., Wang, K.L. Polynomial characteristic method an easy approach to lie symmetry , Thermal Science, 24( 2020 ), 4, pp. 2629 - 2635
  36. Tian , Y . , Wang , K . - L . Conservation laws for partial differential equations based on the polynomial characteristic method , Thermal Science, 24( 2020 ), 4 , pp. 2529 - 25 34
  37. Zhu, L. The Quenching Behavior for a Quasilinear Parabolic Equation with Singular Source and Boundary Flux, Journal of Dynamical and Control Systems, 25(2019), 4, pp. 519 - 526
  38. Zhu , L . Complete quenching phenomenon for a parabolic p - Laplacian equation with a weighted absorption, Journal of Inequalities and Applications, 248( 2018 ), pp.1 - 16.