Abstract
The fundamental objective of this article is to find exact solutions to the stochastic fractionalspace Allen–Cahn equation, which is derived in the Itô sense by multiplicative noise. The exact solutions to this equation are required since it appears in many discipline areas including plasma physics, quantum mechanics and mathematical biology. The tanh–coth method is used to generate new hyperbolic and trigonometric stochastic and fractional solutions. The originality of this study is that the results produced here expand and improve on previously obtained results. Furthermore, we use Matlab package to display 3D surfaces of analytical solutions derived in this study to demonstrate the effect of stochastic term on the solutions of the stochasticfractionalspace Allen–Cahn equation.
1 Introduction
Fractional derivatives have attracted a lot of attention in recent decades due to their possible applications in a variety of fields, such as finance [1,2,3], biology [4], physics [5,6, 7,8], hydrology [9,10] and biochemistry and chemistry [11]. Since derivatives of fractional order allow the memory and heredity qualities of various substances to be described, these fractionalorder equations are more suited than integerorder equations [12].
On the other hand, random perturbations arise from many natural sources in the practically physical system. They cannot be denied and the presence of noise can lead to some statistical properties and important phenomena. As a result, stochastic differential equations were developed, and they began to play an increasingly significant role in modeling phenomena in chemistry, biology, physics, fluid mechanics, oceanography and atmosphere, etc.
Recently, some related research on approximate solutions of fractional differential equations with stochastic term have been explored such as Liu and Yan [13], Mohammed [14], Zou [15,16], Ahmad et al. [17,18], Li and Yang [19], Kamrani [20] and Taheri et al. [21].
In this article, the fractionalspace Allen–Cahn equation induced by multiplicative noise in the Itô sense is taken into account as follows:
where
When
For
The purpose of this article is to find the exact solutions of the stochastic fractionalspace Allen–Cahn Eq. (1) derived by a onedimensional multiplicative white noise by using the tanh–coth method. Furthermore, we expand and improve on some earlier results. The obtained solutions would be quite useful in explaining certain exciting physical phenomena. This is the first work to provide exact solutions to the stochastic fractionalspace Allen–Cahn Eq. (1). Also, we discuss the effect of stochastic term on the exact solutions of the stochastic fractionalspace Allen–Cahn Eq. (1) by utilizing MATLAB program to plot some graphs.
This article is organized as follows. In Section 2, we define the order
2 Modified Riemann–Liouville derivative and properties
The order
where
Now, let us state some significant properties of modified Riemann–Liouville derivative as follows:
and
3 Wave equation of the Allen–Cahn equation
To derive the wave equation of stochastic fractionalspace Allen–Cahn Eq. (1), we apply the next wave transformation:
where
Substituting (3) into Eq. (1), we get the next ODE:
where we put
Since
In the following, we apply the tanh–coth method to attain the solutions of the wave Eq. (6). And we, therefore, get the exact solutions of the stochastic fractionalspace Allen–Cahn Eq. (1).
4 The exact solutions of the Allen–Cahn equation
To find the exact solutions of the stochastic fractionalspace Allen–Cahn Eq. (1), we are using the tanh–oth method that Malfliet proposed [31]. We define the solution
where
Hence, Eq. (7) takes the form:
Substituting Eqs. (9) into (6) we obtain
Hence,
We have by equating each coefficient of
and
We solve these equations by using Mathematica to obtain five cases as follows:
First case:
The solution of wave Eq. (6) in this case is
Therefore, the stochastic fractionalspace Allen–Cahn Eq. (1) has the exact solution:
or
Second case:
In this case Eq. (6) has solution in the following form:
Consequently, the stochastic fractionalspace Allen–Cahn Eq. (1) has the exact solution:
or
Third case:
In this case Eq. (6) has solution in the following form:
Therefore, the stochastic fractionalspace Allen–Cahn Eq. (1) has the exact solution:
or
Fourth case:
In this case, the solitary wave solution of Eq. (6) is
Consequently, the exact solution of the stochastic fractionalspace Allen–Cahn Eq. (1) is
or
Fifth case:
The solution of Eq. (6) in this case is
Therefore, the exact solution of the stochastic fractionalspace Allen–Cahn Eq. (1) is
or
5 The effect of noise on the solutions of Eq. (1)
Here, we investigate the effect of the noise on the exact solutions of the stochastic fractionalspace Allen–Cahn Eq. (1). To describe the behavior of these solutions, we give various graphical representations. We utilize the MATLAB program to plot some figures for different values of
In Figures 1, 2, and 3, when the intensity of the noise is equal to zero, the surface is less flat, as indicated in the first graph in the table. However, when noise appears and the strength of the noise grows (
6 Conclusion
By using the tanh–coth method, we derived the exact solutions of the stochastic fractionalspace Allen–Cahn equation driven in the Itô sense by multiplicative noise. Furthermore, we expanded and enhanced several results, such as those mentioned in refs [22,26]. These solutions play a key role in understanding some fascinating complicated physical phenomena. Finally, we showed the effect of stochastic term on the exact solutions of the stochastic fractionalspace Allen–Cahn equation by using MATLAB package to plot some graphs.

Funding information: This research has been funded by Scientific Research Deanship at University of Ha’il – Saudi Arabia through project number RG21001.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors declare no conflict of interest.
References
[1] Gorenflo R, Mainardi F. Random walk models for spacefractional diffusion processes. Fract Calc Appl Anal. 1998;1:167–91. 10.1016/S03784371(99)000825Search in Google Scholar
[2] Raberto M, Scalas E, Mainardi F. Waitingtimes and returns in highfrequency financial data: an empirical study. Phys A Stat Mech Appl. 2002;314:749–55. 10.1016/S03784371(02)010488Search in Google Scholar
[3] Wyss W. The fractional BlackScholes equation. Fract Calculus Appl Anal. 2000;3:51–61. Search in Google Scholar
[4] Yuste SB, Lindenberg K. Subdiffusionlimited A+A reactions. Phys Rev Lett. 2001;87:118301. 10.1002/9783527622979.ch13Search in Google Scholar
[5] Ahmad H, Khan TA. Variational iteration algorithmI with an auxiliary parameter for wavelike vibration equations. J Low Frequency Noise Vibrat Active Control. 2019;38(3–4):1113–24. 10.1177/1461348418823126Search in Google Scholar
[6] Ahmad H, Seadawy AR, Khan TA, Thounthong P. Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J Taibah Univ Sci. 2020;14(1):346–58. 10.1080/16583655.2020.1741943Search in Google Scholar
[7] Saichev AI, Zaslavsky GM. Fractional kinetic equations: solutions and applications. Chaos. 1997;7:753–64. 10.1063/1.166272Search in Google Scholar
[8] Zaslavsky GM. Chaos, fractional kinetics and anomalous transport. Phys Rep. 2002;6:461–580. 10.1016/S03701573(02)003319Search in Google Scholar
[9] Benson DA, Wheatcraft SW, Meerschaert MM. The fractionalorder governing equation of Lévy motion. Water Resour Res. 2000;36:1413–23. 10.1029/2000WR900032Search in Google Scholar
[10] Liu F, Anh V, Turner I. Numerical solution of the space fractional FokkerPlanck equation. J Comput Appl Math. 2004;166:209–19. 10.1016/j.cam.2003.09.028Search in Google Scholar
[11] Yuste SB, Acedo L, Lindenberg K. Reaction front in an A+B→C reactionsubdiffusion process. Phys Rev E. 2004;69:036126. 10.1103/PhysRevE.69.036126Search in Google Scholar PubMed
[12] Podlubny I. Fractional differential equations. New York: Academic Press; 1999. Search in Google Scholar
[13] Liu J, Yan L. Solving a nonlinear fractional stochastic partial differential equation with fractional noise. J Theor Probab. 2016;29:307–47. 10.1007/s1095901405784Search in Google Scholar
[14] Mohammed WW. Approximate solutions for stochastic timefractional reactiondiffusion equations with multiplicative noise. Math Meth Appl Sci. 2021;44(2):2140–57. 10.1002/mma.6925Search in Google Scholar
[15] Zou G. Galerkin finite element method for timefractional stochastic diffusion equations. Comput Appl Math. 2018;37(4):877–4898. 10.1007/s4031401806093Search in Google Scholar
[16] Zou G. A Galerkin finite element method for timefractional stochastic heat equation. Comput Math Appl. 2018;75(11):4135–50. 10.1016/j.camwa.2018.03.019Search in Google Scholar
[17] Ahmad H, Alam N, Omri M. New computational results for a prototype of an excitable system. Results in Physics. Results Phys. 2021;28:104666. 10.1016/j.rinp.2021.104666. Search in Google Scholar
[18] Ahmad H, Alam N, Rahim A, Alotaibi MF, Omri M. The unified technique for the nonlinear timefractional model with the betaderivative. Results in Physics. 2021;29:104785. 10.1016/j.rinp.2021.104785. Search in Google Scholar
[19] Li X, Yang X. Error estimates of finite element methods for stochastic fractional differential equations. J Comput Math. 2017;35:346–62. 10.4208/jcm.1607m20150329Search in Google Scholar
[20] Kamrani M. Numerical solution of stochastic fractional differential equations. Numer Algor 2015;68:81–93. 10.1007/s1107501498397Search in Google Scholar
[21] Taheri Z, Javadi S, Babolian E. Numerical solution of stochastic fractional integrodifferential equation by the spectral collocation method. J Comput Appl Math. 2017;321:336–47. 10.1016/j.cam.2017.02.027Search in Google Scholar
[22] Mohammed WW, Ahmad H, Hamza AE, ALy ES, ElMorshedy M. The exact solutions of the stochastic GinzburgLandau equation. Results Phys. 2021;23:103988. 10.1016/j.rinp.2021.103988Search in Google Scholar
[23] Bekir A. Multisoliton solutions to Cahn–Allen equation using double expfunction method. Phys Wave Phenom. 2012;20(2):118–21. 10.3103/S1541308X12020045Search in Google Scholar
[24] Taghizadeh N, Mirzazadeh M, Samiei AP, Vahidi J. A exact solutions of nonlinear evolution equations by using the modified simple equation method. Ain Shams Eng J. 2012;3:321–5. 10.1016/j.asej.2012.03.010Search in Google Scholar
[25] Hariharan G. Haar wavelet method for solving Cahn–Allen equation. Appl Math Sci. 2009;3:2523–33. Search in Google Scholar
[26] Wazwaz AM The tanh method for travelling wave solutions of nonlinear equations. Appl Math Comput. 2004;154:714–23. 10.1016/S00963003(03)007458Search in Google Scholar
[27] Tascan F, Bekir A. Travelling wave solutions of the Cahn–Allen equation by using first integral method. Appl Math Comput. 2009;207:279–82. 10.1016/j.amc.2008.10.031Search in Google Scholar
[28] Jumarie G. Modified RiemannLiouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput Math Appl. 2006;51:1367–76. 10.1016/j.camwa.2006.02.001Search in Google Scholar
[29] He JH, Elegan SK, Li ZB. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys Lett A. 2012;376:257–9. 10.1016/j.physleta.2011.11.030Search in Google Scholar
[30] Aksoy E, Kaplan M, Bekir A. Exponential rational function method for spacetime fractional differential equations. Waves Random Complex Media. 2016;26(2):142–51. 10.1080/17455030.2015.1125037Search in Google Scholar
[31] Malfliet W, Hereman W. The tanh method. I. Exact solutions of nonlinear evolution and wave equations. Phys Scr. 1996;54:563–8. 10.1088/00318949/54/6/003Search in Google Scholar
© 2022 Sahar Albosaily et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.