In the present article, the first step is devoted to develop some results about existence and uniqueness of solution to a general problem of fractional order partial differential equations (FPDEs) via classical fixed point theory. In the second step, a novel technique is used to handle the semi-analytical approximate solution for the considered general problem. Then, we extend the said result to fractional order partial Benney equations (FOPBEs) of the second and third order, which are special cases of the general problem we considered. We study the proposed problem under the Caputo-Febrizo fractional derivative (CFFD). With the help of the proposed method, we derive a series type approximate (semi-analytical) solution. Some numerical interpretations and visualizations are also given.
The subject of arbitrary order calculus has been gotten tremendous attraction of researchers in last many years. This branch was founded in the 17th century by Newton and Leibnitz. But due to some complexities in the nature of derivative of arbitrary order, the formal investigation was started in 1819 when for the first time the mentioned derivative was defined for a simple power function, and for detail, see ref. . Later, the respective derivative was defined by various researchers in number of ways in which the definition of Riemann–Liouville, Hadamard, Grownwälad, Fourier, etc. were greatly adopted; see refs. [2,3,4]. In 1967, Caputo modified the Riemann–Liuville definition slightly and the new definition he called Caputo derivative . The mentioned derivative was greatly used in studding of applied problems of biological and physical models, see refs. [6,7,8, 9,10,11] and many others in fractional models and nonlinear dynamical system [12,13, 14,15,16, 17,18].
Fractional calculus are increasingly used by mathematicians for mathematical modeling of most of the real-world problems. Often it cause difficulties in treating fractional derivatives involving singular kernel. To omit this difficulties, recently, Caputo and Fabrizio have defined a new fractional order derivative with non-singular kernel [19,20,22]. Interesting observation is that their fractional integral is the fractional average of the Riemann–Liouville fractional integral of the given function and the function itself. In addition to the aforementioned benefits, the derivative was found very useful in thermal science, material sciences, etc. (see refs. [23,24,25]).
Inspired from the aforesaid work, we investigate nonlinear FOPDE under the aforesaid derivative. To the best of our knowledge, the FOPBE has not yet investigated under the said derivative. To study the considered problem, we utilize Laplace transform coupled with the Adomian decomposition method to form an hybrid method abbreviated as (LADM). In this regard, to investigate a physical problem first, we need to derive its existence. Therefore, we first developed some necessary conditions that guarantee the existence of solution of the considered problem by using some results from the nonlinear analysis. In this view, we first developed sufficient conditions for existence criteria of solution to the given general problem:
where stands for CFFD and , . For the existence theory, we use the classical fixed theory of Krasnoselskii’s and Banach. Then, we come across the solution for the FOPBE under the CFFD.
Classical PBE is provided  as follows:
where is the positive constant called characterizing dispersion and is positive integer, . PBEs describes long waves progression in various problems in fluid dynamics.
Corresponding to , the PBE has been investigated by extended homotopy perturbation method  given by
Here, we remark that Eq. (2) is utilized in plasma for the description of the long waves on a viscous fluid flowing down an inclined plane and the unstable drift waves. If , the said PDE reduces to chemical reaction equations, which is increasingly used in chemical mathematics. Further, if , one obtains
Eq. (3) has many applications in solitons theory, computational fluid mechanics, dynamics, and physics; see refs. [26,27,28,29]. The mentioned equations have been investigated by Wavelets for ordinary fractional derivative in ref. . Therefore, in this article, we study the following FOPBEs in nonsingular derivative for order as well as given by
where . With the help of the proposed technique, we derive the approximate analytical solution to Eqs. (4) and (5). Also some existence results are provided via using the nonlinear analysis. Graphical representations are also provided.
 Let , , and then, the CF derivative is recalled as follows:
where is the normalization function with .
 Let , then CF integral of is recalled as follows:
For some functions, we have 
If is constant then,
If , then = .
If , then
If , then
 The Laplace transform of any function with is given by
Further, if we take , then the Laplace transform of any function is given as follows:
Let define space as with the norm defined by . For existence of at least one result as a solution of the problem (1), we recall the given theorem.
 If be closed nonempty and convex subset, two operators with the conditions
is condensing map;
is compact and continues;
3 Construction of some existence results
In this section, we construct a general procedure regarding existence, uniqueness, and then for iterative solution of the considered problem.
We provide some prior assumptions needed onward as follows:
The nonlinear function satisfies the growth conditions as follows:
constant such that one has
Moreover, for , the result holds.
Inview of hypothesis , if , then problem (1) has at least one solution.
Let us utilize Theorem 2.6, so we define bounded set Thank to the continuity of , operators are continues. To prove that is condensing map, we take in view of and obtain
Hence, is a condensing map. Further to derive the results about compactness and continuity of , we take
Hence, is bounded on . Further, for equicontinuty, we take such that
From Eq. (9), we observe that as . In this way, we come to the conclusion that is compact as well as equi-continuous due to Arzelá-Ascoli’s theorem. Hence, by using Theorem 2.6, the considered problem (1) has at least one result, which lies in .□
Under the hypothesis and if , then the problem (1) under investigation has unique result.
From Lemma 6, we define the operator as follows:
Now consider , we have
Hence, the operator is condensing map, which leads us to the uniqueness result of the problem.□
4 General procedure of solution
Here, we consider a general problems of CFFDEs as follows:
Now let us assume that we want to compute the solution in the form infinite series as and expressing nonlinear terms in terms of Adomian polynomials as follows:
For instance, if then we have the following polynomials for :
and so on. Hence, using the aforementioned representation in (12) and one has on comparing terms:
Evaluating inverse Laplace transform in each step of (14), we obtain following series solution:
The obtained series is in the form of infinite series. Such series are mostly convergent (see ref. ) for ordinary Caputo derivative. In the same manner, we can also prove the convergence for the said derivative.
5 Applications of our method to proposed problem
Consider FOPBE in the second order under initial and boundary conditions as follows:
Inview of the proposed scheme, few terms of solution are as follows:
and so on. In this manner, we compute the remaining terms. Therefore, the series solution becomes
We present the 3D plot in Figure 1, the approximate solutions up to first three terms for different values of and taking We observe as the fractional order approaches to its integer value, the approximate solution tending to the classical order result.
In Figure 2, we provide plots at fixed time and different fractional order .
Consider the FOPBE (5) of third order as follows:
Here, we see that , and . Also
Hence, inview of Eq. (14), we have
and so on. In this way, we obtain the series solution of Example 5.2 as follows:
We present 3D plots in Figure 3, the approximate solutions up to first few terms for different values of and taking . We observe as the fractional order approaches to its integer value, the approximate solution tending to the classical order result.
In Figure 4, we provide plots at fixed time and different fractional order.
Consider another example of FBPDE as follows:
Here, , and On the proposed method, we compute few terms of solution as follows:
and so on. In this fashion, the reaming terms can be computed to get the series type solution as follows:
We present 3D plots in Figure 5, the approximate solutions up to three first terms for different values of and taking . We observe as the fractional order approaches to its integer value, the approximate solution tending to the classical order result of the considered problem.
In Figure 6, we provide plots at fixed time and different fractional order.
Consider another example of FBPDE as follows:
Here, , and Upon utilizing the proposed method, we obtain the following few terms:
and so on. In this manner, all other terms can be computed to get the required series solution. In Figure 7, we provide a 3D plots for the approximate solutions at different fractional order. We observe as the fractional order approaches to its integer value, the approximate solution tending to the classical order result.
In Figure 8, we provide plots at fixed time and different fractional order.
6 Concluding remarks
In this work, we have applied a novel semi-analytical method to compute series solutions to FOBEs of second and third order successfully. The respective method is an hybrid method, which generates solutions in the form of infinite power series, which is rapidly convergent. Here, we have for the first times applied the mentioned method for the mentioned nonlinear equations under CFFD. From the visualizations of solutions through plots, we observed that the mentioned method can be successfully extended to nonlinear CFFODEs.
The author Kamal Shah would like to thank Prince Sultan University for support through research lab TAS.
Funding information: The authors state no funding involved.
Author contributions: Kamal Shah has designed the scheme and established theoretical investigation. Aly R. Seadawy has viewed the results with contribution in analytical part and A. B. Mahmoud: writing - reviewing and editing, validation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
 Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York: Wiley; 1993. Search in Google Scholar
 Kilbas AA, Srivastava H, Trujillo J. Theory and application of fractional differential equations. In: North Holland mathematics studies. Vol. 204. Amsterdam: Elseveir; 2006. Search in Google Scholar
 Podlubny I. Fractional differential equations, mathematics in science and engineering. New York: Academic Press; 1999. Search in Google Scholar
 Lakshmikantham V, Leela S, Vasundhara J. Theory of fractional dynamic systems. Cambridge, UK: Cambridge Academic Publishers; 2009. Search in Google Scholar
 Rossikhin YA, Shitikova MV. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev. 1997;50:15–67. 10.1115/1.3101682Search in Google Scholar
 Li Y, Haq F, Shah K, Shahzad M, Rahman G. Numerical analysis of fractional order Pine wilt disease model with bilinear incident rate. J Maths Comput Sci. 2017;17:420–8. 10.22436/jmcs.017.03.07Search in Google Scholar
 Darvishi MT, Najafiand M, Seadawy AR. Dispersive bright, dark and singular optical soliton solutions in conformable fractional optical fiber Schrodinger models and its applications. Opt Quantum Electronic. 2018;50(181):1–16. 10.1007/s11082-018-1448-6Search in Google Scholar
 Yaro D, Seadawy AR, Lu D, Osafo Apeanti W, Worlanyo Akuamoah S. Dispersive wave solutions of the nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional symmetric regularized long wave equation. Results Phys. 2019;12:1971–9. 10.1016/j.rinp.2019.02.005Search in Google Scholar
 Seadawy AR, Ali KK, Nuruddeen RI. A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations. Results Phys. 2019;12:2234–41. 10.1016/j.rinp.2019.02.064Search in Google Scholar
 Baleanu D, Zibaei S, Namjoo M, Jajarmi A. A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system. Adv Differ Equ. 2021;2021:308. 10.1186/s13662-021-03454-1Search in Google Scholar
 Jajarmi A, Baleanu D, Zarghami Vahid K, Mobayen S. A general fractional formulation and tracking control for immunogenic tumor dynamics. Math Meth Appl Sci. 2022;45(2):667–80. 10.1002/mma.7804Search in Google Scholar
 Baleanu D, HassanAbadi M, Jajarmi A, Zarghami Vahid K, Nieto JJ. A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects. Alexandria Eng J. 2022;61:4779–91. 10.1016/j.aej.2021.10.030Search in Google Scholar
 Jajarmi A, Baleanu D, Zarghami Vahid K, Mohammadi Pirouz H, Asad JH. A new and general fractional Lagrangian approach: a capacitor microphone case study. Results Phys. 2021;31:104950. 10.1016/j.rinp.2021.104950Search in Google Scholar
 Kumar D, Seadawy AR, Joardar AK. Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chinese J Phys. 2018;56:75–85. 10.1016/j.cjph.2017.11.020Search in Google Scholar
 Tariq KU, Seadawy AR, Younis M, Rizvi STR. Dispersive traveling wave solutions to the space-time fractional equal-width dynamical equation and its applications. Opt Quantum Electronic. 2018;50(147):1–16. 10.1007/s11082-018-1400-9Search in Google Scholar
 Seadawy AR. Fractional travelling wave solutions of the higher order extended KdV equations in a stratified shear flow: part I. Comput Math Appl. 2015;70:345–52. 10.1016/j.camwa.2015.04.015Search in Google Scholar
 Abdeljawad T, Baleanu D. On fractional derivatives with exponential kernel and their discrete versions. Reports on Math Phys. 2017;80(1):11–27. 10.1016/S0034-4877(17)30059-9Search in Google Scholar
 Shaikh A, Tassaddiq A, Nisar KS, Baleanu D. Analysis of differential equations involving Caputo-Fabrizio fractional operator and its applications to reaction-diffusion equations. Adv Differ Equ. 2019;2019:178. 10.1186/s13662-019-2115-3Search in Google Scholar
 Jarad F, Abdeljawad T, Baleanu D. On the generalized fractional derivatives and their Caputo modification. J. Nonlinear Sci. Appl. 2017;10(5):2607–19. 10.22436/jnsa.010.05.27Search in Google Scholar
 Toledo-Hernandez R, Rico-Ramirez V, Iglesias-Silva GA, Diwekar UM. A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions. Chem Eng Sci. 2014;117:217–28. 10.1016/j.ces.2014.06.034Search in Google Scholar
 Wang Z, Yang D, Ma T, Sun N. Stability analysis for nonlinear fractional-order systems based on comparison principle. Nonlinear Dynam. 2014;75(1–2):387–402. 10.1007/s11071-013-1073-7Search in Google Scholar
 Al-Refai M, Abdeljawad T. Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel. Adv Differ Equ. 2017;2017(1):315. 10.1186/s13662-017-1356-2Search in Google Scholar
 Shah K, Khalil H, Khan RA. Analytical solutions of fractional order diffusion equations by natural transform method. Iran J Sci Technol (Trans Sci:A). 2018;42(3):1479–90. 10.1007/s40995-016-0136-2Search in Google Scholar
© 2022 Kamal Shah et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.