Rwy Al-Dikh , Omr Au Arqu , , , Mohmm Al-Smi , , Shhr Momni , f

a Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

b Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan

cNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

d Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan

e Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates

f Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates

ABSTRACT In these analyses, we consider the time-fractional Fisher equation in two-dimensional space.Through the use of the Riemann-Liouville derivative approach, the well-known Lie point symmetries of the utilized equation are derived.Herein, we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries.The diminutive equation’s derivative is in the Erdélyi-Kober sense, whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time.The conservation laws for the dominant equation are built using a novel conservation theorem.Several graphical countenances were utilized to award a visual performance of the obtained solutions.Finally, some concluding remarks and future recommendations are utilized.

Keywords:Fractional partial differential equation Time-fractional Fisher equation Lie point symmetry Explicit power series Conservation laws

1.Introduction

FPDEs are extremely used to characterize utmost of the phenomena that emerge in engineering sciences, applied mathematics, quantum mechanics, biology, electricity, and so on [1-5] .As a score, numerous fresh techniques have been effectively utilized,developed, and extended by collections of investigators to obtain approximate or exact analytical solutions for FPDEs in general where the most important goes back to Riemann, Hadamard, Caputo, and Atangana factional derivative approach [6-48] .

The LPS plays a very serious matter in various fields of applied mathematical sciences, particularly in integrable systems, where infinitely many symmetries exist.Thereafter, the LS approach is theorized to be one of the efficacious tactics for obtaining analytical solutions of FPDEs of nonlinear type.Historically, a luge number of treatises are loyal to the theory of the LPS and its implementations to FPDEs [49-51] .The LS analysis of FPDEs is proposed in [52] by utilizing a prolongation shape method for two requisite fractional derivatives which are RLD and Caputo derivatives.Furthermore, the LPS is applied to build conservation laws which is a fundamental rule in the studying of nonlinear real-world phenomena considering LS analyses.The conservation laws are a mathematical shape that announcement that the total amount of a specific physical quantity stays the same through the germination of the physical system and are also applied in the investment of numerical methods to confirm the existence and uniqueness of suggested solutions [53-60] .

During this study, we consider the LS analysis, explicit solutions, and conservation laws of the TFFE in two-dimensional space of the subordinate shape:

where x ∈ R , t ≥0 , 1 ≥α＞ 0 , and λ∈ R .Herein, u ( x, t ) is unknown functions to be determined andis the RLD of order 1 ≥α≥0 and given as

Mathematically, the TFFE describes a wide zone of substantial physical phenomena, it is an FPDE generated from the conventional Fisher equation by ousting the time derivative with an FDO of order 1 ≥α＞ 0 .The TFFE characterizes the propagation of a virile mutant in an infinitely long habitat and is also utilizes a model equation for the evolution of a neutron population in a nuclear reactor and a prototype model for a spreading flame.Theoretically and numerically many researchers have studied TFFE, for more details and futures, the reader is advised to see [61-66] .

Anyhow, the present work is ordered and arranged as the subordinate posterior bullets:

·Section 1 : Introduction: the problem and its importance.

·Section 2 : The LPS representation: formulation and properties.

·Section 3 : The LSs and reduction: characteristic formulas and Erdélyi-Kober.

·Section 4 : Explicit power series analysis: explicit solution and convergence.

·Section 5 : The conservation laws: derivation and discussion.

·Section 6 : 6 Sketches and justifications: 3-D plots and results.

·Section 6 : Conclusion: highlight and futures.

2.The LPS representation

In this part, some main concepts and brief details that revolve around the LS analysis are described and utilized.Hither, the utilizations are related to the formulation of the TFFE in sense of the RLD with 1 ≥α＞ 0 .

Let us first consider the subordinate FPDE of RLD with 1 ≥α＞0 :

The single parameter εLie group representation of conversions is utilized as

In (4) the symbol ε ?1 is the Lie group parameter in which T ,X, and ηare the infinitesimals of the conversions for t, x , and u variables, simultaneously.

Anyhow, the explicit idioms of ηxand ηxxare given as

where Dxis the total derivative concerning x, that are defined by

The corresponding Lie algebra of symmetries consists of a set of vector fields of the form

Depending on the criterion of the basis of the infinitesimal invariance, we obtain

Thereafter, the invariance condition gives

The α-th extended infinitesimal related to RLD with (9) can be represented as

By using the generalized Leibnitz rule, one has

By applying the Leibnitz rule, results in (10) become

Subsequently, the chain rule of the compound function is utilized as

Using rule in (14) and the generalized Leibnitz rule with f(t) =1 , one gets

where m is given as

Posteriorly, the α-th extended infinitesimal in (10) can be expressed as

In the next theorem, a function u = θ( x, t ) is called an invariant surface whenever

Fig.1.The 3-D.plots of approximate simulations for 5-th series term in (47) when ( λ, a 0 , a 1 ) = ( -3 , 1 , 1 ) as: (a) α= 1 , (b) α= 0 .85 , (c) α= 0 .75 , and (d) α= 0 .65 .

Theorem 1.[67] A function u = θ( x, t ) is an invariant solution of(3) only and only if

1 u = θ( x, t ) is an invariant surface,

2 u = θ( x, t ) satisfies (4) .

3.The LSs and reduction

Out of this part, we will find out and derived the characteristic formulas of vector fields, then we use it for obtaining the reduction equations.Herein, we reduce the TFFE (1) to a nonlinear FDE with the Erdélyi-Kober FDO and then solve it in detail.

Let us first suppose that (1) is invariant under a single parameter conversion of (4) .Thus, one can get the subordinate conversed equation:

Using point conversion of (5) in (20) , one gets the subordinate symmetry determining formula as:

By substituting (5) and (17) into (20) , we acquire

Now, for (21) by considering the whole powers of derivatives for u to 0, one has the subordinate system of equations as:

By solving (22) , one has the subordinate infinitesimals:

where c1and c2are arbitrary parameters.

Fig.2.The 3-D plots of approximate simulations for 5-th series term in (47) when ( λ, a 0 , a 1 ) = ( -6 , 1 , 1 ) as: (a) α= 1 , (b) α= 0 .85 , (c) α= 0 .75 , and (d) α= 0 .65 .

Anyhow, the Lie algebra of infinitesimal symmetries of (1) is utilized as

Now, we will use the characteristic formulas of vector fields obtained in (24) for deriving and obtaining the reduction equations as follows:

Phase 1 : The characteristic formula for infinitesimal generator V1couldbesymbolicallyrepresented as

Anyhow, by solving (25) , we gain the zero solution.

Phase 2: The characteristic formula for infinitesimal generator V2couldbesymbolicallyrepresented as

To summarize and by solving (26) , we get

So, by the symmetry V2, one has the group invariant solution as

where fis a qualitative function of z.

By enchanting the similarity variables and similarity solutions,the TFFE (1) can be converted to the subordinate outcome:

Theorem 2.The conversion in (28) reduces the TFFE (1) into the subordinate nonlinear FDE:

with the Erdélyi-Kober FDO which isand defined as

where z ＞ 0 , β＞ 0 , α＞ 0 , and

Fig.3.The 3-D plots of approximate simulations for 5-th series term in (47) when ( α, a 0 , a 1 ) = ( -3 , 1 , 1 ) as: (a) λ= 0 .5 , (b) λ= 0 .25 , (c) λ= -0 .25 , and (d) λ= -0 .5 .

proof.Suppose that n ＞ α＞ n -1 , n = 1 , 2 , 3 , ....Under the similarity conversion of (19) , the RLD becomes as

We now thus simplify the right-hand side of (34) by using the relation z =with φ ∈ C1(0, ∞ ). Thereafter, one has

From this, one can generate the subordinate results:

Recurrent the analogous procedure above for n -1 times, one gets

By the definition of the Erdélyi-Kober FDO, the results in(37) can be written as

Given (39) the results in (1) is converted as below

This completes the proof.

Here, and after a lot of work, we reached the main point through this theorem, which is the conversion of the governing equation to nonlinear FDEs in order to make it easier for us to solve it as in the next section.

4.Explicit power series analysis

After reducing FPDEs into nonlinear FDEs in the pre-last analyses, we will yet study and search for the explicit solution of the nonlinear FDEs through the method of power series.Herein, when we get the explicit solution of FDE, we can easily obtain the power series solutions of the original FPDE.

Now consider the subordinate expansion:

From (40) , we can have

Putting (40) and (41) in (29) , one obtains

Thus, by using some refinement, one gets

Comparing coefficients in (43) , when n = 0 , we get

But when n ≥1 , we get

Thereafter, the explicit solution of (29) may be written as

Consequently, we acquire the explicit power series given by

In (47) the parameters a0and a1 are arbitrary, so all coefficients anwith n = 2 , 3 , 4 , ...can be computed using (44) and (45) .For simplicity, next, we list some of anwith n = 2 , 3 , 4 , ...as

To complete the presented topic further and to facilitate the procedure more, let us now studying and investigating the convergence of the series solution in (40) .Firstly, form (45) , we watch that

in which κ= max { 2 , λ} .

To proceeds, we utilized another power series M(z) as

in which mi= | ai| with i = 0 , 1 , 2 , ....

Thereafter, one has

So, it is easily shown that | mn| ≤ anwith n = 0 , 1 , 2 , ...and M(z) =are the majorant series of (40) .Anyhow, by computations, we get

5.The conservation laws

Conservation laws are of major significance in the FPDEs as they supply conserved quantities for all generated solutions, can expose integrability, explain linearization, and prove the existence and uniqueness of solutions.Conservation laws supply one of the fundamental rules in shape models in mathematics and at certain times FPDEs having a large number of conservation laws depicts a strong implication of its integrability.In this section, the conservation laws of the TFFE (1) are derived and discussed.These explanations can be done depending on the formal Lagrangian and LPSs.

Let us firstly consider a vector C = (Cx, Ct) that admitting the subordinate conservation equation:

Herein, Cx= Cx( x, t, u, ...) and Ct= Ct( x, t, u, ...) are called the conserved vectors for (1) .According to the new conservation theorem due to Ibragimov [68] , the formal Lagrangian of (1) can be given as

where v ( x, t ) is a new dependent variable that is smooth suffi-ciently function.

Based on the Lagrangian, one has an action integral as

The Euler-Lagrange operator is given as

Next, we consider the dependent variable u = u ( x, t ) to obtain the subordinate result:

where R*is defined as

The Lie characteristic function W is defined as

If we use the RLD in (1) , the density component Ctof conservation low is given as

where Jis given by

In fact, the other component is defined as

Next using the basic definitions presented in (68) and (70) , the conservation laws for (1) are utilized.In fact, we generate the components of the conservation laws as follows:

Phase 1:For W1= -uxone may obtain the t- and x -components of the conserved vectors

Phase 1:For W2= ( 3 α-2 ) u -2 αx ux-4 t u t , one may obtain the t- and x -components of the conserved vectors

The conserved vectors involve random solutions of the adjoint formula, herewith inclusion the interminable number of conservation laws.In fact, the conservation laws perform a forcing part in the solution procedure of fractional models.( 11-13 ),(15) , (16) , (18) , (23) , (27) , (28) , (30) , (31) , (32) , (35) , (36) , (38) , (42) , (43) , (46) ,(4 8) , (4 9) , (50) , (51) , 53 ),( 54-67 ),( (69) ,( 71-74 ) theorem1 , theorem2 ,proof

6.Sketches and justifications

This part is important, and it presents some results of the solutions that we obtained from (47) through the evolution process in(44) and (45) to show and clarify the behavior of the explicit series solution concerning the total fractional derivative used in this research.

Through our research, all drawings were executed using the MATHCAD 14 program, and all values were taken into account based on previous research that discusses the TFFE (1) .

Next, some of the physical properties of the utilized explicit series solution are analyzed geometrically throughout the 3-D plots.Anyhow the 5-th series terms in (47) are sketched by using suitable parameter values of λ, a0, a1, and α.Fig.(1) from (a-d) relates to the tribbles ( λ, a0, a1) = ( -3 , 1 , 1 ) , whilst Fig.(2) from (a-d)relates to the tribbles ( λ, a0, a1) = ( -6 , 1 , 1 ) when α= 0 .65 , α=0 .75 , α= 0 .85 , and α= 1 .Fig.(3) from (a-d) relates to the tribbles( α, a0, a1) = ( 0 .85 , 1 , 1 ) when λ= -0 .5 , λ= -0 .25 , λ= 0 .25 , and λ= 0 .5 .

As a quick and insightful look, we can say that the behavers plots in Figs.1 , 2 , and 3 in all shapes are approximately similar in their attitude.This ensures the history of syllabic fractional derivatives used when interchange αand also confirms their convergent explicit series solutions.

7.Conclusion and future recommendation

In this paper, the invariance properties of the general TFFE are presented in the sense of LPS and RLD.The symmetry reductions of the TFFE (1) beside all geometric vector fields are derived and obtained.The reduction of dimension is in the symmetry algebra since the FPDE is invariant under time translation symmetry.Further, the considered fractional model can be converted into FDE,and then one can apply the technique of the power series to find out all explicit analytic solutions.The conservation laws for the main fractional model are symbolically and theoretically computed.Some graphical attitudes are utilized and discussed concerning various parameter’s effects.The gained results witness that the LPS field and technique of power series play a respectable base in the applied mathematical sciences and very efficient powerful tools for obtaining explicit solutions.Our next works will focus on the conformable TFFE in the LS analyses.

Declaration of Competing Interest

The authors declare that they have no conflicts of interest.

Acknowledgment

The authors would like to express their gratitude to the unknown referees for carefully reading the paper and their helpful comments.