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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.67 no.4 México jul./ago. 2021  Epub 14-Mar-2022

https://doi.org/10.31349/revmexfis.67.040704 

Research

Gravitation, Mathematical Physics and Field Theory

Modified exponential function method for nonlinear mathematical models with Atangana conformable derivative

T. Aktürka 

aDepartment of Mathematics and Science Education, Ordu University, 52100 Ordu/Turkey. e-mail: tolgaakturkk@gmail.com


Abstract

In this study, we investigate the exact solutions of the modifie Benjamin-Bona-Mahony and Sharma-Tasso-Olver equations, which are define with Atangana conformable fractional derivative, using the modifie exponential function method. Exact solutions of the modifie Benjamin-Bona-Mahony and Sharma-Tasso-Olver equations were obtained by using the modifie exponential function method. Two and three-dimensional and contour graphics are used to understand the physical interpretations of the resulting exact solutions to the mathematical model. When all these results and graphs are analyzed, it has been shown that the modifie exponential function method is an effective method for obtaining exact solutions for all other nonlinear fractional partial differential equations containing conformable fractional derivatives of Atangana.

Keywords: The modifie exponential function method; The space-time fractional modifie Benjamin-Bona-Mahony equation; Fractional Sharma-Tasso-Olver equation; Atangana conformable derivative; contour surfaces

PACS: 02.30.Jr; 02.60.Cb; 04.20.Jb

1. Introduction

Nonlinear partial differential equations are used in various field of physics, flui mechanics, engineering, health, etc. These equations are widely used for its applications in mathematical modeling of situations encountered across many areas. It is very important to obtain numerical and exact solutions to such mathematical models because with the analysis of the solutions of such mathematical models, a physical interpretation of the situation it represents can be made. For example, in the context of the COVID-19 outbreak, mathematical models have allowed us to give notice of the extent of the epidemic and to measure the rate of spread of the virus, among other useful information; these tools are, and will continue to be, developed in the future. However, there are various methods in the literature about the investigation of the solutions of nonlinear partial differential equations [18]. In recent years, intensive studies have been carried out in the scientifi world to investigate fractional partial differential equations compatible with nonlinear partial differential equations and their solutions. Some fractional derivative operators and nonlinear fractional differential equations have been introduced in the literature [9-24].

A new fractional derivative operator, which is arranged as a fractional derivative, has been determined, and using this definition the exact solutions of the nonlinear fractional differential equation have been obtained [25]. Atangana et al., while giving place to some theorems and properties about conformable derivative, they named this term as beta derivative [26]. The space-time fractional modifie BenjaminBona-Mahony and fractional Sharma-Tasso-Olver equations given with Atanagana’s conformable derivative were solved by the firs integral method [27].

In this article, exact solutions of nonlinear fractional differential equations were investigated with the use of the modifie exponential function method.

2. Beta-derivatives

Definition 1. A new fractional derivative called conformable derivative by Khalil et al. has been brought to the literature [25]. Let, the function f : [0,∞) of the α-th order analyzed as the conformable derivative function is as follows in terms of t > 0, α ∈ (0,1):

Dtα0ft=limε0ft+εt1-α-ftε. (1)

When f is α-differentiable in the range of (0,α), α > 0 and lim ε→0 + f (α)(t) exists, then it can be stated as f (α)(0) = lim ε 0+ f (α)(t).

Definition 2. The beta derivative stated by Atangana et al. is define as follows [26]:

Dtα0Aft=limε0ft+εt+1Γα1-α-ftε. (2)

The most important reason for choosing the fractional derivative of Atangana is that the basic derivatives can provide some useful properties. Some features of the definitio given above are stated below:

  • i) Let g ≠ 0 and f functions be differentiable with respect to beta in the interval β ∈ (0,1]. In this case, the following equation can be written that can be provided for all real numbers α and b.

    Dxα0Aαfx+bg(x)=α0ADxαf(x)+b0ADxαg(x).       (3)

  • ii) Where c is any constant that satisfie the following equation,

    Dxα0Af(t)=0.       (4)

  • iii) Dxα0Afxg(x)=gxDxα0Af(x)+fxDxα0Agx.       (5)

  • iv) Dxα0Af(x)g(x)=g(x)Dxαf(x)+f(x)Dxαg(x)0A0Ag2(x).       (6)

If ε = (x+1/Γ(α)) α−1 h is written instead of ε in Eq. (2) and h → 0, when ε → 0, we get as follows,

Dxα0Af(x)=x+1Γ(α)1-αdf(x)dx, (7)

with

η=δαx+1Γ(α)α, (8)

δ is constant and in this case the following equation can be written

Dxα0Af(η)=δdf(η)dη, (9)

3. The modified exponential function method

In this section, exact solutions of nonlinear fractional partial differential equations define as Atangana derivatives will be introduced in detail by using the modifie exponential function method [28].

The general form of the nonlinear fractional partial differential equation containing the two variables u function and the beta derivative is given as follows:

Pu,Dxα0Au,ux,uxx, =0 (10)

where given x is space and t is time.

The processing steps of the modifie exponential function are as follows:

Step 1. First of all, we can consider the traveling wave transformation to obtain the wave solution of Eq. (10) as follows:

ux,t=uη,     η=kx-λαt+1Γαα, (11)

where k and λ are constants. When the derivative terms required in Eq. (10) are taken from Eq. (11) and replaced, the following nonlinear partial differential equation is obtained,

N(u,u',u'',u''',...) = 0. (12)

Step 2. Let, the exact solution of the nonlinear fractional differential equation analyzed in the article be in the form of (13):

uη=j=0nAje-ϑ(η)ji=0me-ϑ(η)i (13)

where A j ,B i , (0 ≤ jn, 0 ≤ im) are constants.

ϑ'(η) = e-ϑ(η) + keϑ(η) + σ. (14)

When Eq. (14) is integrated according to η the following solution families are obtained [26]:

Family 1. If µ ≠ 0 and σ 2 − 4µ > 0,

ϑη=ln-σ2-4u2μtanh×-σ2-4u2η+E-σ2μ (15)

Family 2. If µ ≠ 0 and σ 2 − 4µ < 0,

ϑη=ln-σ2-4u2μtan ×-σ2-4u2η+E-σ2μ (16)

Family 3. If µ = 0, σ ≠ 0 and σ 2 − 4µ > 0,

ϑη=lnσeσ(η+E)-1. (17)

Family 4. If µ ≠ 0, σ ≠ 0 and σ 2 − 4µ = 0,

ϑη=ln-2ση+E+4σ2η+E. (18)

Family 5. If µ = 0, σ = 0 and σ 2 − 4µ = 0,

ϑ(η) = ln(η + E), (19)

where E, σ, µ are coefficients

Step 3. In this section, the limits of the total symbols should be determined by applying the balance procedure to Eq. (12), which is analyzed as the exact solution of the nonlinear fractional partial differential equation. After the balance procedure is reduced to the form of Eq. (12) under investigation, a relation is obtained between m and n by equating the term with the highest order derivative and the highest order nonlinear term. Then, by determining an arbitrary constant to the value of m in this relation, the constant n is obtained. Then, Eq. (13) and its derivatives are written in Eq. (12) and the algebraic equation system containing e −ϑ(η) is get. With the solution of this system, it will be found in constants such as A 0 ,A 1 ,A 2 ,...,A n ,B 0 ,B 1 ,B 2 ,...,B m . By replacing all these terms, it is checked that they provide the equation and the exact solution function of Eq. (10) is obtained.

4. Applications of nonlinear fractional differential equations with Atangana derivatives

In this section, we investigate exact solutions of the SharmaTasso-Olver equation and modifie Benjamin-Bona-Mahony equation equations using the modifie exponential function method with Atangana’s conformable derivative.

Example 1. Let’s investigate the Sharma-Tasso-Olver equation together with the conformable derivatives of Atangana [27],

Dxα0Au+3αux2+3αu2ux

+ 3auuxx + auxxx=0,0 < α  1. (20)

The equation given above is subject to the following initial conditions

ux,0=-2B0tan2B02x. (21)

where α and B 0 are arbitrary constants and α is a parameter representing the order of the Atangana derivative. We use the following wave transformation to reduce the Eq. (20) to the nonlinear ordinary differential equation form

u(x,t) = u(η)       and

η=x-λαt+1Γαα, (22)

where λ is constant. Then, we substitute Eq. (22) into Eq. (20), we obtain the following nonlinear ordinary differential equation

-λu'+ 3au'2+ 3au2u' + 3auu'' + au''' = 0. (23)

Integrating Eq. (23) with respect to η, we get

c - λu + 3auu' + au3+au'' = 0, (24)

where c is integral constant.

According to the balance procedure, using the highest order nonlinear term u 3 and the highest order term u’’ in Eq. (24), the following equation is obtained

3n - 3m = n - m + 2  n = m + 1. (25)

If m = 1, we then get n = 2.

In this case, Eq. (13) is written as follows.

Uη=Υϕ=α0+α1e-ϑ+A2e-2ϑB0+B1e-ϑ (26)

Let us obtain the necessary derivative terms in Eq. (24) from Eq. (26),

u'η=Υ'ϕ-Υϕ'ϕ2 (27)

u''η=Υ''ϕ3+Υ'ϕ'ϕ2-ϕ2Υ'ϕ'+ϕ2Υϕ''-2ϕϕ'Υ'ϕ-Υϕ'ϕ4 (28)

If Eqs. (26-28) are substituted in Eq. (24), the exact solutions that satisfy Eq. (20) are as follows:

Case 1.

A0=B02A2,     A1=2B0     B1=A2,

μ=-λα+σ2+3B0-σA2+B0A22,

c=σA2-2B0λ-ασ2A22+4ασA2B0-4αB02A23. (29)

If the coefficient in (29) are replaced in Eq. (26), the exact solution functions of Eq. (20) under the terms of solution families are as follows;

Family 1. When, µ ≠ 0 and σ2 - 4µ > 0

u1,1x,t=B0A2-2uσ+-4μ+σ2tanh12-4μ+σ2E+η. (30)

FIGURE 1 Three-dimensional, contour and density plots of the solution (30) for the values E = 0.75, α = 1, α = 0.5, λ = 3, c = −0.432, σ = 2, µ = 0.43, and two-dimensional t = 1. 

Family 2. When, µ ≠ 0 and σ2 - 4µ < 0

u1,2x,t=B0A2-2μσ-4μ-σ2tan124μ+σ2E+η (31)

FIGURE 2 Three-dimensional, contour and density plots of the solution (31) for the values E = 0.75, α = 1, α = 0.5, λ = 3, c = −0.101, σ = 0.1, µ = 1.01, and two-dimensional t = 1. 

FIGURE 3 Three-dimensional, contour and density plots of the solution (32) for the values E = 0.75, α= 1, α = 0.5, λ = 1, c = 0, σ = 1, µ = 0, and two-dimensional t = 0.8. 

Family 3. When, µ = 0, σ ≠ 0 and σ2 - 4µ > 0

u1,3x,t=λ-1+eσ(E+η)+B0A2. (32)

FIGURE 4 Three-dimensional, contour and density plots of the solution (33) for the values E = 0.75, α = 1, α = 0.5, λ = 3, c = −2, σ = 2, µ = 1, and two-dimensional t = 1. 

Family 4. When, µ ≠ 0, σ ≠ 0 and σ2 - 4µ = 0

u1,4x,t=B0A2+λ-12+αα2+(E+x)λ-λx-η. (33)

FIGURE 5 Three-dimensional, contour and density plots of the solution (34) for the values E = 0.75, α = 1, α = 0.5, λ = 1, c = 2/3 3, σ = 0, µ = 0, and two-dimensional t = 1. 

Family 5. When, µ = 0, σ = 0 and σ 2 − 4µ = 0

u1,5x,t=B0A2+1E+x. (34)

Case 2.

A0=B03αB0+3αλ+αμA22-αB023αA2        A1=6αB0+3αλ+αμA22-αB023α

B1=A2,      σ=2B0A2,      c=2λ+4αμA22-4αB02αλ+αμA22-αB0233αA23 (35)

We substitute the coefficient obtained above in Eq. (26). Then, we form the necessary derivative terms for Eq. (20) and write them in their place and obtain exact solution functions under the following solution families.

Family 1. If, µ 6= 0 and σ 2 − 4µ > 0

u2,1x,t=ςσ+3α-4μ-σ+3α+ςρ6ασ+-4μ+σ2ρ. (36)

where, ς=-3α(-12+α and ρ=tanh1/2-4μ+σ2(E+η.

Family 2. When, µ ≠ 0 and σ 2 − 4µ < 0

u2,2x,t=ςσ+3α-4μ-σ-3α+ς4μ-σ2ϖ6ασ-4μ-σ2ϖ (37)

where, ϖ=tan124μ-σ2(E+η)

Family 3. When, µ =0, σ ≠ 0 and σ2 -4µ > 0

u2,3x,t=163+36-3αα+6λ-1+eσ(E+η). (38)

FIGURE 6 Three-dimensional, contour and density plots of the solution (36) for the values E = 0.75, α = 0.2, α = 0.5, λ = 1, c = 0.218222, σ = 2, µ = 0.43, and two-dimensional t = 1. 

FIGURE 7 Three-dimensional, contour and density plots of the solution (37) for the values E = 0.75, α = 0.2, α = 0.5, λ = 1, c = 1.697056, σ = 2, µ = 2, and two-dimensional t = 1. 

FIGURE 8 Three-dimensional, contour and density plots of the solution (38) for the values E = 0.75, α = 0.2, α = 0.5, λ = 1, c = 0.153960, σ = 2, µ = 0, and two-dimensional t = 1. 

FIGURE 9 Three-dimensional, contour and density plots of the solution (39) for the values E = 0.75, α = 0.2, α = 0.5, λ = 1, c = 0.860662, σ = 2, µ = 1, and two-dimensional t = 1. 

FIGURE 10 Three-dimensional, contour and density plots of the solution (40) for the values E = 0.75, α = 0.2, α = 0.5, λ = 1, c = 0.860663, σ = 0, µ = 0, and two-dimensional t = 0.8. 

Family 4. When, µ ≠ 0, σ ≠ 0 and σ 2 − 4µ = 0

u2,4x,t=163+36-3αα+λ-3+62+E+xλ-λ2x-μ. (39)

Family 5. When, µ = 0, σ = 0 and σ 2 − 4µ = 0

u2,5x,t=163+36-3αα+6E+x. (40)

Example 2. Let’s consider the modifie Benjamin-Bona-Mahony equation with the conformable Atangana derivatives [25],

Dtαu0A+Dxαu0A-ku2Dxαu0A+Dx3αu=0.0A (41)

We reduce it to the following nonlinear ordinary differential equation form by applying the wave transformation for Eq. (41),

ux,t= uη,      and     η=Υαx+1Γ(α)α-λαt+1Γαα, (42)

where λ is constant. If we substitute the wave transformation (42) in Eq. (41),

-λu' + γu' - kγu2u' + γ3u'''= 0. (43)

If Eq. (43) is integrated to simplify it,

Υ-λu-Υk3u3+Υ3u''-c=0, (44)

where c is integral constant.

Using the balance procedure, the highest order nonlinear term u 3 and the highest order term u’’ in Eq. (44), the following equation is get,

3n - 3m = n - m + 2  n = m + 1 (45)

If m = 1, we then obtain n = 2.

Accordingly, Eq. (13) can be written as follows.

uη=Υϕ=A0+A1e-ϑ+A2e-2ϑB0+B1e-ϑ (46)

If we obtain the derivative terms required in Eq. (44) from Eq. (46),

u'η=Υ'ϕ-Υϕ'ϕ2, (47)

U''η=Υ''ϕ3+Υ'ϕ'ϕ2-ϕ2Υ'ϕ'+ϕ2Υϕ''-2ϕϕ'Υ'ϕ-Υϕ'ϕ4 (48)

Substituting the terms in Eqs. (46-48) into Eq. (44), we obtain exact solutions that satisfy Eq. (41).

Case 3.

A0=32Υ2B02-σB0B1+2μB12kB1        A1=32Υ2B0+σB1k,        A2=6ΥB1k

c=26Υ4-2B0+σB1B02-σB0B1+μB12kB13,       λ=Υ-12Υ38Υ+σ2-6Υ3B0B0-σB1B12 (49)

Substituting the coefficient obtained above into the Eq. (26), the exact solution functions of the Eq. (20) are written as follows;

Family 1. µ ≠ 0 and σ 2 − 4µ > 0

u3,1x,t=32Υ-σB0+2B02B1+2μB1+8μ2B1Ω2-2μ2B0+σB1ΩkB0-2μB1Ω (50)

FIGURE 11 Three-dimensional, contour and density plots of the solution (50) for the values E = 0.75, α = 0.5, λ = 0.186986, c = 0.000211, σ = 2, µ = 0, k = 4, γ = 0.2, and two-dimensional for t = 0.8. 

where,

Ω=σ+-4μ+σ2tanh-4μ+σ2Eα+η2α.

Family 2. µ ≠ 0 and σ 2 − 4µ < 0

μ3,2x,t=32Υ-σB0+2B02B1+2μB1+8μ2B1Φ2-2μ2μB0+σB1ΦkB0-2μB1Φ (51)

where,

Φ=σ-4μ-σ2tan4μ-σ2Eα+η2α

Family 3. µ = 0 σ ≠ 0 and σ 2 − 4µ > 0

μ3,3x,t=32Υ2λ-1+eσE+η-σ+2B0B1-2λB0-σB1-1+eσE+ηB0+λB1k (52)

Family 4. µ ≠ 0 σ ≠ 0 and σ 2 − 4µ = 0

u3,4x,t=32Υ-2λ+σB0+4B02B1+λ2+4μ-λσB1+4α2λ2B1υ2+2αλ2B0+-2λ+σB1υ2kB0+12λB1-1+2αυ (53)

FIGURE 12 Three-dimensional, contour and density plots of the solution (51) for the values E = 0.75, α = 0.5, λ = 0.186986, c = 0.000211, σ = 2, µ = 0, k = 4, γ = 0.2, and two-dimensional for t = 0.8. 

FIGURE 13 Three-dimensional, contour and density plots of the solution (52) for the values E = 0.75, α = 0.5, λ = 0.198986, c = 0.000211, σ = 1, µ = 0, k = 4, γ = 0.2, and two-dimensional for t = 0.8. 

FIGURE 14 Three-dimensional, contour and density plots of the solution (53) for the values E = 0.75, α = 0.5, λ = 0.153900, c = −0.0466588, σ = 2, µ = 1, k = 0.1, γ = 0.2, and two-dimensional for t = 1. 

FIGURE 15 Three-dimensional, contour and density plots of the solution (54) for the values E = 0.75, α = 0.5, λ = 0.199980, c = 0.00396593, σ = 0, µ = 0, k = 0.1, γ = 0.2, and two-dimensional for t = 1. 

Family 5. µ = 0 σ = 0 and σ 2 − 4µ = 0

u3,5x,t=32Υ2θ2B02-θ-2+θσB0B1+2+θσB12kB1B0+B1 (54)

where θ = (E + x).

Case 4.

A0=A22Υ-λ+Υ32μ+σ2B0-6σΥ3Υ-λ+Υ3-4μ+σ2B0212Υ3B0

A1=σA2-A2Υ3Υ-λ+Υ3-4μ+σ2B026Υ3B0

B1=6Υ3B023Υ3σB0-6Υ3Υ-λ+Υ3-4μ+σ2B02

c=2Υ-2λ+Υ34μ-σ2A22-Υ+λ+Υ34μ-σ2B0+6σΥ3Υ-λ+Υ3-4μ+σ2B0236Υ3B02

k=-72Υ5B0-2Υ+2λ+Υ38μ-5σ2B0-26Υ3Υ-λ+Υ3-4μ+σ2B02-2Υ+2λ+Υ38μ+σ22A22 (55)

We substitute the coefficient calculated above in Eq. (26). After findin the required derivative terms in Eq. (20), we obtain the exact solution functions according to the following solution families by typing the resulting exact solution function model.

FIGURE 16 Three-dimensional, contour and density plots of the solution (56) for the values E = 0.75, α = 0.5, λ = 0.186986, c = 0.596486, σ = 2, µ = 0.1, k = 0.005894, γ = 0.2, and two-dimensional t = 0.8. 

FIGURE 17 Three-dimensional, contour and density plots of the solution (57) for the values E = 0.75, α = 0.5, λ = −3, c = −1.37344, σ = 1, µ = 1, k = 7.127265, γ = 1, and two-dimensional for t = 0.8. 

Family 1. While, µ ≠ 0 and σ 2 − 4µ > 0

μ4,1(x,t)A22Υ-λ+Υ32μ+σ2B0-σZ12Υ3B0+4μ2A2Λ2+μA2-6Υ3σB0+Z3Υ3B0ΛB0+12Υ3μB02-3Υ3σB0+ZΛ (56)

where,

z=6Υ3Υ-λ+Υ3-4μ+σ2B02

and

Λ=σ+-4μ+σ2tanh12α-4μ+σ2E+η

Family 2. While, µ ≠ 0 and σ 2 − 4µ < 0

μ4,2x,t=A22Υ-λ+Υ32μ+σ2B0-σZ12Υ3B0+4μ2A2Π+2μσA2A2Z6Υ3B0-ΠB0+12Υ3μB02-3Υ3σB0+ZΠ (57)

where,

Π=σ-4μ+σ2tan12α4μ+σ2E+η

FIGURE 18 Three-dimensional, contour and density plots of the solution (58) for the values E = 0.75, α = 0.5, λ = 0.1, c = −1.341331, σ = 0.1, µ = 0, k = 0.001235, γ = 0.2, and two-dimensional for t = 800. 

Family 3. While, µ = 0, σ ≠ 0 and σ 2 − 4µ > 0

μ4,3=λ2A2-1+eσ(E+η)2+A22Υ-λ+Υ3σ2B0-σZ12Υ3B0+λσA2-A2Z6Υ3B0-1+eσ(E+η)B0-6Υ3λB02-1+eσ(E+η)-3Υ3σB0+Z (58)

Family 4. While, µ ≠ 0, σ ≠ 0 and σ 2 − 4µ = 0

μ4,4=A22Υ-λΥ3+2μ+σ2-σZΥ3B0+λ2-6Υ3σB0+ZEα+ηΥ3B0O+3λ4Eα+η2O212B0-3Υ3λ2B02(Eα+η)3Υ3σB0-ZO (59)

where O = (α[2 + ] + λη).

Family 5. While, µ = 0, σ = 0 and σ 2 − 4µ = 0

μ4,5x,t=A226(E+x)2+1Υ2+6σ(E+x)+σ2-62+(E+x)σΥ4(1+Υ2σ2)B0E+xΥ3B012B0-6Υ3B02E+x-3Υ3σB0+6Υ41+Υ2σ2B02 (60)

FIGURE 19 Three-dimensional, contour and density plots of the solution (59) for the values E = 0:75, α = 0:5, λ = 0:1, c = (4-2030 /3000), σ = 2, µ = 1, k = -(12500(-3004-4030 )/561001), γ = 0.2, and two-dimensional for t = 800. 

FIGURE 20 Three-dimensional, contour and density plots of the solution (40) for the values E = 0.75, α = 0.5, λ = 0.1, c = 1.38, σ = 0, µ = 0, k = 0.001152, γ = 0.2, and two-dimensional t = 800. 

Remark The exact solutions of the Eqs. (18) and (40) are obtained by the modifie exponential function method. Graphs representing the physical interpretations of all calculations and solution functions used during the solutions were checked and drawn using Mathematica 12. Exact solutions obtained in this study as a result of the necessary searching processes have not been found in the literature.

5. Conclusions

In this study, the modifie exponential function method is applied to obtain new exact solutions of the modifie BenjaminBona-Mahony and Sharma-Tasso-Olver equations given by the congruent Atangana derivative. In addition, different exact solution functions calculated for these equations are classified With this method, hyperbolic and rational function solutions of the resulting functions are obtained. Parameters suitable for exact solution function models were determined and physical motion models on three-dimensional, contour, density and two-dimensional graphs were analyzed for these values. It is understood that the physical behaviors of the resulting graphics and exact solution functions have similar characters. Thus, it was concluded that the modifie exponential function method gave very effective results in findin exact solutions of other nonlinear fractional partial differential equations define by the Atangana derivative. In subsequent studies, we will investigate new exact solutions of these models by applying them to other nonlinear fractional partial differential equations define by the Atangana derivative of the modifie exponential function method.

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Received: February 05, 2021; Accepted: March 10, 2021

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