Original article

Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives: A comparative study

  • Ismail Onder , a ,
  • Melih Cinar , a, b ,
  • Aydin Secer , a, c ,
  • Mustafa Bayram , c
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  • a Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
  • b Graduate School of Natural and Aplied Sciences, Yildiz Technical University, Istanbul, Turkey
  • c Department of Computer Engineering, Biruni University, Istanbul, Turkey
*E-mail addresses: (I. Onder),
(M. Cinar),
(A. Secer),

Received date: 2022-05-03

  Revised date: 2022-06-04

  Accepted date: 2022-06-06

  Online published: 2024-08-17

Abstract

This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived. We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system. Furthermore, some 2D and 3D graphical illustrations of the obtained solutions have been presented. The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted. The derived solutions may contribute to comprehending wave propagation in shallow water. So, the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.

Highlights

● Simplified modified Camassa-Holm equation with conformable and M- fractional derivative order is investigated.

● The novel solutions of considered equations are obtained analytically.

● The solutions of the conformable and M truncated model are graphically compared in the figures for different values of α and β that are in order of the derivative operator.

● The considered method suggests trigonometric functions producing dark, singular, and trigonometric solitons etc.

Cite this article

Ismail Onder , Melih Cinar , Aydin Secer , Mustafa Bayram . Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives: A comparative study[J]. Journal of Ocean Engineering and Science, 2024 , 9(3) : 240 -250 . DOI: 10.1016/j.joes.2022.06.012

1. Introduction

Modeling the wave motions in shallow water waves has been one of the areas that attracted the attention of researchers in the last eight decades [1], [2]. The mathematical model of the motion of the waves as other real phenomena is commonly described by partial differential equations [3], [4], [5], [6], [7]. Some of the phenomenon wave equations are Korteweg-de-Vries [8], Benjamin-Bona-Mahony [9], Boussinesq-like [10], [11], [12], [13], Davey-Stewartson [14], Novikov-Veselov [15], Whitham [16], Kadomstev-Petviashvili [17]. Besides, Roberto Camassa and Darryl D. Holm introduced a completely integrable dispersive shallow-water equation in 1993 [18]:
Θt+2kΘxΘxxt+3ΘΘx=2ΘxΘxx+ΘΘxxx,
where Θ=Θ(x,t) is the fluid velocity, k is a constant associated with the critical speed of wave of the shallow water. CH equation models the soliton interaction and wave breaking and is used for simulating shallow waters, coastal and harbors in ocean engineering [19], [20]. There are some studies on global conservative solutions of the Camassa-Holm (CH) equation [21], generalization of the equation [22], inverse scattering [23], conservation laws [24], long-time asymptotics [25], peakons of CH [26], finite propagation speed, [27] well-posedness, [28] etc. Besides, some researchers carried out some studies on the modifications and simplifications of the CH equation. In 2006, Abdul-Majid Wazwaz studied the modified Camassa-Holm (MCH) equation as follows [29]:
ΘtΘxxt+3Θ2Θx2ΘxΘxxΘΘxxx=0.
As a result of simplifying the equation above, the simplified modified Camassa-Holm equation (SMCH) [30] is obtained:
Θt+2kΘxΘxxt+ωΘ2Θx=0,
where k and ω are non-zero constants.
Fractional calculus has become quite well-known in a number of areas of science and engineering. It has been used to model a wide variety of dynamical processes and complicated nonlinear physical phenomena in the fields of physics, engineering, chemistry, viscoelasticity, and electromagnetics, etc [31], [32], [33], [34], [35]. This subject has risen to prominence in recent decades as a result of its extensive applicability in the aforementioned fields. Riemann-Liouville [36], [37], Caputo [38], and Caputo-Fabrizio [39] fractional derivative operators are the most frequently used operators in the literature. Besides, beta, M truncated and conformable derivative operators have been introduced recently. There are some recent studies in the literature on conformable and M-truncated derivative models in ocean engineering field. These studies [40], [41], [42], [43] are about obtaining soliton solutions of the models with local derivatives. On the other hand, the number of articles in which these conformable and M truncated derivative operators are used together and which are compared is very few. We believe that this study will help partially fill this gap in the literature. Therefore, it has become extremely important to solve the differential equations containing these kinds of derivatives analytically or numerically.
This paper mainly aims to derive the analytical solutions of the SMCH equations with conformable and M- truncated derivatives and compare the obtained solutions. To our best knowledge, an extended rational sinecosine and sinhcosh methods [44], [45] have been successfully applied to MCH equation including various derivative operators for the first time. The considered methods were used for the perturbed nonlinear Schrodinger equation [44], Boussinesq-like equations, [46], nonlinear fractional ϕ4 function [47], coupled nonlinear Schrdinger equation [48], Hirota-Satsuma coupled KdV equation [49], coupled Maccaris system [50] and Radhakrishnan-Kundu-Lakshmanan equation [51]. Some studies on SMCH equation which we consider are modified simple equation method [30], exp-function method on the fractional type [52], Hes semi-inverse method [53], Darboux transformation and multi-soliton solution [54], exp(ϕ(η)) -expansion method [55], novel (G/G) expansion method [56], sine-gordon expansion method [57], new auxiliary method [58] and unified solver method [59].
The remaining parts of the article are organized as: Some preliminaries are considered in the Secton 2. The considered methods and their algorithms are dealt with in Section 3. In Section 4, the governing models with various derivative operators are presented. In Section 5, we apply the method to SMCH equation with different definitions of the derivatives and compare the results in 2D and 3D figures. The results and discussions are included in Section 6. The conclusion is given in the final section.

2. Preliminary information

2.1. Conformable derivative

Definition 2.1.1. Let z:[0,)R be a function. Local conformable derivative of z(y) with α order is defined as follows [60]:
Dyα(z(y))=limh0z(y+hy1α)z(y)h,
where α(0,1],y>0 .
Theorem 2.1.1 [60] Let ϕ(y) and ψ(y) be α differentiable functions for α(0,1],y>0 . Then,
(i) Dyα(mϕ(y)+nψ(y))=mDyαϕ(y)+nDyαψ(y), m,nR,
(ii) Dyα(yn)=nynα,nR ,
(iii) If ϕ(y)=m in which m is a constant, then Dyα(m)=0,
(iv) Dyα(ψ(y)ϕ(y))=ψ(y)Dyαϕ(y)+ϕ(y)Dyαψ(y),
(v) Dyα(ϕ(y)ψ(y))=ψ(y)Dyαϕ(y)ϕ(y)Dlαψ(y)ψ2(y),ψ(y)0,
(vi) If the first derivative of ϕ(y) exists, then Dyα(ϕ(y))=y1αdϕ(y)dy.

2.2. Local M-truncated derivative

Definition 2.2.1. The truncated Mittag-Leffler function (TMLF) is defined as follows [61]:
iEβ(f)=j=0ifjΓ(βj+1),
in which β>0 and fC .
Definition 2.2.2. Let v:[0,)R be a function, the local M-truncated derivative of v of order α(0,1) with respect to (w.r.t.) y is given [61]:
DM,yα,βv(y)=limh0v(y+iEβ(hyα))v(y)h,
in which β,y>0 and iEβ(·) is a TMLF.
Theorem 2.2.1 Let v(y) be α order differentiable function at y0>0 where α(0,1] and β>0 . Then, v(y) is continuous at y0 [61] .
Theorem 2.2.2 [61] Let 0<α1,β>0,r,sR and ϕ,ψ be α differentiable at a point y>0. Then,
(1) DM,yα,β(rϕ+sψ)(y)=rDM,yα,βϕ(y)+sDM,yα,βψ(y) where r,s are real constants,
(2) DM,yα,β(ϕψ)(y)=ϕ(y)DM,yα,βψ(y)+ψ(y)DM,yα,βϕ(y),
(3) DM,yα,β(ϕψ)(y)=ϕ(y)DM,yα,βψ(y)ψ(y)DM,yα,βϕ(y)ψ(y)2,
(4) DM,yα,β(ϕψ)(y)=ϕ(y)DM,yα,βψ(y)ψ(y)DM,yα,βϕ(y)ψ(y)2,
(5) If ϕ is differentiable, then DM,yα,β(ϕ)(y)=y1αΓ(β+1)dϕ(y)dy .

3. Analysis of the method

(i) Consider the general form of the conformable PDE:
F(DxαΘ,DtαΘ,DxαDtαΘ,)=0,
and the traveling wave transformation:
Θ(x,t)=Y(η),η=1α(xαtα).
(ii) Consider the local M-truncated PDE in general form:
F(DM,xα,βΘ,DM,tα,βΘ,DM,xα,βDM,tα,βΘ,)=0,
and the traveling wave transformation:
Θ(x,t)=Y(η),η=Γ(1+β)α(xαtα).
where Θ(x,t) is the unknown function, D*αΘ and DM,*α,βΘ denote α order conformable and M-truncated derivative of Θ w.r.t. *(x or t) , respectively. Inserting the wave transformation in Eq. (8) into the PDE in Eq. (7) turns into the following nonlinear ODE in general form:
K(Y,Y,Y,)=0
where Y is the unknown functions of η and the superscripts represent ordinary differential operator ddη .

3.1. Extended rational sinecosine technique

Step 1: Assume that Eq. (11) has the solution as:
Y(η)=δ0sin(μη)δ2+δ1cos(μη),cos(μη)δ2δ1,
or
Y(η)=δ0cos(μη)δ2+δ1sin(μη),sin(μη)δ2δ1,
where μ is the wave number and δi are parameters to be found for i=0,1,2 .
Step 2: Substitute Eq. (12) or Eq. (13) to Eq. 11, collect the all terms including the same powers of cos(μη) or sin(μη) and then equate to zero the coefficients of trigonometric functions which are cos(μη) or sin(μη) gives a system of equations. When the system is solved, the unknowns (δ0,δ1,δ2 and μ ) can be found.
Step 3: With substitution δ0,δ1,δ2 and μ to Eq. (12) or Eq. (13), the solutions of Eq. (11) can be found.

3.2. Extended rational sinhcosh technique

Step 1: Assume that Eq. (11) has the solution as:
Y(η)=δ0sinh(μη)δ2+δ1cosh(μη),cosh(μη)δ2δ1,
or
Y(η)=δ0cosh(μη)δ2+δ1sinh(μη),sinh(μη)δ2δ1.
where μ is the wave number and δi are parameters to be found for i=0,1,2 .
Step 2: Substitute Eq. (12) or Eq. (13) to Eq. (11), collect the all terms including the same powers of cosh(μη) or sinh(μη) and then equate to zero the coefficients of trigonometric functions which are cosh(μη) or sinh(μη) gives a system of equations. When the system is solved, the unknowns (δ0,δ1,δ2 and μ ) can be found.
Step 3:By substituting δ0,δ1,δ2,μ and η to Eq. (14) or Eq. (15), the solutions of Eq. (11) can be found.

4. Governing model

In this section, we deal with the simplified modified Camassa-Holm equation (SMCH) w.r.t. different definitions of derivatives that are conformable and M-truncated [59]:

4.1. Conformable SMCH equation

Conformable SMCH equation can be written as follows [59]:
DtαΘ+λDxαΘDxxt3αΘ+ωDtαΘ3=0,
where Θ=Θ(x,t) and λ,ω are non-zero real parameters. We use following wave transformations:
Θ(x,t)=Y(η),η=xααγtαα.

4.2. M-truncated SMCH equation

M-truncated SMCH equation can be written as follows [59]:
iDM,tα,βΘ+λiDM,xα,βΘiDM,xxt3α,βΘ+ωiDM,tα,βΘ3=0.
For M-truncated derivative, we use the following wave transformations:
Θ(x,t)=Y(η),η=Γ(1+β)α(xαγtα).

4.3. Solving the SMCH equations with conformable and M-truncated derivatives

Inserting the wave transformations in (17), (19) to the Eq. (16) and Eq. (18), respectively, we obtain the following nonlinear ODE:
γY(η)+ωY3(η)+(λγ)Y(η)=0.

5. Application

5.1. Conformable SMCH equation

5.1.1. Solving via extended rational sinecosine technique

Let us suppose that the solutions of the Eq. (20) are:
Y(η)=δ0sin[μη]δ2+δ1cos[μη].
Let us insert Eq. (21) into the Eq. (20) and collect all terms that includes the same power of cos[μη]m , one can find the following algebraic equation system:
cos(μη)2:δ12δ0γ+δ12δ0λ+δ03(ω)=0,cos(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,cos(μη)0:8δ12δ0γμ24δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved:
Set 1.
{μ=±γλ2γ,δ0=±δ12(λγ)ω,δ1=δ1,δ2=0}.
Choosing the set 1, we get the solutions of Eq. (20) as:
Y1(η)=±λγωtan[ηγλ2γ].
Considering set 1 and Eq. (24), one can derive:
Θ11(x,t)=λγωtan[(xαγtα)αγλ2γ],
Θ12(x,t)=λγωtan[(xαγtα)αγλ2γ].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 2.
{μ=±2γλγ,δ0=±δ12(λγ)ω,δ2=δ2,δ1=±δ2}.
Choosing the set 2, we get the solutions of Eq. (20) as:
Y2(η)=±λγωsin[2γλγη]δ2+δ1cos[2γλγη].
Considering set 2 and Eq. (28), one can derive:
Θ21(x,t)=λγωsin[2γλγ(xαγtα)α]1+cos[2γλγ(xαγtα)α],
Θ22(x,t)=λγωsin[2γλγ(xαγtα)α]1+cos[2γλγ(xαγtα)α],
Θ23(x,t)=λγωsin[2γλγ(xαγtα)α]1cos[2γλγ(xαγtα)α],
Θ24(x,t)=λγωsin[2γλγ(xαγtα)α]1cos[2γλγ(xαγtα)α].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Suppose that the solutions of the Eq. (20) are:
Y(η)=δ0cos[μη]δ2+δ1sin[μη].
Let us insert Eq. (33) into the Eq. (20) and collect all terms that includes the same power of sin[μη]m , one can find the following algebraic equation system:
sin(μη)2:δ12δ0γδ12δ0λ+δ03ω=0,sin(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,sin(μη)0:8δ12δ0γμ24δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved:
Set 3.
{μ=±γλ2γ,δ0=±δ12(λγ)ω,δ1=δ1,δ2=0}.
Choosing the set 3, we get the solutions of Eq. (20) as:
Y3(η)=±(λγ)ωcot[γλ2γη].
Considering set 3 and Eq. (36), one can derive:
Θ31(x,t)=(λγ)ωcot[γλ2γ(xαγtα)α],
Θ32(x,t)=(λγ)ωcot[γλ2γ(xαγtα)α],
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 4.
{μ=±2γλγ,δ0=±δ12(λγ)ω,δ2=δ2,δ1=±δ2}.
Choosing the set 4, we get the solutions of Eq. (20) as:
Y4(η)=±(λγ)ωcos[2γλγη]δ2±δ1sin[2γλγη].
Considering set 4 and Eq. (40), one can derive:
Θ41(x,t)=(λγ)ωcos[2γλγ(xαγtα)α]1+sin[2γλγ(xαγtα)α],
Θ42(x,t)=(λγ)ωcos[2γλγ(xαγtα)α]1+sin[2γλγ(xαγtα)α],
Θ43(x,t)=(λγ)ωcos[2γλγ(xαγtα)α]1sin[2γλγ(xαγtα)α],
Θ44(x,t)=(λγ)ωcos[2γλγ(xαγtα)α]1sin[2γλγ(xαγtα)α].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .

5.1.2. Solving via extended rational sinhcosh technique

Let us suppose that the solutions of the Eq. (20) are:
Y(η)=δ0sinh[μη]δ2+δ1cosh[μη].
Let us insert Eq. (45) into the Eq. (20) and collect all terms that includes the same power of cosh[μη]m , one can find the following algebraic equation system:
cosh(μη)2:δ12δ0γ+δ12δ0λ+δ03ω=0,cosh(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,cosh(μη)0:8δ12δ0γμ2+4δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ3δ03ω=0.
The set below can be derived when the system above is solved:
Set 5.
{μ=±λγ2γ,δ0=±δ1γλω,δ1=δ1,δ2=0}.
Choosing the set 5, we get the solutions of Eq. (20) as:
Y5(η)=±γλωtanh[λγ2γη].
Considering set 5 and Eq. (48), one can derive:
Θ51(x,t)=γλωtanh[λγ2γ(xαγtα)α],
Θ52(x,t)=γλωtanh[λγ2γ(xαγtα)α],
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 6.
{μ=±2λγγ,δ0=±δ1γλω,δ1=δ1,δ2=0}.
Choosing the set 6, we get the solutions of Eq. (20) as:
Y6(η)=±γλωsinh[2λγγη]δ2±δ1cosh[2λγγη].
Considering set 6 and Eq. (52), one can derive:
Θ61(x,t)=γλωsinh[2λγγ(xαγtα)α]1+cosh[2λγγ(xαγtα)α],
Θ62(x,t)=γλωsinh[2λγγ(xαγtα)α]1+cosh[2λγγ(xαγtα)α],
Θ63(x,t)=γλωsinh[2λγγ(xαγtα)α]1cosh[2λγγ(xαγtα)α],
Θ64(x,t)=γλωsinh[2λγγ(xαγtα)α]1cosh[2λγγ(xαγtα)α].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Suppose that the solutions of the Eq. (20) are:
Y(η)=δ0cosh[μη]δ2+δ1sinh[μη].
Let us insert Eq. (57) into the Eq. (20) and collect all terms that includes the same power of sinh[μη]m , one can find the following algebraic equation system:
sinh(μη)2:δ12δ0γ+δ12δ0λ+δ03ω=0,sinh(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,sinh(μη)0:8δ12δ0γμ2+4δ22δ0γμ2+δ12δ0γ4δ22δ0γδ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved: Set 7.
{μ=±λγ2γ,δ0=±a1γλω,δ1=δ1,a2=0}.
Choosing the set 7, we get the solutions of Eq. (20) as:
Y7(η)=±γλωcoth[λγ2γη].
Considering set 7 and Eq. (60), one can derive:
Θ71(x,t)=γλωcoth[λγ2γ(xαγtα)α],
Θ72(x,t)=γλωcoth[λγ2γ(xαγtα)α].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 8.
{μ=±2λγγ,δ0=±a1γλω,δ2=δ2,δ1=±δ2i}.
Choosing the set 8, we get the solutions of Eq. (20) as:
Y8(η)=±γλωcosh[2λγγη]δ2±δ1isinh[2λγγη].
Considering set 8 and Eq. (64), one can derive:
Θ81(x,t)=γλωcosh[2λγγ(xαγtα)α]1+isinh[2λγγ(xαγtα)α],
Θ82(x,t)=γλωcosh[2λγγ(xαγtα)α]1+isinh[2λγγ(xαγtα)α],
Θ83(x,t)=γλωcosh[2λγγ(xαγtα)α]1isinh[2λγγ(xαγtα)α],
Θ84(x,t)=γλωcosh[2λγγ(xαγtα)α]1isinh[2λγγ(xαγtα)α].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .

5.2. M-truncated SMCH equation

5.2.1. Solving via extended rational sinecosine technique

Let us suppose that the solutions of the Eq. (20) are:
Y(η)=δ0sin[μη]δ2+δ1cos[μη].
Let us insert Eq. (69) into the Eq. (20) and collect all terms that includes the same power of cos[μη]m , one can find the following algebraic equation system:
cos(μη)2:δ12δ0γ+δ12δ0λ+δ03(ω)=0,cos(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,cos(μη)0:8δ12δ0γμ24δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved: Set 1.
{μ=±γλ2γ,δ0=±δ12(λγ)ω,δ1=δ1,δ2=0}.
Choosing the set 1, we get the solutions of Eq. (20) as:
Y1(η)=±λγωtan[ηγλ2γ].
Considering set 1 and Eq. (72), one can derive:
Θ11(x,t)=λγωtan[γλ2γ(Γ(β+1)(xαγtα)α)],
Θ12(x,t)=λγωtan[γλ2γ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 2.
{μ=±2γλγ,δ0=±δ12(λγ)ω,δ2=δ2,δ1=±δ2}.
Choosing the set 2, we get the solutions of Eq. (20) as:
Y2(η)=±λγωsin[2γλγη]δ2+δ1cos[2γλγη].
Considering set 2 and Eq. (76), one can derive:
Θ21(x,t)=λγωsin[2γλγ(Γ(β+1)(xαγtα)α)]1+cos[2γλγ(Γ(β+1)(xαγtα)α)],
Θ22(x,t)=λγωsin[2γλγ(Γ(β+1)(xαγtα)α)]1+cos[2γλγ(Γ(β+1)(xαγtα)α)],
Θ23(x,t)=λγωsin[2γλγ(Γ(β+1)(xαγtα)α)]1cos[2γλγ(Γ(β+1)(xαγtα)α)],
Θ24(x,t)=λγωsin[2γλγ(Γ(β+1)(xαγtα)α)]1cos[2γλγ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Suppose that the solutions of the Eq. 20 are:
Y(η)=δ0cos[μη]δ2+δ1sin[μη].
Let us insert Eq. (81) into the Eq. (20) and collect all terms that includes the same power of sin[μη]m , one can find the following algebraic equation system:
sin(μη)2:δ12δ0γδ12δ0λ+δ03ω=0,sin(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,sin(μη)0:8δ12δ0γμ24δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved:
Set 3.
{μ=±γλ2γ,δ0=±δ12(λγ)ω,δ1=δ1,δ2=0}.
Choosing the set 3, we get the solutions of Eq. (20) as:
Y3(η)=±(λγ)ωcot[γλ2γη].
Considering set 3 and Eq. (84), one can derive:
Θ31(x,t)=(λγ)ωcot[γλ2γ(Γ(β+1)(xαγtα)α)],
Θ32(x,t)=(λγ)ωcot[γλ2γ(Γ(β+1)(xαγtα)α)],
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 4.
{μ=±2γλγ,δ0=±δ12(λγ)ω,δ2=δ2,δ1=±δ2}.
Choosing the set 4, we get the solutions of Eq. (20) as:
Y4(η)=±(λγ)ωcos[2γλγη]δ2±δ1sin[2γλγη].
Considering set 4 and Eq. (40), one can derive:
Θ41(x,t)=(λγ)ωcos[2γλγ(Γ(β+1)(xαγtα)α)]1+sin[2γλγ(Γ(β+1)(xαγtα)α)],
Θ42(x,t)=(λγ)ωcos[2γλγ(Γ(β+1)(xαγtα)α)]1+sin[2γλγ(Γ(β+1)(xαγtα)α)],
Θ43(x,t)=(λγ)ωcos[2γλγ(Γ(β+1)(xαγtα)α)]1sin[2γλγ(Γ(β+1)(xαγtα)α)],
Θ44(x,t)=(λγ)ωcos[2γλγ(Γ(β+1)(xαγtα)α)]1sin[2γλγ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .

5.2.2. Solving via extended rational sinhcosh technique

Assume that the solutions of the Eq. (20) are:
Y(η)=δ0sinh[μη]δ2+δ1cosh[μη].
Let us insert Eq. (93) into the Eq. (20) and collect all terms that includes the same power of cosh[μη]m , one can find the following algebraic equation system:
cosh(μη)2:δ12δ0γ+δ12δ0λ+δ03ω=0,cosh(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,cosh(μη)0:8δ12δ0γμ2+4δ22δ0γμ2δ12δ0γ4δ22δ0γ+δ12δ0λ+4δ22δ0λ3δ03ω=0.
The set below can be derived when the system above is solved: Set 5.
{μ=±λγ2γ,δ0=±δ1γλω,δ1=δ1,δ2=0}.
Choosing the set 5, we get the solutions of Eq. (20) as:
Y5(η)=±γλωtanh[λγ2γη].
Considering set 5 and Eq. (96), one can derive:
Θ51(x,t)=γλωtanh[λγ2γ(Γ(β+1)(xαγtα)α)],
Θ52(x,t)=γλωtanh[λγ2γ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 6.
{μ=±2λγγ,δ0=±δ1γλω,δ1=δ1,δ2=0}.
Choosing the set 6, we get the solutions of Eq. (20) as:
Y6(η)=±γλωsinh[2λγγη]δ2±δ1cosh[2λγγη].
Considering set 6 and Eq. (100), one can derive:
Θ61(x,t)=γλωsinh[2λγγ(Γ(β+1)(xαγtα)α)]1+cosh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ62(x,t)=γλωsinh[2λγγ(Γ(β+1)(xαγtα)α)]1+cosh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ63(x,t)=γλωsinh[2λγγ(Γ(β+1)(xαγtα)α)]1cosh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ64(x,t)=γλωsinh[2λγγ(Γ(β+1)(xαγtα)α)]1cosh[2λγγ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 . Suppose that the solutions of the Eq. (20) are:
Y(η)=δ0cosh[μη]δ2+δ1sinh[μη].
Let us insert Eq. (105) into the Eq. (20) and collect all terms that includes the same power of sinh[μη]m , one can find the following algebraic equation system:
sinh(μη)2:δ12δ0γ+δ12δ0λ+δ03ω=0,sinh(μη)1:2δ0δ1δ2γμ24δ0δ1δ2γ+4δ0δ1δ2λ=0,sinh(μη)0:8δ12δ0γμ2+4δ22δ0γμ2+δ12δ0γ4δ22δ0γδ12δ0λ+4δ22δ0λ+3δ03ω=0.
The set below can be derived when the system above is solved:
Set 7.
{μ=±λγ2γ,δ0=±a1γλω,δ1=δ1,a2=0}.
Choosing the set 7, we get the solutions of Eq. (20) as:
Y7(η)=±γλωcoth[λγ2γη].
Considering set 7 and Eq. (108), one can derive:
Θ71(x,t)=γλωcoth[λγ2γ(Γ(β+1)(xαγtα)α)],
Θ72(x,t)=γλωcoth[λγ2γ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .
Set 8.
{μ=±2λγγ,δ0=±a1γλω,δ2=δ2,δ1=±δ2i}.
Choosing the set 8, we get the solutions of Eq. (20) as:
Y8(η)=±γλωcosh[2λγγη]δ2±δ1isinh[2λγγη].
Considering set 8 and Eq. (112), one can derive:
Θ81(x,t)=γλωcosh[2λγγ(Γ(β+1)(xαγtα)α)]1+isinh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ82(x,t)=γλωcosh[2λγγ(Γ(β+1)(xαγtα)α)]1+isinh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ83(x,t)=γλωcosh[2λγγ(Γ(β+1)(xαγtα)α)]1isinh[2λγγ(Γ(β+1)(xαγtα)α)],
Θ84(x,t)=γλωcosh[2λγγ(Γ(β+1)(xαγtα)α)]1isinh[2λγγ(Γ(β+1)(xαγtα)α)].
The solutions exist for ω(γλ)>0 and γ(γλ)>0 .

6. Results and discussion

In this paper, the powerful sinecosine and sinhcosh techniques have been used for obtaining some analytical solutions of the SMCH equations with two new kinds of derivative operators which are conformable and M-truncated. A variety of solutions have been obtained using the methods and the derived solutions for two different derivative operators have been compared in 3D and 2D graphics by using Mathematica 12.
In Fig. 1, we depict the 3D and 2D graphs of the solutions Θ24(x,t) of the conformable and M-truncated SMCH equation. In Fig. 1a, 3D comparison between Eq. (32) (conformable) and Eq. (80) (M-truncated) are plotted in three dimensional for α=0.5 and β=0.6 . In Fig. 1b, we present a 2D comparison between the conformable solution with α=0.7 and M-truncated solution with α=0.7,β=0.4 for SMCH equation. The solution Θ24(x,t) in Fig. 1 demonstrates a singular-periodic solution.
Fig.1 Comparison of solutions of conformable in Eq. (32) and M-truncated derivative in Eq. (80) with γ=4,λ=2, and ω=3 (Singular-periodic solution).
In Fig. 2, we depict the 3D and 2D graphs of the solutions Θ51(x,t) . For SMCH equation with conformable derivative, the solutions Θ51(x,t) are given in Eq. (49) and the solutions Θ51(x,t) are given in Eq. (97) for SMCH equation with M-truncated derivative. In Fig. 2a, comparison between Eq. (49) and Eq. (97) are plotted in three dimensional for α=0.4 and β=0.6 . In Fig. 2b, we depicted the 2D plots of the solution with γ=0.7 , λ=70 , ω=0.2 , α=0.4 for conformable solution and α=0.4,β=0.6 for solutions for SMCH with M-truncated derivative solution of Θ51(x,t) . The solution Θ51(x,t) in Fig. 2 represents a kink solution.
Fig.2 Comparison of solutions of conformable in Eq. (49) and M-truncated derivative in Eq. (97) with γ=0.7,λ=70, and ω=0.2 (Kink soliton).
Fig. 3 shows the effects of changing of the parameters on the wave propagation. In Fig. 2b, we illustrate the effect of the ω parameter on M-truncated and conformable, respectively. The parameters ω=0.2,0.3,0.4 and fixed γ=0.7 , λ=70 are used for the solution Θ51(x,t) in (49), (97) where α=0.4 , β=0.5 . Since the parameter ω is in the denominator of the obtained solutions, the wave amplitude increases when ω decreases. In Fig. 2, Fig. 3 a, we demonstrate the effect of the γ parameter on M-truncated and conformable, respectively where γ=0.7,0.8,0.9 and fixed ω=0.2 , λ=70 . When the parameter γ increases, the wave moves to the left along the t-axis. In Fig. 3c, we give the effect of the parameter λ on solutions with M-truncated and conformable, respectively where λ=70,80,90 and fixed ω=0.2 , γ=0.7 . There are a positive correlation between the parameter λ and the wave amplitude.
Fig.3 The effects of the parameters to solutions Conformable in Eq. (49) with M-truncated and Eq. (97) with fixed α=0.4 and β=0.6 (Kink soliton).
In Fig. 4, the some plots of the solutions Θ64(x,t) are illustrated. For SMCH equation with conformable derivative, the solutions Θ64(x,t) are given in Eq. (56) and the solutions Θ64(x,t) are given in Eq. (104) for SMCH equation with M-truncated derivative. Fig. 4a shows the 3D plots of the conformable with α=0.5 and solutions for SMCH with M-truncated derivative with α=0.5,β=0.6 . The solution is known as singular solution. Fig. 4b shows the 2D plots of conformable with α=0.4,0.5 and solutions for SMCH with M-truncated derivative with fixed β=0.1 and α=0.4,0.5 . The figure of the solution Θ64(x,t) illustrates a singular solution.
Fig.4 Comparison of solutions of conformable in Eq. (56) and M-truncated derivative in Eq. (104) with γ=2,λ=4, and ω=3 (Singular solution).
In Fig. 5, we present the three-dimensional and two-dimensional graphs of the solution Θ71(x,t) . In Fig. 5a, we depict the 2D plots of solutions in Eq. (109) for SMCH with M-truncated derivative where α=0.5 and various β=1,2,3 and β=4 . Fig. 5a represents a comparison between the solutions Θ71(x,t) of SMCH equations with the conformable and M-truncated derivative. Fig. 5b shows that the solutions of SMCH equations with the conformable and M-truncated derivative overlap for β=1 . The solution Θ71(x,t) represent a singular solution.
Fig.5 Various 2D comparisons where γ=2,λ=4, and ω=3.
In [58], the SMCH equation with classical derivative was studied and some closed-form wave solutions for the equation were derived. When our results are compared with the reference [58], our solutions in more general solutions than the solutions in the reference. Because we have used the models with the conformable and M-truncated, which are one of the general forms of the classical derivative. In addition, our method is powerful and effective such that it presents many solutions which include kinds of solitons. Also, our study is one of the rare studies examining the effect of these two local derivative operators on the same model. Not only the effect of the operators is examined, but also the effect of the parameters in the model on the wave motion. We expect that our results might be helpful for future studies in ocean engineering and science.
The figures include various kinds of solitons and simulate the propagation and breaking of shallow water waves in oceans, lakes, and rivers. Understanding the wave propagation and interactions contributes to important topics in the ocean and coastal engineering such as the development of autonomous ships/underwater vehicles and designing coastal defense schemes, which the engineers must consider. So, it is expected that the results might be helpful for the ocean and coastal engineering.

7. Conclusion

In this paper, we have dealt with the SMCH including conformable or M-truncated derivatives instead of the model with the classical derivative, which has been commonly studied in the literature. We have applied the extended rational sinecosine and sinhcosh methods to construct analytical solutions for SMCH equation modeling the unidirectional propagation of shallow-water waves. As a main contribution of the work, a variety of exact solutions for the SMCH equation with various derivative operators have been successfully extracted in the rational forms of trigonometric and hyperbolic functions. Some of the obtained solutions using the considered methods have been compared in 3D surface and 2D line plots. In the figures, the effect of the used different derivative operators in the SMCH equations has been illustrated. It can be deduced that M- truncated derivative is reduced to conformable derivative when β=1, and it is reduced to classical derivative for β=α=1. The derived result shows that the techniques are efficient and easily applicable for producing analytical solutions of non-linear PDEs, including various kinds of derivatives. The obtained solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.

Declaration of Competing Interest

Authors declare that there is no conflict of interest whatsoever.

Acknowledgment

The second author (MC) would like to thank the Scientific and Technological Research Council of Turkey (TUBITAK) for the financial support of the 2211-A Fellowship Program.
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