Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

Abstract

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

1. Introduction

Fractional calculus appeared in the middle of the 17th century. However, it is now attracting substantial interest from scientists due to its applications in many fields [1,2,3,4,5,6]. Many researchers have directed their studies to fractional calculus. Several different definitions of fractional operators have been presented in the literature since the middle of the 17th century. These operators play an important role in understanding the characteristic properties of real-world problems. One of the most significant operators of the fractional derivatives is the Caputo operator [7,8]. This operator satisfies the basic rules of classical calculus. In this regard, Brzezinski presented the comparisons of fractional definitions [9]. Youssef and his team applied the Haar wavelet to extract the solutions of Poisson’s Equation in [10]. Eslami and his team observed the general features of the Wu–Zhang system, including a conformable operator [11]. The fundamental properties of hepatitis E virus were observed via the Caputo–Fabrizio operator in [12]. Many important models and their deep properties were investigated by using a conformable operator in [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].

In this paper, firstly, we consider the nonlinear Gardner equation containing a conformable operator in the following form [33,34,35,36,37]:

$$u_t^{\gamma }(x,t)+6[u(x,t)-{\lambda }^{2}u{(x,t)}^2]u_x(x,t)+u_{xxx}(x,t)=0,t\ge 0,0<\gamma \le 1,$$

(1)

where $\lambda$ is a nonzero real number, $u(x,t)$ is a dependent function of x and t, the terms $u{u}_x$ and $u^2{u}_x$ are used to represent the nonlinear wave, and $u_{xxx}$ is used to explain the spreading of waves. Equation (1), formed by combining KdV and mKdV equations, is used to describe the interior shallow water solitary waves.

Secondly, the nonlinear (3+1)-dimensional mKdV-ZKE containing a conformable operator given by [38]

$$u_t^{\gamma }(x,y,z,t)+pu{(x,y,z,t)}^2{u}_x(x,y,z,t)+u_{xxx}(x,y,z,t)+u_{xyy}(x,y,z,t)+u_{zzx}(x,y,z,t)=0,$$

(2)

is studied. In Equation (2), $t>0,0<\gamma \le 1$ and also p is a nonzero real number. $u(x,y,z,t)$ is a dependent function of $x,y,z$ and t, the term $u^2{u}_x$ is used to represent the nonlinear waves, and $u_{xxx}$ is used to explain the spreading of waves.

The rest of the paper is organized as follows. In Section 2, we give some definitions and theorems related to the conformable operator. In Section 3, we present the general properties of the rational sine-Gordon expansion method (RSGEM). In Section 4, we apply the RSGEM to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator to obtain analytical solutions such as periodic, singular, trigonometric, and traveling solutions. Section 5 contains the discussion and physical meanings of the results reported in this paper. Finally, we present a conclusion along with ideas about future work regarding this framework in Section 6.

2. General Properties of Conformable Operator

This section presents the definition and theorem about the conformable operator as follows [7].

Definition 1. 

Given a function $f:[0,\infty )⟶R,$ then the conformable operator definition of $f\left(t\right)$ order α is defined as

$$T_{\alpha }\left(f\right)\left(t\right)=\underset{\epsilon \to 0}{lim}\frac{f(t+\epsilon t^{1-\alpha })-f\left(t\right)}{\epsilon }$$

for all $t>0$, $\alpha \in (0,1).$ If f is α-differentiable in $(0,a),a>0,$ and ${lim}_{t\to 0^{+}}{f}^{\left(\alpha \right)}\left(t\right)$ exists, then we define$f^{\left(\alpha \right)}\left(0\right)={lim}_{t\to 0^{+}}{f}^{\left(\alpha \right)}\left(t\right).$

A conformable operator satisfies some properties given in the following theorem.

Theorem 1. 

Let $\beta \in (0,1]$ and $f,g$ be β-differentiable at point $t>0$. Then,

(1) $T_{\beta }(af+bg)=a{T}_{\beta }\left(f\right)+b{T}_{\beta }\left(g\right)$, for all $a,b\in R$;

(2) $T_{\beta }\left(t^p\right)=p{t}^{p-\beta }$ foll all $p\in R$;

(3) $T_{\beta }\left(\chi \right)=0$, for all constant functions $f\left(t\right)=\chi$;

(4) $T_{\beta }\left(fg\right)=f{T}_{\beta }\left(g\right)+g{T}_{\beta }\left(f\right)$;

(5) $T_{\beta }\left(\frac{f}{g}\right)=\frac{g{T}_{\beta }\left(f\right)-f{T}_{\beta }\left(g\right)}{g^2}$;

(6) If f is differentiable, then $T_{\beta }\left(f\right)\left(t\right)=t^{1-\beta }\frac{df\left(t\right)}{dt}.$

3. General Properties of RSGEM

In this section, we introduce the general properties of RSGEM. Before presenting the RSGEM based on the sine-Gordon equation, we need to investigate the sine-Gordon equation.

3.1. The Sine-Gordon Equation

The sine-Gordon equation is given by [39,40,41]

$$u_{xx}-u_{tt}=m^{2}sin\left(u\right),$$

(3)

where $u=u(x,t)$, m is a nonzero real number. Applying the wave transformation given as $u=u(x,t)=U\left(\xi \right),\xi =\mu (x-ct)$ to Equation (3) yields

$$U^{″}=\frac{m^2}{{\mu }^2(1-c^2)}sin\left(U\right),$$

(4)

where $U=U\left(\xi \right)$, $U^{″}=\frac{d^{2}U}{d{\xi }^2}$, and c is the velocity of the wave. After some calculations, we obtain

$${(\frac{U^{\prime }}{2})}^2=\frac{m^2}{{\mu }^2(1-c^2)}{sin}^2(\frac{U}{2})+r,$$

(5)

where r is an integral constant and a nonzero real number. Taking $r=0$, $w\left(\xi \right)=\frac{U\left(\xi \right)}{2}$ and $a^2=\frac{m^2}{{\mu }^2(1-c^2)}$, Equation (5) becomes

$$w^{\prime }=asin\left(w\right).$$

(6)

In (6), if $a=1$, we reach

$$w^{\prime }=sin\left(w\right).$$

(7)

Solving (7) by using the separating variable method

$$\frac{dw}{d\xi }=sin\left(w\right)\Rightarrow \frac{1}{sin\left(w\right)}dw=d\xi ,$$

we obtain the following two important properties:

$$sin\left(w\right)=sin\left(w\left(\xi \right)\right)=\frac{2p{e}^{\xi }}{p^2{e}^{2\xi }+1}\mid p_{=1}=\sec \mathrm{h}\left(\xi \right),$$

(8)

$$cos\left(w\right)=cos\left(w\left(\xi \right)\right)=\frac{p^2{e}^{2\xi }-1}{p^2{e}^{2\xi }+1}\mid p_{=1}=\tan \mathrm{h}\left(\xi \right).$$

(9)

where p is a nonzero real number.

3.2. The RSGEM

RSGEM is the generalized version of the sine-Gordon expansion method (SGEM). Let us consider the nonlinear partial differential equation given by

$$P(u,u_x,u_t,u_{xx},u_{tt},u^2,\cdots )=0.$$

(10)

If we apply $u=u(x,t)=U\left(\xi \right),\xi =\mu (x-ct)$ into Equation (10), we get the following nonlinear ordinary differential equation (NODE):

$$N(U,U^{\prime },U^{″},U^2,\cdots )=0,$$

(11)

where $U=U\left(\xi \right),U^{\prime }=\frac{dU}{d\xi },U^{″}=\frac{d^{2}U}{d{\xi }^2}$. The test function of solution formula for Equation (11) is considered as [42]

$$U\left(\xi \right)=\frac{{\sum }_{i=1}^n{\tan \mathrm{h}}^{i-1}\left(\xi \right)[A_i\sec \mathrm{h}\left(\xi \right)+C_i\tan \mathrm{h}\left(\xi \right)]+A_0}{{\sum }_{j=1}^m{\tan \mathrm{h}}^{j-1}\left(\xi \right)[B_j\sec \mathrm{h}\left(\xi \right)+D_j\tan \mathrm{h}\left(\xi \right)]+B_0}.$$

(12)

Integrating Equations (8) and (9) into Equation (12), it can be rewritten in the following form:

$$U\left(w\right)=\frac{{\sum }_{i=1}^n{cos}^{i-1}\left(w\right)[A_{i}sin\left(w\right)+C_{i}cos\left(w\right)]+A_0}{{\sum }_{j=1}^m{cos}^{j-1}\left(w\right)[B_{j}sin\left(w\right)+D_{j}cos\left(w\right)]+B_0},$$

(13)

where $A_0,A_i,C_i,B_0,B_j,D_j$ are nonzero real numbers to be determined later. It is known that the rational functions are more general than normal polynomial functions with SGEM. If we consider the solution function as the rational function, this means that we have one more parameter. This parameter produces more different solutions to the model studied. Putting Equation (13) into Equation (11), we can obtain the values of $\lambda ,\mu ,A_0,A_i,C_i,B_0,B_j,D_j$. When we integrate these values of parameters into Equation (12), we find the solutions of Equation (10).

4. Applications of RSGEM

This part applies the RSGEM to the Equations (1) and (2) to obtain some traveling wave solutions such as periodic, trigonometric, traveling, complex and hyperbolic solutions.

4.1. RSGEM to the Gardner Equation including a Conformable Operator

Considering the wave transformation formula given as

$$u(x,t)=U\left(\xi \right),\xi =\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma },$$

(14)

where $\alpha$ and $\kappa$ are nonzero real numbers, we convert Equation (1) into NODE given by

$${\alpha }^3{U}^{‴}-\kappa U^{\prime }+3\alpha {\left(U^2\right)}^{\prime }-2\alpha {\lambda }^2{\left(U^3\right)}^{\prime }=0.$$

(15)

Integrating (15) with respect to $\xi$ yields

$${\alpha }^3{U}^{″}-\kappa U+3\alpha U^2-2\alpha {\lambda }^2{U}^3=0.$$

(16)

In (16), the integral constant is zero. Especially, if we take $n=m$ in (13), by the balance principle, we have

$$U\left(w\right)=\frac{A_{1}sin\left(w\right)+C_{1}cos\left(w\right)+A_0}{B_{1}sin\left(w\right)+D_{1}cos\left(w\right)+B_0}.$$

(17)

where $A_1\ne B_1,C_1\ne D_1,A_0\ne B_0$ in the same time. Substituting (17) into (16), the following solutions are obtained.

Case 1. If $A_0=-C_1,B_0=-\frac{2C_1}{{\alpha }^2}+D_1,B_1=\frac{A_1^2+C_1^2-{\alpha }^2{C}_1{D}_1}{{\alpha }^2{A}_1},\lambda =\frac{1}{\alpha },\kappa ={\alpha }^3$, we get

$$u_1(x,t)=\frac{{\alpha }^2{A}_1}{A_1+(-C_1+{\alpha }^2{D}_1)e^{\alpha x-\frac{t^{\gamma }}{\gamma }{\alpha }^3}}.$$

(18)

Figure 1 shows 3D and 2D graphs of (18) under the suitable values of parameters.

Case 2. When $A_1=0,A_0=-C_1,B_0=-\frac{C_1}{2{\alpha }^2}+D_1,B_1=0,\lambda =-\frac{\sqrt{-\frac{C_1}{{\alpha }^2}+4D_1}}{2\sqrt{-C_1+4{\alpha }^2{D}_1}}$ and $\kappa =4{\alpha }^3$, it gives

$$u_2(x,t)=\frac{-C_1+C_1\tan \mathrm{h}(\alpha x-\frac{4t^{\gamma }}{\gamma }{\alpha }^3)}{\frac{-C_1}{2{\alpha }^2}+D_1+D_1\tan \mathrm{h}(\alpha x-\frac{4{\alpha }^3}{\gamma }{t}^{\gamma })}.$$

(19)

Figure 2 presents 3D and 2D graphs of (19) under the suitable values of parameters.

Case 3. Taken as $A_1=-\frac{i\sqrt{2}\sqrt{{\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2}\sqrt{B_0^2-D_1^2}}{3\alpha },C_1=\frac{D_1}{3\alpha }({\alpha }^3+2\kappa ),A_0=\frac{({\alpha }^3+2\kappa )B_0}{3\alpha }$, $B_1=\frac{i({\alpha }^3+2\kappa )\sqrt{{\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2}\sqrt{B_0^2-D_1^2}-3\sqrt{-{\alpha }^6({\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2)(B_0^2-D_1^2)}}{\sqrt{2}({\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2)},\lambda =-\frac{3\sqrt{\alpha ({\alpha }^3+\kappa )}}{\sqrt{2}\sqrt{{({\alpha }^3+2\kappa )}^2}}$, we obtain

$$u_3(x,t)=\frac{\frac{\tau B_0}{3\alpha }-\frac{i\sqrt{2}\sqrt{{\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2}\sqrt{B_0^2-D_1^2}}{3\alpha }\sec \mathrm{h}(\alpha x-\frac{t^{\gamma }}{\gamma }\kappa )+\frac{\tau D_1}{3\alpha }\tan \mathrm{h}(\alpha x-\frac{t^{\gamma }}{\gamma }\kappa )}{B_0+\frac{\sec \mathrm{h}(\alpha x-\frac{t^{\gamma }}{\gamma }\kappa )(i\tau \chi -3\sqrt{-{\alpha }^6\chi (B_0^2-D_1^2)})}{\sqrt{2}({\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2)}+D_1\tan \mathrm{h}(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma )}},$$

(20)

where $\tau ={\alpha }^3+2\kappa ,\chi =\sqrt{{\alpha }^6+{\alpha }^3\kappa -2{\kappa }^2}\sqrt{B_0^2-D_0^1}$. We plot the several graphs of (20) as Figure 3, Figure 4 and Figure 5.

Case 4. Considering $A_0=C_1,B_0=\frac{2C_1}{{\alpha }^2}-D_1,B_1=\frac{A_1^2+C_1^2-{\alpha }^2{C}_1{D}_1}{{\alpha }^2{A}_1},\lambda =-\frac{\sqrt{-C_1+{\alpha }^2{D}_1}}{\sqrt{-{\alpha }^2{C}_1+{\alpha }^4{D}_1}}$, $\kappa ={\alpha }^3$, we find

$$u_4(x,t)=\frac{{\alpha }^2{A}_1[cosh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })+sinh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })]}{A_{1}cosh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })+A_{1}sinh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })+C_1-{\alpha }^2{D}_1}.$$

(21)

It is observed that the breath surfaces of (21) are presented in Figure 6.

Case 5. It is selected from the algorithm that when $D_1=\frac{2C_1}{{\alpha }^2},A_0=-C_1,B_0=0,B_1=\frac{A_1^2-C_1^2}{{\alpha }^2{A}_1},\lambda =\frac{1}{\alpha },\kappa ={\alpha }^3$. These coefficients produce

$$u_5(x,t)=\frac{{\alpha }^2{A}_1}{A_1+C_{1}cosh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })+C_{1}sinh(\alpha x-\frac{{\alpha }^3}{\gamma }{t}^{\gamma })}.$$

(22)

With the suitable values of parameters in (22), the graphs are plotted in Figure 7.

Case 6. If it is selected as $A_1=-\frac{\sqrt{A_0-{\alpha }^2{B}_0}\sqrt{A_0^2-C_1^2}}{\sqrt{A_0}},D_1=\frac{B_0{C}_1}{A_0},\kappa =-\frac{{\alpha }^3}{2}+\frac{3\alpha A_0}{2B_0},\lambda =-\frac{\sqrt{B_0}\sqrt{3A_0+{\alpha }^2{B}_0}}{2A_0},B_1=\frac{-A_0^{\frac{5}{2}}{B}_0\sqrt{A_0-{\alpha }^2{B}_0}\sqrt{A_0^2-C_1^2}+\sqrt{{\alpha }^4{A}_0^3{B}_0^4(A_0-{\alpha }^2{B}_1)(A_0^2-C_1^2)}}{A_0^3(A_0-{\alpha }^2{B}_0)},$ we have

$$u_6(x,t)=\frac{A_0-\frac{\sqrt{A_0-{\alpha }^2{B}_0}\sqrt{A_0^2-C_1^2}}{\sqrt{A_0}}\sec \mathrm{h}[\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma }]+C_1\tan \mathrm{h}(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })}{\frac{\sec \mathrm{h}(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })\vartheta }{A_0^3(A_0-{\alpha }^2{B}_0)}+B_0[1-\frac{\sec \mathrm{h}(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })\sqrt{A_0^2-C_1^2}}{\sqrt{A_0}\sqrt{A_0-{\alpha }^2{B}_0}}+\frac{C_1\tan \mathrm{h}(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })}{A_0}]},$$

(23)

where $\vartheta =\sqrt{{\alpha }^4{A}_0^3{B}_0^4(A_0-{\alpha }^2{B}_0)(A_0^2-C_1^2)}$ and $A_0-{\alpha }^2{B}_0>0$ for valid solution.

Case 7. Taking $D_1=\frac{C_1(-A_0^2+A_1^2+C_1^2)}{{\alpha }^2(-A_0^2+C_1^2)},\lambda =-\frac{1}{2}\sqrt{\frac{(4A_0^2-A_1^2-4C_1^2)(A_0^2-A_1^2-C_1^2)}{{\alpha }^2{(A_0^2-C_1^2)}^2}},$ $B_0=\frac{A_0(A_0^2-A_1^2-C_1^2)}{{\alpha }^2(A_0^1-C_1^2)},B_1=\frac{(-A_0^2+A_1^2+C_1^2)(-2A_0^2+A_1^2+2C_1^2)}{{\alpha }^2{A}_1(A_0^2-C_1^2)},\kappa ={\alpha }^3-\frac{3{\alpha }^3{A}_1^2}{2(-A_0^2+A_1^2+C_1^2)}$ gives the other breath solution

$$u_7(x,t)=\frac{{\alpha }^2{A}_1(cosh(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })A_0+A_1+sinh(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })C_1)(A_0^2-C_1^2)}{\theta (-2A_0^2-cosh(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })A_0{A}_1+A_1^2-sinh(\alpha x-\frac{\kappa }{\gamma }{t}^{\gamma })A_1{C}_1+2C_1^2)},$$

(24)

where $\theta =-A_0^2+A_1^2+C_1^2$. Figure 8 and Figure 9 present some simulations of (24).

4.2. RSGEM for the mKdV-ZK Model with Conformable

This part applies RSGEM to the Equation (2) to extract some traveling wave solutions. The wave transformation formula is defined as

$$u(x,y,z,t)=U\left(\xi \right),\xi =\alpha x+\beta y+\theta z-\frac{k}{\gamma }{t}^{\gamma },$$

(25)

where $\alpha ,\beta ,\theta ,k$ are nonzero real numbers and $0<\gamma \le 1$. Putting Equation (25) into Equation (2), the following NODE is obtained:

$$-k{U}^{\prime }+\frac{p\alpha }{3}{\left(U^3\right)}^{\prime }+({\alpha }^3+\alpha {\beta }^3+\alpha {\theta }^2)U^{‴}=0.$$

(26)

Integrating (26) twice with respect to $\xi$ and getting to the zero for both integral constants, we obtain

$$-3kU+p\alpha U^3+3({\alpha }^3+\alpha {\beta }^2+\alpha {\theta }^2)U^{″}=0.$$

(27)

Specially, if we take $n=m=1$, we have

$$U\left(w\right)=\frac{A_{1}sin\left(w\right)+C_{1}cos\left(w\right)+A_0}{B_{1}sin\left(w\right)+D_{1}cos\left(w\right)+B_0}.$$

(28)

By substituting (28) into (27), we find the following solutions of (2).

Case 1. Considering $A_1=-\frac{i{C}_1\sqrt{B_0^2-B_1^2-D_1^2}}{B_0},A_0=\frac{C_1{D}_1}{B_0},p=\frac{3k{B}_0^2}{\alpha C_1^2},\theta =\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\beta }^2)}}{\sqrt{\alpha }}$, we find

$$u_1=\frac{C_1\left(D_1-i\sec \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)\sqrt{B_0^2-B_1^2-D_1^2}+B_0\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)\right)}{B_0\left(B_0+\sec \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)B_1+D_1\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)\right)},$$

(29)

where $B_0^2-B_1^2-D_1^2>0$ for a valid solution. Taking some values of parameters under the strain conditions, we plot its surfaces in Figure 10, Figure 11 and Figure 12.

Case 2. If $A_1=i{C}_1,B_1=i{D}_1,A_0=\frac{C_1{D}_1}{B_0},p=\frac{3k{B}_0^2}{\alpha C_1^2},\theta =\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\beta }^2)}}{\sqrt{\alpha }},$ we obtain

$$u_2=\frac{C_1\left(D_1+B_0(i\sec \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)+\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z))\right)}{B_0\left(B_0+D_1(i\sec \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)+\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z))\right)},$$

(30)

where $-2k-\alpha ({\alpha }^2+{\beta }^2)>0$ for a valid solution.

Case 3. If $A_0=-\frac{A_1{D}_1}{\sqrt{-B_0^2+B_1^2+D_1^2}},C_1=-\frac{A_1{B}_0}{\sqrt{-B_0^2+B_1^2+D_1^2}},p=-\frac{3({\alpha }^2+{\beta }^2+{\theta }^2)(-B_0^2+B_1^2+D_1^2)}{2A_1^2}$, $k=-\frac{\alpha ({\alpha }^2+{\beta }^2+{\theta }^2)}{2}$, we extract

$$\begin{array}{cc}\hfill u_3=& \frac{A_1\left(-sinh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma }\right)B_0}{\left(cosh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma })B_0+B_1+sinh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma })D_1\right)\tau }\hfill \\ \hfill & -\frac{A_1\left(cosh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma })D_1+\sqrt{-B_0^2+B_1^2+D_1^2}\right)}{\left(cosh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma })B_0+B_1+sinh(\alpha x+\beta y+\theta z+\frac{t^{\gamma }\alpha ({\alpha }^2+{\beta }^2{\theta }^2)}{2\gamma })D_1\right)\tau },\hfill \end{array}$$

(31)

where $\tau =\sqrt{-B_0^2+B_1^2+D_1^2}$, and also, $-B_0^2+B_1^2+D_1^2>0$ for a valid solution.

Case 4. In case of selecting $A_1=0,B_1=0,A_0=\frac{C_1{D}_1}{B_0},p=\frac{3k{B}_0^2}{\alpha C_1^2},\theta =\frac{\sqrt{-k-2\alpha ({\alpha }^2+{\beta }^2)}}{\sqrt{2}\sqrt{\alpha }}$, Equation (2) has the following hyperbolic function solution:

$$u_4=\frac{C_1{D}_1+C_1{B}_0\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)}{B_0^2+B_0{D}_1\tan \mathrm{h}(\alpha x+\beta y-\frac{k}{\gamma }{t}^{\gamma }+\theta z)},$$

(32)

where $C_1\ne 0,B_0\ne 0,D_1\ne 0$ for a valid solution.

Case 5. If we consider $A_1=-\sqrt{A_0^2-C_1^2},B_1=0,B_0=\frac{C_1{D}_1}{A_0},p=-\frac{3({\alpha }^2+{\beta }^2+{\theta }^2)D_1^2}{2A_0^2}$, $k=-\frac{\alpha ({\alpha }^2+{\beta }^2+{\theta }^2)}{2},$ we obtain

$$u_5=\frac{A_0^2-A_0\sec \mathrm{h}(\alpha x+\beta y-\frac{k{t}^{\gamma }}{\gamma }+z\theta )\sqrt{A_0^2-C_1^2}+A_0{C}_1\tan \mathrm{h}(\alpha x+\beta y-\frac{k{t}^{\gamma }}{\gamma }+z\theta )}{D_1{C}_1+D_1{A}_0\tan \mathrm{h}(\alpha x+\beta y-\frac{k{t}^{\gamma }}{\gamma }+z\theta )D_1},$$

(33)

where $A_0^2-C_1^2>0$ for a valid solution.

Case 6. When $A_0=-\frac{\sqrt{3}\sqrt{k}{D}_1}{\sqrt{p}\sqrt{\alpha }},A_1=-\frac{i\sqrt{p\alpha C_1^2-3k(B_1^2+D_1^2)}}{\sqrt{p}\sqrt{\alpha }},B_0=-\frac{\sqrt{p}\sqrt{\alpha }{C}_1}{\sqrt{3}\sqrt{k}},\beta =-\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\theta }^2)}}{\sqrt{\alpha }}$, we find

$$u_6=\frac{\left(3\sqrt{k}(\sqrt{3}\sqrt{k}{D}_1+i\sec \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)\omega -\sqrt{p}\sqrt{\alpha }{C}_1\tan \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y))\right)}{\left(\sqrt{3}p\alpha C_1-3\sqrt{3}\sqrt{k}\sqrt{p}\sqrt{\alpha }\sec \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)(B_1+sinh(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)D_1)\right)},$$

(34)

where $\omega =\sqrt{p\alpha C_1^2-3k(B_1^2+D_1^2)},\mathsf{\Xi }=\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\theta }^2)}}{\sqrt{\alpha }}$ for a valid solution. In Figure 13, Figure 14 and Figure 15, several simulations are plotted.

Case 7. If $A_0=\frac{\sqrt{3}\sqrt{k}{D}_1}{\sqrt{p}\sqrt{\alpha }},A_1=i{C}_1,B_0=\frac{\sqrt{p}\sqrt{\alpha }{C}_1}{\sqrt{3}\sqrt{k}},B_1=i{D}_1,\beta =-\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\theta }^2)}}{\sqrt{\alpha }}$ results in

$$u_7=\frac{\frac{\sqrt{3}\sqrt{k}{D}_1}{\sqrt{p}\sqrt{\alpha }}+C_1\left(i\sec \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)+\tan \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)\right)}{\frac{\sqrt{p}\sqrt{\alpha }{C}_1}{\sqrt{3}\sqrt{k}}+D_1\left(i\sec \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)+\tan \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\mathsf{\Xi }y)\right)},$$

(35)

where $\mathsf{\Xi }=\frac{\sqrt{-2k-\alpha ({\alpha }^2+{\theta }^2)}}{\sqrt{\alpha }}.$ Figure 16, Figure 17 and Figure 18 present the graphs of (35).

Case 8. Coefficients such as $A_0=\frac{\sqrt{3}\sqrt{k}{D}_1}{\sqrt{p}\sqrt{\alpha }},A_1=0,B_0=-\frac{\sqrt{p}\sqrt{\alpha }{C}_1}{\sqrt{3}\sqrt{k}},B_1=0,\beta =-\frac{i\sqrt{k+2{\alpha }^3+2\alpha {\theta }^2}\sqrt{-p\alpha C_1^2+3k{D}_1^2}}{\sqrt{-2p{\alpha }^2{C}_1^2+6k\alpha D_1^2}}$ produce

$$u_8=\frac{\frac{3\sqrt{3}\sqrt{k}{D}_1}{\sqrt{p}\sqrt{\alpha }}-3\sqrt{k}{C}_1\tan \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\frac{iy\sqrt{k+2{\alpha }^3+2\alpha {\theta }^2}\sqrt{-p\alpha C_1^2+3k{D}_1^2}}{\sqrt{-2p{\alpha }^2{C}_1^2+6k\alpha D_1^2}})}{\sqrt{3}\sqrt{p}\sqrt{\alpha }{C}_1-3\sqrt{k}{D}_1\tan \mathrm{h}(\alpha x-\frac{k{t}^{\gamma }}{\gamma }+\theta z-\frac{iy\sqrt{k+2{\alpha }^3+2\alpha {\theta }^2}\sqrt{-p\alpha C_1^2+3k{D}_1^2}}{\sqrt{-2p{\alpha }^2{C}_1^2+6k\alpha D_1^2}})}.$$

(36)

5. Discussion and Physical Meanings

By using RSGEM, we found some traveling wave solutions of the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. These solutions are in the forms of the rational, hyperbolic, periodic, trigonometric, complex and mixed hyperbolic function solutions. Figure 1 symbolizes the exponential surfaces of (18) when $D_1=1.5,\alpha =0.2,C_1=-0.5,A_1=0.32,\gamma =0.99,-50<x<50,$$0<t<150$ for 3D and $t=0.21$ for 2D. Figure 2 represents the hyperbolic function graphs of (19) when $D_1=1.5,\alpha =0.2,C_1=-0.5,\gamma =0.99,-50<x<50,0<t<150$ for 3D and $-150<x<150,t=0.21$ for 2D. Figure 3 explains the 3D graphs in $-35<x<35,0<t<35$, and Figure 4 investigates the 2D with $-13<x<13,t=0.1$. Figure 5 represents the contour surfaces with $0<t<35$ of the complex hyperbolic function solution of (20) if it is selected as $D_1=0.5,\alpha =-0.65,C_1=0.1,B_0=2,\gamma =0.99,\kappa =-0.2.$ Figure 6 presents the 3D and 2D graphs of the hyperbolic function solution of (21) for $D_1=1.5,\alpha =0.2,C_1=0.5,A_1=-2,\gamma =0.5,t=0.5,-150<x<150,$ for 3D and $0<t<150$, $-150<x<150$ for 2D solutions. Figure 7 is used to explain the 3D and 2D hyperbolic function solution of (22) for $\alpha =0.2,C_1=0.5,A_1=2,\gamma =0.99,0<t<150-50<x<50,$ for 3D and $t=0.12-150<x<150$ for 2D solutions. Figure 8 and Figure 9 symbolize the singular wave distributions of (24) under the $\alpha =2$, $A_0=7,C_1=-0.2,A_0=2,\gamma =0.5,0<x<50,50<t<50,-50<x<50,0<t<50$, and $t=1$ for 2D. Figure 10, Figure 11 and Figure 12 are plotted to observe the 3D, 2D and contour surfaces of the mixed hyperbolic function solution of (29) under $C_1=0.3,D_1=0.12,B_0=3,\alpha =4,\beta =3,\gamma =0.5,k=0.3,z=1.5,y=2.5,\theta =3.4,B_1=0.13,-20<x<20,-10<t<10$ for 3D and $t=0.01$ for 2D solutions. Figure 13, Figure 14 and Figure 15 are plotted to explain 3D, 2D and contour surfaces of the mixed complex hyperbolic function solution (34) under the terms of $C_1=0.3,D_1=0.12,\alpha =1.4,\gamma =0.5,k=0.3,z=1.5,y=2.5,\theta =1.4,A_0=1.3,p=2,B_1=1.2,-20<x<20,-160<t<20,-20<x<20,-40<t<40$ for 3D, and, $-40<x<40,-50<t<10,-20<x<20$ for contour graph and $t=0.2-10<x<10,-15<x<15$ for 2D graph. Figure 16, Figure 17 and Figure 18 introduce the singular wave properties of (35) under the values of $C_1=0.2,D_1=0.12,\alpha =1.4,\gamma =0.5,k=-3,z=1.5,y=2.5,\theta =1.4,p=2,-60<x<20,-60<t<20,$ for 3D graphs and $-40<x<40,-50<t<10,-60<t<50$ for the contour surface, as well as $t=0.1,-15<x<15$ for the 2D graph.

6. Conclusions

In this paper, we have successfully applied RSGEM to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. We extracted some solutions such as complex, rational, exponential, complex hyperbolic and mixed complex function solutions. We have chosen suitable values of the parameters, and some graphical simulations are also plotted. Necessary strain conditions are also reported in detail. When we consider these results and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, it may be observed that these solutions are used to explain the wave distributions for the governing models. Moreover, it is observed that these findings produce the estimated behaviors of models. When we compare these solutions with [38], it may be seen that these are new wave function solutions.

In this paper, we considered $n=m=1$ in particular. If we consider other equalities of n and m, this will produce more sophisticated solutions to the models studied. This newly presented method can be also used to find many entirely new traveling, singular and complex solutions to the nonlinear partial differential equations arising in real-world problems.