Novel Solutions to Fractional Nonlinear Equations for Crystal Dislocation and Ocean Shelf Internal Waves via the Generalized Bernoulli Equation Method

Nonlinear phenomena encompass a diverse range of applications, spanning fields such as atmospheric science, condensed matter physics, theoretical particle physics, and cosmic structures. These phenomena are particularly relevant in disciplines including fluid mechanical engineering, electrical engineering, solid-state physics, plasma physics, plasma waves, chemical physics, and optical fibers. Nonlinear waves and the structures they elucidate are central to contemporary theories across various domains. Nonlinear partial differential equations (PDEs) play a critical role in characterizing nonlinear wave phenomena, accounting for dispersion, dissipation, convection, and response. Current research efforts are primarily focused on exploring exact or numerical solutions to leverage the insights derived from these studies. Investigating the precise traveling wave solutions of nonlinear fractional PDEs is crucial for deepening our understanding of the underlying factors.

A myriad of methods has been employed to solve nonlinear fractional PDEs, including the Riccati-Bernoulli sub-ODE method [1, 2], Kudyashov [3, 4], first integral method [5, 6], generalized Kudryashov method [7, 8], modified Kudryashov method [9, 10], functional variable method [11, 12], G′/G-expansion method [13, 14], fractional sub-equation method [15, 16], and simple equation method [17, 18].

In 2006, the Riemann-Liouville derivative of the Jumarie equation [19] was described as follows:

$D_{t}^{\alpha }\Upsilon \left( t \right)=\left\{ \begin{array}{*{35}{l}}   \Upsilon \left( t \right) & ,\alpha =0  \\   \frac{\frac{d}{dt}\int\limits_{0}^{t}{\frac{\left[ \Upsilon \left( \delta  \right)-\Upsilon \left( 0 \right) \right]}{{{\left( t-\delta  \right)}\quad ^{\alpha }}}\quad d\delta }}{\Gamma \left( 1-\alpha  \right)} & ,0<\alpha <1  \\   \frac{{{d}^{n}}}{d{{t}^{n}}}D_{t}^{\alpha -n}\Upsilon \left( t \right) & \begin{align}  & ,n\le \alpha <n+1, \\ & n\ge 1, \\\end{align}  \\\end{array} \right.$     (1)

where, a is an order of the fractional derivative.

In 2009, the following was presented as the Riemann-Liouville derivative of the Jumarie [20]:

$D_{t}^{\alpha }{{t}^{\varphi }}=\frac{\Gamma \left( \varphi +1 \right)}{\Gamma \left( \varphi -\alpha +1 \right)}{{t}^{\varphi -\alpha }},\,\,\varphi \ge 0,$      (2)

$D_{t}^{\alpha }\left[ \Upsilon \left( t \right)\Psi \left( t \right) \right]=\Upsilon \left( t \right)D_{t}^{\alpha }\Psi \left( t \right)+\Psi \left( t \right)D_{t}^{\alpha }\Upsilon \left( t \right),$    (3)

$D_{t}^{\alpha }\Upsilon \left[ \Psi \left( t \right) \right]=D_{\Psi }^{\alpha }\Upsilon \left[ \Psi \left( t \right) \right]{{\left[ {\Psi }'\left( t \right) \right]}^{\alpha }}={{{\Upsilon }'}_{\Psi }}\left[ \Psi \left( t \right) \right]D_{t}^{\alpha }\Psi \left( t \right).$      (4)

The (2+1)-dimensional cKG equation is one of the second-order non-linear evolution equations. This equation has been employed to form a variety of different non-linear phenomena [21], including the propagation of crystal dislocation, the action of elementary particles, and the proliferation of fluxions in Josephson junctions, to name a few examples. The cKG equation has a non-linearity that indicates that the constant value with ω=ω(x, y, t) and β, ε are not zero, as seen below:

${{\omega }_{xx}}+{{\omega }_{yy}}-{{\omega }_{tt}}+\beta \omega +\varepsilon {{\omega }^{3}}=0.$      (5)

The exact solution [21] of Eq. (5) is:

$\omega =\pm I\sqrt{\frac{\beta }{\varepsilon }}\left( 1-\frac{2{{b}_{1}}\cosh ({{b}_{2}})+sinh({{b}_{2}})}{({{b}_{1}}-2{{a}_{2}}\alpha )\cosh ({{b}_{2}})+({{b}_{1}}+2{{a}_{2}}\alpha )sinh({{b}_{2}})} \quad\right),$       (6)

where, $b_1=a_1\left(\lambda^2-2\right), b_2=\sqrt{\frac{\alpha}{-2\left(\lambda^2-2\right)}}(x+y-\lambda t)$ and a₁, a₂ are constants of integration.

On the ocean shelf, powerful nonlinear internal waves are described by the (2+1) dimensional GKP Equation [22]. This is the form of the non-linear positive GKP Equation [23] of the fourth-order with ω=ω(x, y, t):

${{\left( {{\omega }_{t}}+6\omega {{\omega }_{x}}+6\omega {}^{2}{{\omega }_{x}}+{{\omega }_{xxx}} \right)}_{x}}+{{\omega }_{yy}}=0.$     (7)

The explicit form of exact travelling wave solutions of positive GKP Equation [23] is:

$\omega=\frac{4(c+1) e^{\sqrt{-c-1 \,\,\xi}}}{4 c e^{2 \sqrt{-c-1} \,\,\,\,\xi}\,\,\,\,-4 e^{\sqrt{-c-1} \,\,\,\,\xi}\,\,\,\,\,-1}$,    (8)

where, $\xi =\int{\frac{d\omega }{\varphi }},\,\,\varphi =\frac{d\omega }{d\xi }$ and c is wave speed.

The fractional cKG equation and the fractional GKP equation have both been solved in this article by using Jumarie's Riemann-Liouville derivative and the generalized Bernoulli equation method. We were able to get 13 solutions for each equation, and these solutions came in the form of hyperbolic functions, exponential functions, and rational functions. Using two-dimensional graphs and three-dimensional graphs, we were able to demonstrate the exact solutions as well as the physical waves which are kink waves, solitary waves and periodic waves. These results can be compared with the numerical solutions. Moreover, it can also be applied to initial and boundary value problems. Finding the traveling wave solutions in this research demonstrates the efficiency of the generalized Bernoulli equation method for solving the non-linear fractional PDEs.

Solving the two non-linear space-time fractional equations that are presented below allows us to show the generalized Bernoulli equation method.

3.1 The non-linear space and time fractional of (2+1)-dimensional cKG equation

The results of applying the non-linear space and time fractional cKG equation to traveling waves are shown below:

$D_{x}^{2\alpha }\omega +D_{y}^{2\alpha }\omega -D_{t}^{2\alpha }\omega +\beta \omega +\varepsilon {{\omega }^{3}}=0,$             (27)

when, t>0, 0<α≤1, ω=ω(x, y, t) and the value of β, ε are constants. Taking the assumption that ω(x, y, t)=$\Theta$(δ) is the correct result and applying transformation:

$\delta =\frac{r{{x}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}+\frac{s{{y}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}-\frac{c{{t}^{\alpha }}}{\Gamma \left( \alpha +1 \right)},$      (28)  

where, the constants r, s and c do not have the value zero. Eq. (28) was used to develop an ODE:

${{r}^{2}}{\Theta }''+{{s}^{2}}{\Theta }''-{{c}^{2}}{\Theta }''+\beta \Theta +\varepsilon {{\Theta }^{3}}=0.$            (29)

Eq. (10) was the structure that was used in order to provide the solution to the issue by using the generalized Bernoulli equation method. After that, we made certain that the highest order derivative and the non-linear components of Eq. (29) were in balance with one another. Thus K=1. In the end, Eq. (12) was transformed into:

$\Theta \left( \delta  \right)={{b}_{0}}+{{b}_{1}}G\left( \delta  \right).$      (30)

Eq. (30) will take the place of Eq. (29) in this application. In order to make each coefficient equal to zero, we collected all of the terms that belonged to the same power of G(δ), and then we did the following:

${{G}^{0}}\left( \delta  \right)\,\,\,\,\,:\,\,\,\,\,{{b}_{0}}\beta +b_{0}^{3}\varepsilon =0,$       (31)

${{G}^{1}}\left( \delta  \right)\,\,\,\,\,:\,\,\,\,\,{{b}_{1}}{{r}^{2}}{{\eta }^{2}}+{{b}_{1}}{{s}^{2}}{{\eta }^{2}}-{{b}_{1}}{{c}^{2}}{{\eta }^{2}}+{{b}_{1}}\beta +3b_{0}^{2}{{b}_{1}}\varepsilon =0,$               (32)

${{G}^{2}}\left( \delta  \right)\,\,\,\,\,:\,\,\,\,\,3{{b}_{1}}{{r}^{2}}\eta \xi +3{{b}_{1}}{{s}^{2}}\eta \xi -3{{b}_{1}}{{c}^{2}}\eta \xi +3{{b}_{0}}b_{1}^{2}\varepsilon =0,$       (33)

${{G}^{3}}\left( \delta  \right)\,\,\,\,\,:\,\,\,\,\,2{{b}_{1}}{{r}^{2}}{{\xi }^{2}}+2{{b}_{1}}{{s}^{2}}{{\xi }^{2}}-2{{b}_{1}}{{c}^{2}}{{\xi }^{2}}+b_{1}^{3}\varepsilon =0.$           (34)

After working out the solution to the system of Eqns. (31)-(34), we obtain:

${{b}_{0}}=\pm \sqrt{-\frac{\beta }{\varepsilon }},$       (35)

${{b}_{1}}=\pm \xi \sqrt{-\frac{2\left( {{r}^{2}}+{{s}^{2}}-{{c}^{2}} \right)}{\varepsilon }},$      (36)

$c=\pm \frac{\sqrt{{{r}^{2}}{{\eta }^{2}}+{{s}^{2}}{{\eta }^{2}}-2\beta }}{\eta }.$       (37)

Eqns. (14)-(26), (28), and (35)-(37) provide for the representation of the exact traveling wave solutions of the non-linear space-time fractional cKG equations in a total of 13 different cases which are distinct from the one that was found in the first solution indicated in Eq. (6).

For real and non-zero η, η²>0 and ξ≠0a and let ${{b}_{1}}=\pm \xi \sqrt{-\frac{2\left( {{r}^{2}}+{{s}^{2}}-{{c}^{2}} \right)}{\varepsilon }},$ we obtained:

${{\omega }_{1}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{{{b}_{1}}}{2\xi }\left( -\eta -\eta \tanh \left( \frac{\eta \delta }{2} \right) \right),$         (38)

${{\omega }_{2}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{{{b}_{1}}}{2\xi }\left( -\eta -\eta \coth \left( \frac{\eta \delta }{2} \right) \right),$          (39)

${{\omega }_{3}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{{{b}_{1}}\left( -\eta -\eta \tanh \left( \eta \delta  \right)\mp i\eta \operatorname{sech}\left( \eta \delta  \right) \right)}{2\xi },$            (40)

${{\omega }_{4}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{{{b}_{1}}\left( -\eta -\eta \coth \left( \eta \delta  \right)\mp \eta \operatorname{csch}\left( \eta \delta  \right) \right)}{2\xi },$       (41)

${{\omega }_{5}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{{{b}_{1}}}{4\xi }\left( -2\eta -\eta \tanh \left( \frac{\eta \delta }{4} \right)-\eta \coth \left( \frac{\eta \delta }{4} \right) \right),$             (42)

${{\omega }_{6}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+{{b}_{1}}\left( \frac{\eta \sqrt{{{Q}^{2}}+{{R}^{2}}}-\eta Q\cosh \left( \eta \delta  \right)}{2\xi Q\sinh \left( \eta \delta  \right)+2\xi R}-\frac{\eta }{2\xi } \right),$             (43)

${{\omega }_{7}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+{{b}_{1}}\left( \frac{-\eta \sqrt{{{R}^{2}}-{{Q}^{2}}}-\eta Q\sinh \left( \eta \delta  \right)}{2\xi Q\cosh \left( \eta \delta  \right)+2\xi R}-\frac{\eta }{2\xi } \right),$      (44)

where, Q and R are two non-zero real constant that satisfy ${{R}^{2}}-{{Q}^{2}}>0.$

${{\omega }_{8}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{\pm {{b}_{1}}\eta {{e}^{\eta \delta }}}{\xi \mp \xi {{e}^{\eta \delta }}},$           (45)

${{\omega }_{9}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{\pm {{b}_{1}}\eta {{e}^{\eta \delta }}}{\xi i\mp \xi {{e}^{\eta \delta }}},$                 (46)

${{\omega }_{10}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+{{b}_{1}}\frac{\eta \sqrt{{{Q}^{2}}-{{R}^{2}}}\pm \eta \left( Q{{e}^{\eta \delta }}-iR \right)}{\mp \xi Q\left( {{e}^{\eta \delta }}-{{e}^{-\eta \delta }} \right)\pm 2\xi iR},$            (47)

where, Q and R are two non-zero real constant that satisfy ${{Q}^{2}}-{{R}^{2}}>0.$

${{\omega }_{11}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{-{{b}_{1}}\eta \phi {{e}^{\eta \delta }}}{\xi +\xi \phi {{e}^{\eta \delta }}},$             (48)

${{\omega }_{12}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{-{{b}_{1}}\eta {{e}^{\eta \delta }}}{\xi \phi +\xi {{e}^{\eta \delta }}}\quad,$                  (49)

where, ϕ is an arbitrary constant.

For η=0 and ξ≠0 and arbitrary constant ψ:

${{\omega }_{13}}=\pm \sqrt{-\frac{\beta }{\varepsilon }}+\frac{-{{b}_{1}}}{\xi \delta +\psi },$             (50)

where, $\delta =\frac{r{{x}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}+\frac{s{{y}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}\mp \frac{{{t}^{\alpha }}\sqrt{{{r}^{2}}{{\eta }^{2}}+{{s}^{2}}{{\eta }^{2}}-2\beta }}{\eta \Gamma \left( \alpha +1 \right)}.$

We have presented some physical graphs which are comparisons between α=0.4 and α=0.7. Figures 1, 2, 3 and 4 present the physical graphs of Eqns. (38), (39), (42) and (45) respectively when we set the parameters β=1, ε=-1, r=-1, s=1, η=1, ξ=1, t=100, 1≤x≤30, 1≤y≤30. We set:

β=1, ε=-1, r=-1, s=1, η=1.2, ξ=1, t=100, 1≤x≤30, 1≤y≤30 in Eq. (41) which shown by Figure 5. When we substitute β=1, ε=-1, r=-1, s=1,η=1, ξ=1, t=100, Q=1, R=2, 1≤x≤30, 1≤y≤30 into Eqns. (43) and (44), the physical graphs are presented as Figures 6 and 7. Graphs of the physical wave are shown in Figures 8 and 9 when β=1, ε=-1, r=-1, s=1, η=1, ξ=1, t=100, ϕ=1, 1≤x≤30, 1≤y≤30 are substituted into Eqns. (48) and (49). Substituting β=1, ε=-1, r=-1, s=1, η=1, ξ=1, t=100, ψ=1, 1≤x≤30, 1≤y≤30 into Eq. (50), we get the physical graphs shown in Figure 10.

1.png

Figure 1. Kink wave of ω₁

2.png

Figure 2. Solitary wave of ω₂

3.png

Figure 3. Solitary wave of ω₅

4.png

Figure 4. Solitary wave of ω₈

5.png

Figure 5. Kink wave of ω₄

6.png

Figure 6. Periodic wave of ω₆

7.png

Figure 7. Kink wave of ω₇

8.png

Figure 8. Kink wave of ω₁₁

9.png

Figure 9. Kink wave of ω₁₂

10.png

Figure 10. Periodic wave of ω₁₃

3.2 The non-linear space and time fractional of (2+1)-dimensional GKP equation

Below are the exact results of traveling waves through the non-linear space and time fractional GKP equation:

$D_{x}^{\alpha }\left( D_{t}^{\alpha }\omega +6\omega D_{x}^{\alpha }\omega +6{{\omega }^{2}}D_{x}^{\alpha }\omega +D_{x}^{3\alpha }\omega  \right)+D_{y}^{2\alpha }\omega =0,\,$           (51)

where, t>0, 0<α≤1 and ω=ω(x, y, t). Assuming that ω(x, y, t)=$\Theta$(δ) and using transformation in Eq. (24) again:

$r(-c{\Theta }'+6r\Theta {\Theta }'+6r{{\Theta }^{2}}{\Theta }'+{{r}^{3}}{\Theta }'''{)}'+{{s}^{2}}{\Theta }''=0.$          (52)

when, Eq. (52) is integrated twice with a constant of zero, the result is:

$-cr\Theta +3{{r}^{2}}{{\Theta }^{2}}+2{{r}^{2}}{{\Theta }^{3}}+{{r}^{4}}\frac{{{d}^{2}}\Theta }{d{{\delta }^{2}}}+{{s}^{2}}\Theta =0.$       (53)

After bringing Eq. (53) into balance, we found that K=1. As a result, Eq. (10) evolved into:

$\Theta \left( \delta  \right)={{b}_{0}}+{{b}_{1}}G\left( \delta  \right).$         (54)

Eq. (54) will be used in replacement of Eq. (53). In order to ensure that each coefficient was equivalent to zero, we first gathered up all of the words that belonged to the same power of G(δ) and then proceeded to do the following:

${{G}^{0}}\left( \delta  \right)\,\,\,:\,\,\,-{{b}_{0}}cr+3b_{0}^{2}{{r}^{2}}+2b_{0}^{3}{{r}^{2}}+{{b}_{0}}{{s}^{2}}=0,$              (55)

${{G}^{1}}\left( \delta  \right)\,\,\,:\,\,\,-{{b}_{1}}cr+6{{b}_{0}}{{b}_{1}}{{r}^{2}}+6b_{0}^{2}{{b}_{1}}{{r}^{2}}+{{b}_{1}}{{r}^{4}}{{\eta }^{2}}+{{b}_{1}}{{s}^{2}}=0,$                (56)

${{G}^{2}}\left( \delta  \right)\,\,\,:\,\,\,3b_{1}^{2}{{r}^{2}}+6{{b}_{0}}b_{1}^{2}{{r}^{2}}+3{{b}_{1}}{{r}^{4}}\eta \xi =0,$              (57)

${{G}^{3}}\left( \delta  \right)\,\,\,:\,\,2b_{1}^{3}{{r}^{2}}+2{{b}_{1}}{{r}^{4}}{{\xi }^{2}}=0.$           (58)

We reach the following result after calculating out the solution to the system of Eqns. (55)-(58):

${{b}_{0}}=\frac{-1\pm r\eta i}{2},$            (59)

${{b}_{1}}=\pm r\xi i,$          (60)

$c=\frac{{{s}^{2}}-2b_{0}^{2}{{r}^{2}}-{{r}^{4}}{{\eta }^{2}}}{r}.$             (61)

Eqns. (12)-(14), (26), and (59)-(61) give for the description of the different exact 13 solutions of the non-linear space and time fractional cKG equations. This generates all answers distinct from that presented in Eq (8).

For η, η²>0 and ξ≠0 these are real and not zero, then we get:

${{\omega }_{1}}=\frac{-1\pm r\eta i\pm ri\left( -\eta -\eta \tanh \left( \frac{\eta \delta }{2} \right) \right)}{2},$        (62)

${{\omega }_{2}}=\frac{-1\pm r\eta i\pm ri\left( -\eta -\eta \coth \left( \frac{\eta \delta }{2} \right) \right)}{2},$          (63)

${{\omega }_{3}}=\frac{-1\pm r\eta i\pm ri\left( -\eta -\eta \tanh \left( \eta \delta  \right)\mp i\eta \operatorname{sech}\left( \eta \delta  \right) \right)}{2},$        (64)

${{\omega }_{4}}=\frac{-1\pm r\eta i\pm ri\left( -\eta -\eta \coth \left( \eta \delta  \right)\mp \eta \operatorname{csch}\left( \eta \delta  \right) \right)}{2},$            (65)

${{\omega }_{5}}=\frac{-1\pm r\eta i}{2}\pm \frac{ri}{4}\left( -2\eta -\eta \tanh \left( \frac{\eta \delta }{4} \right)-\eta \coth \left( \frac{\eta \delta }{4} \right) \right),$         (66)

${{\omega }_{6}}=\frac{-1\pm r\eta i}{2}\pm ri\left( \frac{\eta \sqrt{{{Q}^{2}}+{{R}^{2}}}-\eta Q\cosh \left( \eta \delta  \right)}{2Q\sinh \left( \eta \delta  \right)+2R}-\frac{\eta }{2} \right),$        (67)

${{\omega }_{7}}=\frac{-1\pm r\eta i}{2}\pm ri\left( \frac{-\eta \sqrt{{{R}^{2}}-{{Q}^{2}}}-\eta Q\sinh \left( \eta \delta  \right)}{2Q\cosh \left( \eta \delta  \right)+2R}-\frac{\eta }{2} \right),$           (68)

where, Q and R are two non-zero real constant that satisfy ${{R}^{2}}-{{Q}^{2}}>0.$

${{\omega }_{8}}=\frac{-1\pm r\eta i}{2}\pm \frac{ri\eta {{e}^{\eta \delta }}}{1\mp {{e}^{\eta \delta }}},$             (69)

${{\omega }_{9}}=\frac{-1\pm r\eta i}{2}\pm \frac{ri\eta {{e}^{\eta \delta }}}{i\mp {{e}^{\eta \delta }}},$         (70)

${{\omega }_{10}}=\frac{-1\pm r\eta i}{2}\pm ri\left( \frac{\eta \sqrt{{{Q}^{2}}-{{R}^{2}}}\pm \eta \left( Q{{e}^{\eta \delta }}-iR \right)}{\mp Q\left( {{e}^{\eta \delta }}-{{e}^{-\eta \delta }} \right)\pm 2iR} \right),$             (71)

where, Q and R are two non-zero real constant that satisfy ${{Q}^{2}}-{{R}^{2}}>0.$

${{\omega }_{11}}=\frac{-1\pm r\eta i}{2}\mp \frac{ri\eta \phi {{e}^{\eta \delta }}}{1+\phi {{e}^{\eta \delta }}},$       (72)

${{\omega }_{12}}=\frac{-1\pm r\eta i}{2}\mp \frac{ri\eta {{e}^{\eta \delta }}}{\phi +{{e}^{\eta \delta }}},$           (73)

where, ϕ is an arbitrary constant.

For η=0 and ξ≠0 and arbitrary constant ψ, the solution to Eq. (10) is:

${{\omega }_{13}}=\frac{-1\pm r\eta i}{2}\mp \frac{r\xi i}{\xi \delta +\psi },$              (74)

where, $\delta =\frac{r{{x}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}+\frac{s{{y}^{\alpha }}}{\Gamma \left( \alpha +1 \right)}\mp \frac{\left( {{s}^{2}}-2b_{0}^{2}{{r}^{2}}-{{r}^{4}}{{\eta }^{2}} \right){{t}^{\alpha }}}{r\Gamma \left( \alpha +1 \right)}.$