Abstract
In this manuscript, we implement the travelling wave solutions of the fractional (3+1) generalized computational nonlinear wave equation with gas bubbles via application of five mathematical methods. Liquids with gas bubbles primarily arise in various applications like science, engineering, and mathematical physics. The obtained solitary waves solutions have fruitful applications in engineering, science, life, nature and physics. Several novel soliton solutions of concerned model are established in the form of hyperbolic, trigonometric, exponential and rational functions. To handle all calculations and verification of obtained results, computational software Mathematica 12.1 is used. For the demonstration of the physical behaviour of concern model, some solutions are plotted graphical in 2-dimensional and 3-dimensional by imparting specific values to the parameters under constrain conditions. Finally, we intrigue both two and three dimensional to explain the physical behavior of the model.
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Introduction
The exploration of nonlinear computational partial differential equation solutions is perilous in understanding numerous physical situations in several scientific and engineering applications. As a result, numerous logical and numerical methods have been used to tackle a diversity of such problems, including the generalized Kudryashov1, sine-cosine2, sine-Gordon expansion, extended auxiliary equation3,4, direct algebra5 , Safdar sub-equation6, generalized Riccati method7 and many more8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. These approaches intensely depend on wave transformation techniques. However, other analytical methods didn’t depend on the wave transforms approach, invariant subspace method12,13, Lie symmetry method8,11, reduction method14,15, etc.32,33,34,35,36,37,38,39,40,41. The study of bubbling liquids and their applications in numerous disciplines of engineering and medical sciences has annoyed the curiosity of several scholars for periods. Most bubbles with uniform radius are elucidated by a fourth-order linear partial differential equation for convinced physical phenomena in isothermal bubbly liquids27,28,29,30,31,42,43,44,45,46,47,48,49.
Consider the liquid with the gas bubbles model as27,28;
where \(Q_{t}\) and \(QQ_{x}\) is used to perform a role in the evolution of time and the steepening of the wave and Q is wave amplitude. The parameters \(\delta _1,\delta _2,\delta _3,\delta _4\) and \(\delta _5\) are represented the bubble-liquid-nonlinearity, the bubble liquid-viscosity The y transverse perturbation and the z transverse perturbation.The scrutiny of a generalized (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that transpire in liquids with gas bubbles will be accomplished.
Furthermore, almost studies on the logical and numerical blend (solution) of generalized nonlinear model Eq. (1) with gas bubbles have been explored in the literature, for example, the bilinear formalism and soliton solutions using Hirota bilinear method21, Assemble mixed rogue wave-stripe solitons and mixed lump-stripe solitons23, the binary Bell polynomials obtaining the bilinear form of this model25, and the solitons and lumps solution for the generalized nonlinear wave26. There are numerous fractional derivative operators in fractional calculus, such as the Caputo derivative, Grunwald derivative, Riemann–Liouville derivative, and so on.
Therefore, the chief focus of this research is to accomplish waves solutions of Eq. (1) via five mathematical methods50,51,52,53,54, these methods are called ESE method50, modified extended AEM approach51, \((G'/G)\)-expansion scheme52, \(Exp(-\Psi (\phi ))\)-expansion method53 and modified F-expansion method54 respectively. The investigated our solutions are in different types like exponential, trigonometric, hyperbolic and rational forms and are totally d new solutions as compared to exist in previous literature by using the different techniques of distinct authors on this model21,23,25,26.
The remaining arrangement of research work as: In “Description of accordant fractional derivatives” some definition of common fractional derivatives, properties of the accordant derivative. In “Description of mathematical methods”, proposed mathematical methods are described. In “Applications”, soliton solutions of Eq. (1) are established by application of five mathematical method. In “Discussion of the results”, Discussion of results has been been mentioned. In “Conclusion”, conclusion of the work has been discussed.
Description of accordant fractional derivatives
This segment will start by defining the bulk common partial (fractional) derivative definitions, like: the Riemann–Liouville, Caputo, and Grunwald–Letnikov definitions42.
Definition 1
(Riemann Liouville)
Definition 2
(Caputo )
Definition 3
(Grunwald–Letnikov)
Definition 4
(Accordant divided (fractional) derivative ))
Let a function \(Q:[0,\infty ]\rightarrow ()\) then the Accordant divided (fractional) derivative of Q order \(\delta\) is as follow:
Theorem 2.1
(42)
Let R(z),Q(z) are \(\mu\)-difference at a point \(z>0\) and \(\mu \epsilon (0,1)\). Then
1: \(L_z^{\mu } \left( \alpha _2 Q z+\alpha _1 R z\right) =\alpha _2 Q z L_z^{\mu }+\alpha _1 R z L_z^{\mu }\)
2: \(z^h L_z^{\mu }=h \overset{h-\mu }{z}\)
3: \(L_z^{\mu }(\lambda )=0\) for all constant function \(Q(z)=\lambda\)
4: \(L_z^{\mu }\left( R(z) (Q (z)\right)\)=R(z) \(L_z^{\mu }\left( (Q (z)\right)\)+Q(z) \(L_z^{\mu }\left( (R (z)\right)\)
5: If Q(z) is differentile then \(L_z^{\mu }(R)z=\frac{(\text {dR} (z)) z^{1-\mu }}{\text {dz}}\)
Description of mathematical methods
Consider the general nonlinear FDEs
Let the fractional transformation,
Extended simple equation method
Let (8) has solutiion,
Let \(\varPsi\) satisfy,
The general solutions of new simple ansatz Eq. (10) are as following
If \(c_0=0,~c_3=0\) , then simple ansatz Eq. (10) reduces to Bernoulli equation, which has the following solutions:
If \(c_1=0,~c_3=0\), then the ansatz Eq. (10) reduces to Riccati equation, which has the following solutions:
Substitute Eq. (9) along with Eq. (10) into Eq. (8), obtained a system of equations which can be solved to achieve the required solution of Eq. (8) with help of Eqs. (11–15).
Modified extended auxiliary equation mapping method
Let solution of Eq. (8) is
Let \(\Psi '\) satisfiy,
The ansatz Eq. (17) has the following solutions as;
Substitute Eq. (16) along with Eq. (17) into Eq. (8), obtained a system of equations which can be solved to achieve the required solutionof Eq. (8) with help of Eqs. (18–20).
The \((G'/G)\)-expansion method
Let Eq. (8) has solutiion,
L Let \(G(\xi )\) obeys the second order ODE as
Put Eq. (21) with Eq. (22) in Eq. (8), obtained a system of equations having the following solutions cases.
CASE I: When \(\lambda _1^2-4 \mu _1 >0\)
CASE II: When \(\lambda _1^2-4 \mu _1 <0\)
CASE III: When \(\lambda _1^2-4 \mu _1 =0\)
By substituting all solutions of Eq. (22) into Eq. (23), we obtained the required solutions of Eq. (8)
The \(Exp(-\varPsi (\xi ))\)-expansion method
Let solution of Eq. (8) is
Let
When \(\lambda _1^2-4 \mu _1 >0\), \(\mu _1\ne 0\) then Eq. (27) has the following solution
When \(\lambda _1^2-4 \mu _1 >0\), \(\mu _1=0\), then Eq. (27) has the following solution
When \(\lambda _1^2-4 \mu _1 =0\), \(\lambda _1\ne 0\), then Eq. (27) has the following solution
When \(\lambda _1^2-4 \mu _1 =0\), \(\mu _1, \lambda _1=0\), then Eq. (27) has the following solution
When \(\lambda _1^2-4 \mu _1 <0\), then Eq. (27) has the following solution
By substituting all solutions of Eq. (27) into Eq. (26), we obtained the required solutions of Eq. (8)
Modified F-expansion method
Let solution of (8) is:
Let F obliges,
The relation between A, B, C corresponding values of \(F(\xi )\) in Eq. (34) is given as;
Values of A, B, C | \(F(\xi )\) |
---|---|
\(A=0\), \(B=1\), \(C=-1\) | \(\frac{1}{2}+\frac{1}{2}\tanh (\frac{1}{2}\xi )\) |
\(A=0\), \(B=-1\), \(C=1\) | \(\frac{1}{2}-\frac{1}{2}\coth (\frac{1}{2}\xi )\) |
\(A=\frac{1}{2}\), \(B=0\), \(C=-\frac{1}{2}\) | \(\coth (\xi )\pm \csc h(\xi )\) |
\(A=1\), \(B=0\), \(C=-1\) | \(\tanh (\xi ),\coth (\xi )\) |
\(A=\frac{1}{2}\), \(B=0\), \(C=\frac{1}{2}\) | \(\sec (\xi )+\tan (\xi )\) |
\(A=-\frac{1}{2}\), \(B=0\), \(C=-\frac{1}{2}\) | \(\sec (\xi )-\tan (\xi )\) |
\(A=1(-1)\), \(B=0\), \(C=1(-1)\) | \(\tan (\xi )(\cot (\xi ))\) |
\(A=0\), \(B=0\), \(C\ne 0\) | \(-\frac{1}{C\xi +\epsilon }, (\epsilon\) is arbitary constant) |
\(A\ne 0\), \(B=0\), \(C=0\) | \(A\xi\) |
\(A\ne 0\), \(B\ne 0\), \(C=0\) | \(\frac{\exp (B\xi )-A}{B}\) |
By substituting all solutions of Eq. (34) into Eq. (33), we obtained the required solutions of Eq. (8)
Applications
This segment will present several type results for the fractional (3+1)-dimensional (aspect) generalized (computational) wave model of liquids with gas bubbles Eq. (1) via applications of five mathematical methods. Now, utilizing the definition of the accordant fractional derivative in Eq. (1) to achieve
Consider wave transformation,
Subsitute Eq. (36) in Eq. (35),
Application of extended simple equation method
Let Eq. (37) has solution as;
Putting Eq. (38) with (10) into (37),
CASE 1: \(c_3=0\),
FAMILY-I
FAMILY-II
CASE 2: \(c_0=c_3=0\),
CASE 3: \(c_1=0,~~c_3=0\),
FAMILY-I
FAMILY-II
FAMILY-III
Application of modified extended auxiliary equation mapping method
Let Eq. (37) has solution as;
CASE I:
CASE II:
where \(\Omega = \left( -\frac{\sqrt{\frac{\beta _1}{\beta _3}} \left( \frac{\sqrt{\beta _1} \epsilon \cosh \left( \sqrt{\beta _1} (\xi +\chi )\right) }{\cosh \left( \sqrt{\beta _1} (\xi +\chi )\right) +\eta }-\frac{\sqrt{\beta _1} \epsilon \sinh ^2\left( \sqrt{\beta _1} (\xi +\chi )\right) }{\left( \cosh \left( \sqrt{\beta _1} (\xi +\chi )\right) +\eta \right) {}^2}\right) }{2 \left( -\sqrt{\frac{\beta _1}{4 \beta _3}} \left( \frac{\epsilon \sinh \left( \sqrt{\beta _1} (\xi +\chi )\right) }{\cosh \left( \sqrt{\beta _1} (\xi +\chi )\right) +\eta }+1\right) \right) }\right) ^2\)
CASE III:
Application of \(G'/G\)-expansion method
Let Eq. (37) has solution as;
CASE I: \(\lambda _1^2-4 \mu _1 >0\)
CASE II: \(\lambda _1 ^2-4 \mu _1<0\)
CASE III: \(\lambda _1 ^2-4 \mu _1=0\)
Application of \(Exp(-\Psi (\xi ))\)-expansion method
Let Eq. (37) has solution as;
CASE I: \(\lambda _1^2-4 \mu _1 >0\), \(\mu _1\ne 0\)
CASE II: \(\lambda _1^2-4 \mu _1 >0\), \(\mu _1=0\)
CASE III: \(\lambda _1^2-4 \mu _1=0\), \(\mu _1,~\lambda _1\ne 0\)
CASE IV: \(\lambda _1^2-4 \mu =0\), \(\mu _1=\lambda _1=0\)
CASE V: \(\lambda _1^2-4 \mu _1 <0\)
Application of modified F-expansion method
Let Eq. (37) has as;
For A = 0, B = 1, C = −1,
For A=0, C=1, B=-1,
For A=1/2, B=0, C=-1/2
FAMILY-I
FAMILY-II
FAMILY-III
For A=1, B=0, C=-1
FAMILY-I
FAMILY-II
FAMILY-III
For A=C=1/2, B=0,
FAMILY-I
FAMILY-II
FAMILY-III
For A=C=-1/2, B=0,
FAMILY-I
FAMILY-II
FAMILY-III
For A=C=-1 B=0,
FAMILY-I
FAMILY-II
FAMILY-III
For A=0, B=1
B=C=0
For C=0
Discussion of the results
In this section we have compared our investigated solutions with others solutions in different latest articles obtained solutions by using different techniques. Due to derive different values of \(A{i},(i=-2,-1,0,1,2)\) in Eq. (38)and \(A_{i}, B_{i}, C_{i}, D_{i} ,(i = 0,1,2)\) in Eq. (55) and \(A_{N}, (N=1,2)\) in Eq. (60) and Eq. (65) respectively. Furthermore \(a_{i},b_{i},(i = 1,2)\) in Eq. (72), we have achieved serval types results in the form of trigonometric, rational, exponential and rational functions. However, some of our constructed solutions are likely similar to due to the following pints.
\(\bullet\) Our solutions \({U_1}\) and \(U_{2}\) are likely similar to the solutions mentioned in Eqs. (16) and (18) respectively in55.
\(\bullet\) Our solutions \({U_{14}}\) and \(U_{15}\) are likely similar to the solutions mentioned in Eqs. (44) and (45) respectively in56.
\(\bullet\) Our solutions \({U_{27,1}}\) and \(U_{27,2}\) are likely similar to the solutions mentioned in Eqs. (73) and (81) respectively in57.
The residual overall constructed results are novel and have not been explored in any research literature. It is shown that our proposed methods provide an effective and a more powerful mathematical tool for solving nonlinear evolution equations in physical sciences. It is reliable and endorses a assortment of exact solutions NFPDEs.
Conclusion
We have built abundant various exact solutions of Eq. (1) by employing the five mathematical approaches. All calculations and figures are handling by using the Mathematica 12.1 software. We conspired both 2-dimensional and 3-dimensional plots to understand physical behaviour of concern model by assigned certain value to the parameters in Figs. 1, 2, 3, 4, 5, 6 and 7. The offered mathematical techniques are more powerful and investigated results have many application in nonlinear science. The obtained results are in the form of trignometric, hypberbolic, exponential and rational forms. Engineers and researchers need to carefully evaluate the properties of the materials used and refine them to optimize the performance and efficiency of traveling wave solutions in various applications. Further research and development in this field can lead to improvements in hydrodynamics and fluids. We intend to use a variety of techniques in the future to study these nonlinear fractional wave equations in mathematical physics and find the lump, soliton, and breather solutions.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2023/01/24001).
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Seadawy, A.R., Ali, A., Altalbe, A. et al. Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles. Sci Rep 14, 1862 (2024). https://doi.org/10.1038/s41598-024-52249-3
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DOI: https://doi.org/10.1038/s41598-024-52249-3
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