A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution’s existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.


Basic definitions
Here, we elucidate some basic notions related to fractal-fractional (FF) calculus.

Existence of the initial value problems
The existence and uniqueness of the initial value problems are studied in this section by using σ-type F-contraction 36 .For this purposes, Suppose (Z, d) be a complete metric space and ς be the family of strictly increasing functions F : R + → R having the following properties: • there exist υ ∈ (0, 1) such that lim a→0 + a υ F(a) = 0.
Definition 12 Let T : Z → Z be self mapping and σ : Z × Z → [0, ∞) , if for all Y, V ∈ Z , then T is called σ-admissible.
Theorem 1 Let (Z, d) be a CMS and T : Z → Z be an σ-type F-contraction such that 1. there exist Then T has a fixed point where E the space of all continuous functions Y : , then we can write the IVP (1) in CF fractional derivative sense as: with IC: where The theorem provided below gives the existence of solution of the problem (3) Theorem 2 There exist G : R 2 → R such that: Then there exist at least one fixed point of T which is the solution of the problem (3).Proof To prove that T has a fixed point, therefore Thus for Y, U ∈ Z with G(Y, U) ≥ 0, we obtain Taking ln on both sides, we have: (d(TY, TU)) 2 + d(TY, TU) b + ln[d(TY, TU)) The following theorem show the existence of solution of the problem (4) Then there exist at least one fixed point of T which is the solution of the problem (3).Proof Vol:.( 1234567890 there exist Y * ∈ Z such that TY * .Hence Y * is the solution of the initial value problem (4).

Proposed method
Here, we develop the Laplace transform of the FF operators with different kernels.Next, we'll use LADM to roughly solve the system under consideration.

Scheme for the proposed model with CFO
Equation (1) in terms of the Caputo operator, which is provided by.
with IC, Equivalent form of Eq. ( 5) is: Applying LT to Eq. ( 7), we get: In "Basic definitions" section on the power law kernel, we discussed the definition of the LT.
The series solution can be expressed as: The decomposed non-linear terms are as follows: where E n denotes Adomian polynomials ω 0 , ω 1 , ω 2 , . .., Applying L −1 to Eq. ( 9) together with Eq. ( 10), we get: The series solution is obtained by comparing the terms on both sides of Eq. ( 11).
The series can written as:

Scheme for the proposed model with CFFO
with IC, Equivalently Eq. ( 12) gets the form: Applying LT to Eq. ( 14), we get: In "Basic definitions" section on the exponential decay kernel, we addressed the definition of the LT.
The whole series solution can be scripted as, the non-linear terms are decomposed with Adomian-polynomial discussed above.Applying L −1 to Eq. ( 16), we get Equating terms on both sides in Eq. ( 17), we get: , Vol:.( 1234567890)

Scheme for the proposed model with ABC operator
with IC: Equivalently Eq. ( 18) can written as: Applying LT to Eq. ( 20), we obtain: Using the definition of LT discussed in "Basic definitions" section on Mittag-Leffler kernel, we get: The whole series solution can be scripted as: the non-linear terms are decomposed with Adomian-polynomial discussed above.Applying L −1 to Eq. ( 22), we get Equating terms on both sides in Eq. ( 23), we get: The whole series solution can be scripted as ω(x, t) = ∞ n=0 ω n (x, t).

Validation of the proposed method
Here in this section we solve some of the examples by the proposed method discussed in "Existence of the initial value problems" section.
Example 1 Consider Eq. (1) in Caputo sense which is given by We find ω 0 , ω 1 , ω 2 and so on in order to solve Eq. ( 24) .Since we know that: after some calculation we get: Thus, the solution is: The above table shows the error in approximate vs Exact solution of the considered model with CFO for parameters taken as , ρ = −72 169 , η = 105 169 , and c = 36 169 .From the Table 1, it is observable that the absolute error decreases as space variable x increases, at small time t.
Here, in Fig. 1 the parameters taken is same as above for the Table 1. Figure 1 shows the 3-dimensional graph of Error in Approximate solution of considered model with CFO.Anyone can get an Idea at a glance that how much the proposed method with CFO is efficient by giving such a negligible error vs Exact solution of considered model.www.nature.com/scientificreports/ Figure 2 is the 3d behavior of approximate solution (Eq.27) for the parameters taken as same as above.
In the right plot, we illustrate the 2D behavior of the solution Eq. ( 27) of the model considered, in a fractalfractional sense, comparing it with the exact solution Eq. ( 26) with integer order, for a fixed fractional variable σ = 1 and varying fractal parameter ϑ = 0.6, 0.2.
This analysis pertains to the fractal fractional Kawahara equation, which describes the evolution of wave phenomena in complex media with fractal and fractional characteristics.
We observe that for small values of t, the waves exhibit a close proximity to each other, indicating a subtle interplay between the fractional and fractal effects.Moreover, as time (t) increases, the system's behavior becomes more pronounced, revealing intricate patterns and dynamics.Notably, we note a convergence of the approximate soliton solution towards the exact solution of the considered model, underscoring the robustness of the theoretical framework in capturing the underlying physics of the system.
In Fig. 4, the left plot is the illustration of Eq. ( 27) vs time t for x = 6.5 with varying fractional parameter σ = 0.9, σ = 0.8, σ = 0.6 with fixed fractal variable ϑ = 1 , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. ( 27 25) and ( 26) are the initial condition and exact solution, respectively.The following formula can be used to calculate Eq. ( 28) approximate solution: after simplification we get: Same like above, we can also calculate ω 2 and so on.The solution is describes as .

The above table shows the error in approximate vs Exact solution of the considered model with CFO for parameters taken as
, ρ = −72 169 , η = 105 169 , and c = 36 169 .From the Table 2, it is observable that the absolute error decreases as space variable x increases, at small time t.
Here, in Fig. 1 the parameters taken is same as above for the Table 2. Figure 5 shows the 3-dimensional graph of Error in Approximate solution of considered model with CFO.Anyone can get an Idea at a glance that how much the proposed method with CFO is efficient by giving such a negligible error vs Exact solution of considered model.
Figure 6 is the 3d behavior of approximate solution (Eq.29) for the parameters taken as same as above.
We explore the dynamics of the Caputo-Fabrizio fractal fractional operator in the framework of the Kawahara equation in Fig. 7.
As for the right figure, it explores the 2D behavior of the precise solution Eq. ( 26) under integer order in comparison to the solution Eq. ( 29) under the Caputo-Fabrizio fractal fractional operator.In this case, the fractal parameter ( ϑ = 0.6, 0.2 ) is varied while the fractional variable is fixed at σ = 1 .These studies illuminate wave processes in complicated media, capturing fractional and fractal properties in the context of the Kawahara equation.After investigation, we find that the waves show an impressive coherence at tiny time steps, suggesting complex interactions between fractional and fractal constituents.The system's behavior becomes increasingly    www.nature.com/scientificreports/apparent with time, exposing complex dynamics and changing patterns.We see that the approximation soliton solution converges noticeably to the precise solution, demonstrating the ability of the theoretical framework to accurately describe the physics underlying the system.In Fig. 8, the left plot is the illustration of Eq. ( 29) vs time t for x = 6.5 with varying fractional parameter σ = 1.0, σ = 0.9, σ = 0.8 with fixed fractal variable ϑ = 1 , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. ( 29) for different value of time in order with fixed fractal and fractional variables i − e σ = 1 = ϑ.
Example 3 Consider Eq. (1) with Mittag Leffler kernel Equations ( 25) and ( 26) are the initial condition and exact solution, respectively.The following formula can be used to calculate Eq. ( 26) approximate solution: Same like above we can find other terms.The solution is describes as: .The above table shows the error in approximate vs Exact solution of the considered model with ABC fractional operator for parameters taken as , ρ = −72 169 , η = 105 169 , and c = 36 169 .From the Table 3, it is observable that the absolute error decreases as space variable x increases, at small time t.
Here, in Fig. 9 the parameters taken is same as above for the Table 3. Figure 9 shows the 3-dimensional graph of Error in Approximate solution of considered model with ABC.Anyone can get an Idea at a glance that how much the proposed method with ABC fractional operator is efficient by giving such a negligible error vs Exact solution of considered model.
Figure 10 is the 3d behavior of approximate solution (Eq.31) for the parameters taken as same as above.
(    31) and exact Eq. ( 26) solutions.We examine how the fractional order variable σ behaves at various values, namely σ = 0.7 and σ = 0.3 , while   26) and (31) for different values of σ and ϑ respectively.maintaining the constant value of the fractal variable ϑ at ϑ = 1 .The temporal evolution is fixed at t = 5 in this case.However, when compared to the integer-order exact solution Eq. ( 26), the right plot reveals the complex 2D trajectory of the solution presented in Eq. (31).The fractal parameter ϑ assumes different values: ϑ = 0.7 and ϑ = 0.3 , while the fractional variable σ stays constant at σ = 1 .During our investigation, we find remarkable physical insights.First, at shorter time intervals, the waves show a remarkable closeness, highlighting the interaction of the system's aspects.Furthermore, the system exhibits a measurable progression throughout time, with each instant enhancing its dynamic diversity.Especially, we see a strong convergence: over time, the approximation soliton solution smoothly approaches the precise solution, demonstrating the stability of our model and its ability to describe real-world processes.
In Fig. 12, the left plot is the illustration of Eq. ( 31) vs time t for x = 5 with varying fractional parameter σ = 1.0, σ = 0.9, σ = 0.8 with fixed fractal variable ϑ = 1 , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. (31) for different value of time in order with fixed fractal and fractional variables i − e σ = 1 = ϑ.

Comparative analysis
Here in this section we shows the results obtained on all of the above mentioned operator.
While fractal fractional operators such as the Caputo, Caputo Fabrizio, and ABC operators are scrutinized by means of Laplace Adomian decomposition techniques, a comparison of the outcomes displays in Table 4 and graphically represented in Fig. 13, we observe that the ABC operator accomplishes better than the others.Even though all operators are convenient for simulating sophisticated systems with fractional derivatives, the ABC operator is more accurate and efficient in capturing the complex behavior of fractal events.The ABC operator's supremacy arises from its exceptional capacity to provide a more adaptable framework for explaining anomalies and non-local behaviors seen in fractal systems.In contrast to the Caputo and Caputo Fabrizio operators, the ABC operator adds a new parameter that allows for more precise modifications to the fractional order, improving its flexibility to adapt to a wider range of fractal concepts.As a result, the ABC operator is the go-to option for

Ethical declaration
In this study, human data has not been used for modeling.

Conclusion
This research study has given a way for solving the Kawahara equation using the Laplace transform Adomian decomposition method (LADM) under three different fractal fractional differential operators: Caputo, Caputo-Fabrizio, and ABC.We have proved the solution's existence and uniqueness of solution via advanced fixed point theorems.Our results show that the suggested approach offers an effective and precise solution for nonlinear partial differential equations like the Kawahara problem which has been observed in absolute error in the tables.
A comparison between results under three different have been presented via tables and graphs to figure out that ABC operator provide good results due to nonsingular and nonlocal kernel.Contributions of this paper include a new approach to partial differential equations solution based on fractal fractional differential operators and the LADM.The LADM can be used to other nonlinear issues in physics, engineering, and other disciplines.It also has the benefit of offering effective numerical solutions that work for a variety of issues.We hope that our results will stimulate additional investigation into the application of fractal fractional differential operators and the LTADM for resolving complex issues and promote the creation of more effective and precise numerical methods for solving partial differential equations.By means of our investigation, we provide a valuable contribution to the rapidly developing domain of fractional calculus and establish a foundation for forthcoming research paths that aim to fully use fractal fractional approaches in the modeling of intricate physical processes.Now a days delay differential equations, neural network approach and fractional calculus have many applications in the different fields of sciences such as bifurcation and BAM neural nework 37,38 , predator-prey and Lotka-Volterra system 39,40 , plankton-oxygen model 41 , and others 42,43 .Using these approaches, one can find soliton solutions for the considered system.
then L −1 is: Definition 9 The Laplace transform (LT) of CFO as: Definition 10 The LT of CFFO is: Definition 11 The LT of ABC sense is: where r = [σ ] + 1.

( 2 ,Figure 1 .
Figure 1.The surface plot of Error analysis for exact versus approximate with CFO.

Figure 4 .
Figure 4. Time behavioral plots of considered model with CFO.

Figure 5 .
Figure 5.The surface plot of Error analysis for exact versus approximate with CFFO.

Figure 8 .
Figure 8.Time behavioral plots of considered model with CFFO.

Figure 9 .
Figure 9.The surface plot of Error analysis between exact and approximate with ABC fractional operator.

Figure 12 .
Figure 12.Time behavioral plots of considered model with ABC fractional operator.

Figure 13 .
Figure 13.Comparison analysis of different fractional order.

Table 4 .
Comparison between Caputo fractal fractional, Caputo-Fabrizio, ABC operator.www.nature.com/scientificreports/both researchers and practitioners in various domains, including engineering and the natural sciences, since it produces more accurate findings and makes it easier to comprehend the underlying dynamics of fractal systems.