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The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear timefractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When ζ = 1, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems. [ABSTRACT FROM AUTHOR]

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system's approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. [ABSTRACT FROM AUTHOR]

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations. [ABSTRACT FROM AUTHOR]

This article investigates the seventh-order Lax's Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax's Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance. [ABSTRACT FROM AUTHOR]

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article's results possess complimentary properties that relate to the symmetry of partial differential equations. [ABSTRACT FROM AUTHOR]