An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform

In this article, the ρ-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique procedure is straightforward and appealing, and it may be used to solve non-linear fractional-order partial differential equations effectively. The Caputo operator is used to express the fractional derivatives. Four numerical problems involving fractional-order third-order dispersive partial differential equations are presented with their analytical solutions. The graphs determined that their findings are in excellent agreement with the precise answers to the targeted issues. The solution to the problems at various fractional orders is achieved and found to be correct while comparing the exact solutions at integer-order problems. Although both problems are the non-linear fractional system of partial differential equations, the present technique provides its solution sophisticatedly. Including both integer and fractional order issues, solution graphs are carefully drawn. The fact that the issues’ physical dynamics completely support the solutions at both fractional and integer orders is significant. Moreover, despite using very few terms of the series solution attained by the present technique, higher accuracy is observed. In light of the various and authentic features, it can be customized to solve different fractional-order non-linear systems in nature.


Introduction
Fractional calculus (FC) has been a significant field of applied sciences for a few decades. They model actual phenomena with fractional-order integral and derivative answers better than classical derivatives. In modelling many physical phenomena, certain signification implementation can be traced, especially genetics algorithm, signal processing, visco-elasticity damping, transport schemes, electronics, biology, communication, physics, robotics, finance, and chemistry. In the area of FC, scientists are focusing on many significant contributions and discoveries [1][2][3][4]. Due to its appealing applications, FC is a significant study subject for most scholars, and the analysis of fractional-order partial differential equations (PDEs) has drawn special interest from several areas. As a result, various approaches to solving linear and non-linear fractional PDEs have been developed. The local meshless technique, for example, is used to solve fractional-order and anomalous mobile-immobile result transport producers [5][6][7][8][9].

Definition 4.
The continuous function of a ρ-Laplace transform g : [0, +∞] → R is defined as [39]: The ρ-Laplace transform Caputo generalized fractional-order derivative of a continuous function g is given as [39]:

Definition 5.
The generalized Mittag-Leffler function is define as:

The General Implementation of Methodology
Consider the functional equation where g is a source function, G is linear and N non-linear terms, respectively. Let us consider analysis of Equation (1) is given as: where m = 1, 2, 3 · · · , Now, linear and non-linear terms can be expressed as [30]: We obtain the solution of (1) as: Consider the fractional-order Caputo derivative equation is given as: the initial condition ϕ(ε, 0) = g(ε), where N is a non-linear term. Now applying ρ-Laplace transform of Equation (6), we get: applying ρ-Laplace transform, we achieve: Simplify (8), we get: Using inverse Laplace transform of (10), we have: Applying new iterative method, we get The recurrence relation is defined as: where m = 1, 2, 3 · · · , Hence, the solution of Equation (6) is define as,

Convergence of NITM
Now, we discus the condition for convergence of NITM.
As it is difficult to show bounded ness of u i , for all i, a more useful result is proved in the following theorem, where conditions on N (k) (u 0 ) are given which are sufficient to guarantee convergence of the series.
for all n, then the series ∑ ∞ n=0 G n is absolutely convergent [40]. (13), (11), and the hypothesis of Theorem 2, we observe that

Graphical Discussion
The aim this article to investigate an approximate result of the third-order fractional dispersive PDEs, implemented the analytical technique. The new iterative transformation method is applied to analysis of the given problems. The validity show of the suggested method, the result to some illustrative equations are suggested. In Figure 1 Figures 6-9. It has been demonstrated that the suggested methods are similarly accurate It is analyzed whether fractional-order problems, like fractional-order observation, converge to an integer-order result. In Table 1 show that absolute error of different fractional order represent close contact with each other. The convergence of fractional-order solutions to integral-order approaches is observed in the same way.

Conclusions
The iterative transform approach is used to solve fractional-order third-order dispersive partial differential equations in this paper. The resulting findings' graphical representations have been completed. The improved accuracy of the recommended procedure is clearly demonstrated by this depiction of the acquired findings. The solutions obtained for fractional systems are closely akin to their exact result. It has been demonstrated that fractional answers may be converted to integer-order solutions. The proposed method's key themes include fewer computations and improved precision. It was later developed to solve various fractional-order linear and nonlinear partial differential equations by the researchers.