Anew Technique to Obtain Analytical Solutions for The Fractional Foam Drainage Formula

In the present study, an innovative approach is proposed to solve a special case of fractional foam drainage using the Laplace residual power series technique (LRPS) in conjunction with the Caputo operator for determining the fractional derivative. The study provides extensive guidelines for utilizing this approach to solve time-fractional nonlinear formulas. The effectiveness and validity of the proposed method are investigated and established by comparing the obtained results with the accurate responses using graphs. The study also confirms that the accuracy of the proposed technique increases with the number of items in the combined solution of the problems, as demonstrated by the convergence of the correlation between the obtained solutions and the actual solutions for the special case of fractional foam drainage formula. The research findings suggest that the proposed technique is not only accurate and uncomplicated but also highly adaptable, making it suitable for addressing both linear and nonlinear situations. MSC..


Introduction
In recent years, great progress has been made in the field of partial differential equations as pioneers in the field, such as [1][2][3][4][5][6][7] were given a comprehensive introduction to fractional of differential equations concepts and gained an organized comprehension of partial calculus.Considering the existence and originality of solutions.In 2010 [8] a paper was published on recent achievements in the concept of differential equation with fractional derivatives.Furthermore, as demonstrated by the studies of [9][10], scientists have investigated multiple uses of calculus in multidisciplinary domains including the processing of images and control theories.These contributions have laid a strong foundation for continued research and development in the field of fractional calculus and its applications.
There are no approaches in the research literature that generate precise answers to nonlinear differential equations with fractions.Just approximate solutions can be obtained through linearization, consecutive or perturbation approaches.These approaches include the technique of variational iteration [11], the technique of Adomian decomposition [12][13][14][15], the Homotopy method of analysis [16] and the Iteration Laplace transformation technique [17].
The primary goal of this research is to investigate the results for the special case of fractional Foam drainage concept stated by using a new unique approach known as the Laplace residual power series approach.

𝐷 𝜂
(, ) =   (, ),  ∈ ℝ,  > 0,  > 0, 0 <  ≤ 1, And initial condition as: The present article follows the following pattern: Section 2 contains descriptions and results about Caputo's derivatives and fractions power series.We develop a LRPS approach to the special case of fractional foam drainage issue in Section 3. Section 4 presents graphical findings for the special case of the foam drainage concept.

Preliminaries
In this section, we also go through the Laplace transformation's outcomes and the essential concept of fractions calculus: Definition 1 [18].The fractional derivative in the Caputo meaning defined as: where J  represents the Riemann-Liouville (RL) integrating operations as: and  ∈ ℕ.

Definition 2. [18]
The transform of Laplace (LT) given on the function (x, ) is And the inverse LT defined as:

Remark 5. [19]
Upon applying the inverse Laplace transform to equation ( 6), we obtain the following expression: The fractional Taylor's equation introduced in reference [19] bears resemblance to the equation under consideration.

The Proposed Method for Special Case of Fractional Foam Drainage Formula
To exemplify how the LRPS approach can be utilized to generate a series answer to the FPDEs, firstly we take LT of both sides of Eq. ( 1) we obtain: The construction of equation ( 8) can be achieved by utilizing Lemma 3 as follows: where (, ) = ℒ[(, )] and   (, ) = ℒ[  (, )].
If we assume that U 1 (, ) = () s +  1 ()  1+ , then Eq. ( 14) can be interpreted as: By multiplying both sides of Eq. ( 16) with  1+ , we obtain: By solving the formula given for  1 ()and applying the assumption in Eq. ( 15), as well as taking the limit as s approaches infinity for both sides of Eq. ( 17), we can determine the value of  1 (): By plugging in the value of  1 ()into the algebraic formula (18), we can obtain the following result: To determine the value of the next unknown parameter  2 (), we can substitute the 2nd-truncated series of Eq. ( 16), 1+2 , into the 2nd-LRF and apply the following equation: + ,  > 0. ( We obtain the following equation by multiplying both sides of Eq. ( 20) with  1+2 : To derive the following formula, take the limit as s approaches infinity for both sides of Eq. ( 21) and apply Eq. ( 15): We can obtain the value of  2 () by solving the resulting algebraic equation: As same way, we can find  3 ()   4 () as follows The factor   () can be determined by examining the pattern of the computed factors, which continues as follows: Eq. ( 12) can be expressed as an infinite series using the following representation: By utilizing the inverse Laplace transform of Eq. ( 26) in the provided simplified format, we can obtain the solution for Eqs. ( 6) and (7):

Numerical Issues
This section is dedicated to exploring how LRPSM can be utilized to obtain an approximate solution for the FPDEs: Issue 1.Consider with regard to the shape's FPDEs: And initial condition as: The Laplace transform of Eq. (29) yields Eq. (28) as the resulting equation.The kth-truncated series is purported to be: Therefore, the kth LRFs can be expressed as: By inserting the kth-truncated series (Eq.31) into the kth LRF (Eq.32), we can derive   ().We can then derive the relationship by multiplying the resulting expression by  1+ as Lim →∞ ( +1 LRes  (, )) = 0,  = 1,2,3, ..
The result can be obtained by taking the inverse Laplace transform of the equation.
The accurate solution of Eq. ( 34) is: Figure 1 provides visual evidence of the precision of the proposed method in generating accurate outcomes for the given problem.The figure showcases the graphs of the exact solution and the 5th-order approximation of Eqs. ( 28) and (29) over the interval [0,2].Based on the results, it can be inferred that the proposed approach is a dependable analytical and numerical technique for obtaining exact solutions to fractional partial differential equations.
And initial condition as: The Laplace transform of Eq. (37) yields Eq. (36) as the resulting equation.
The kth-truncated series is purported to be: Therefore, the kth LRFs can be expressed as: By entering the values of   ().into(39) for  = , , , . .., we can derive their specific expressions.
The result can be obtained by taking the inverse Laplace transform of the equation.
The accurate solution of Eq. ( 42) is: Figure 1 provides visual evidence of the precision of the proposed method in generating accurate outcomes for the given problem.The figure showcases the graphs of the exact solution and the 5th-order approximation of Eqs. ( 36) and (37) over the interval [0,2].Based on the results, it can be inferred that the proposed approach is a dependable analytical and numerical technique for obtaining exact solutions to fractional partial differential equations.

Conclusion
This article presents a novel analytical iterative technique that employs the Laplace residual power series to estimate the solution of a nonlinear special case of fractional foam drainage formula.The study investigates the impact of two distinct initial conditions of the special case Foam model on the physical behavior of the system.The results demonstrate that when  is in proximity to 0, the solutions exhibit a bifurcation phenomenon that generates wavelike patterns, whereas when approaches 1, no detectable pattern emerges.This finding provides a new perspective on the relationship between time-fractional derivatives and real-life phenomena.The accuracy of the proposed method is evaluated by analyzing the absolute errors of the approximations of the Foam model, as displayed in Figures 1 and 2. The simplicity and precision of the LRPS approach suggest its potential as a valuable tool for the fractional theory and computations field.In future research, the authors intend to expand the LRPS method's application to linear and nonlinear time-fractional physical problems.