Research Analytical Investigation of Nonlinear Fractional Harry Dym and Rosenau-Hyman Equation via a Novel Transform

We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we ﬁ rst construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to ﬁ nd an error analysis that demonstrates that our ﬁ nal result converges to the exact and approximate result. The convergent series form solution demonstrates the method ’ s e ﬃ ciency in resolving several types of fractional di ﬀ erential equations. Furthermore, the solutions obtained in this study agree well with the exact solutions; thus, this strategy is powerful and e ﬃ cient as an alternate way for obtaining approximate solutions to both linear and nonlinear fractional di ﬀ erential equations.


Introduction
Fractional calculus FC history dates back 300 years. FC originated with Leibniz's usage of the nth derivative notation in his papers in 1695. L'Hopital raises a query from Leibniz about the result of his nth derivative notation if the order of "n" is 1 /2 [1]. Many phenomena in engineering and other fields can be effectively represented by models based on fractional calculus, that is, the theory of fractional derivatives and integrals of fractional noninteger order. Respectable interest in fractional calculus has been utilised in several studies in recent years, such as regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, viscoelasticity, electrical circuits, electro-analytical chemistry, biology, and control theory [2][3][4][5].
Due to their prevalence in a wide range of applications and accurate description of nonlinear processes, researchers are increasingly focusing on fractional order differential equations, particularly fractional partial differential equations (FPDEs). FPDEs are the most common mathematical tools used to simulate diverse physical phenomena in applied sciences such as physics, engineering, and other social sciences.
In the present study, we implement the Elzaki transform in connection with the CFD and ABC operators to solve two nonlinear problems. We consider fractional Harry Dym equation and fractional Rosenau-Hyman equation of the form having initial source and having initial source The Harry Dym is a crucial dynamical equation that is used in a variety of physical systems. The Harry Dym equation was initially published in Kruskal and Moser [31] and is credited to Harry Dym in an unpublished study from 1973-1974. It denotes a system in which dispersion and nonlinearity are inextricably linked. Harry Dym is a totally integrable nonlinear evolution equation that obeys an infinite number of conversion rules but lacks the Painleve property. The Harry Dym equation is closely related to the Korteweg-de Vries equation, and this equation has been used to hydrodynamic problems [32]. The Sturm-Liouville operator is linked to the Lax pair of the Harry Dym equation. This operator is spectrally transformed into the Schrodinger operator by the Liouville transformation [33]. Rosenau and Hyman [34] found the Rosenau-Hyman equation, which arises in the creation of patterns in liquid drops with compaction solutions. The Rosenau-Hyman equation compact on investigations is useful in applied sciences and mathematical physics [35][36][37][38].
The following is how the rest of the paper is structured: we begin with basic preliminaries and definitions of fractional calculus in Section 2. The proposed method's general methodology is introduced in Section 3. Section 4 focuses on applying the approach to a set of test problems, using graphs and tables to demonstrate the technique's efficiency. The discussion and conclusion of this work were delivered in Section 5.
Definition 3. The fractional integral operator in ABC manner is given as [40] ABC Definition 4. For exponential function in set A, the Elzaki transform is given as [41,42] G is a finite number, but p 1 and p 2 may be finite or infinite for a function selected in the set.

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Proof. From Definition 2, we get Then, from Elzaki transform definition and its convolution, we obtain

Description of the Technique via a New Integral Transform
In this part, we presented the general methodology used in this article to solve fractional nonlinear PDE as with initial source and the boundary sources Here, known functions are κ z , θ, γ 0 , and γ 1 . In Eq. (15), D ρ τ ψðυ, τÞ represents the Caputo or ABC fractional derivatives whereas Lð:Þ and Nð:Þ are linear and nonlinear terms.
(1)-(2) and (3)-(4) represent the problems to be solved. By means of Elzaki transform of CFD in Eq. (11) and ABC in Eq. (12), we take Efψðυ, τÞgðμÞ =ζðυ, μÞ in Eq. (15). Thus, by means of Caputo fractional derivative, we get Also by means of ABC derivative, we get Here, E½θðυ, τÞ = e θðυ, μÞ. Now by taking the Elzaki transform of the boundary conditions, we obtain We get the solution of Eqs. (15)-(17) by means of perturbation techniquẽ In Eq. (15), the nonlinear terms are calculated as and the terms υ E ðυ, τÞ are taken in [45] as For Caputo operator, the solution is determined as by putting Eqs. (21) and (22) into Eq. (18), Also for Atangana-Baleanu operator, the solution is determined as by putting Eqs. (21) and (22) into Eq. (19), Then, by solving (24) and (25) in terms of X, the given Caputo homotopies are obtained:

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In addition, the ABC homotopies are obtained as given: When X ⟶ 1, we get Eqs. (26) and (27) approximate solution for Eqs. (24) and (25) as Now by taking inverse ET of Eq. (28), we get the approximate solution of Eq. (15)

Applications
In this part, we will solve problems in Eqs. (1)-(4) by implementing Elzaki transform. First, we implement Elzaki transform technique in combination with Caputo derivative to solve problem (1) having initial source (2). By taking the Elzaki transform, we get Now applying Elzaki perturbation transform technique in Eq. (30), we obtain On taking Elzaki inverse transform of Eq. (31), we get In Eq. (43), the υ E ð:Þ denotes the nonlinear terms given in Eq. (24), Thus by considering powers of X, we get Caputo operator solution as The series form solution of the problem is given as which gives the solution at ðρ = 1Þ as ða − 3 ffiffi ffi b p /2ðυ + bτÞÞ 2/3 . Now, we implement Elzaki transform technique in combination with Atangana-Baleanu operator to solve same problem. By taking the Elzaki transform, we get Now applying Elzaki perturbation transform technique to (36), we obtain On taking Elzaki inverse transform of Eq. (37), we get Journal of Function Spaces In Eq. (38), υ E ð:Þ denotes the nonlinear terms given in Eq. (23). By repeating the same process for nonlinear terms, we obtain the following terms: On taking Elzaki inverse transform of Eq. (42), we get Thus by considering powers of X, we get Caputo operator solution as c cos 2 υ 4 , The series form solution of the problem is given as which gives the solution at ðρ = 1Þ as, −8/3c cos 2 ð1/4ðυ − cτÞÞ. Now, we implement Elzaki transform technique in combination with Atangana-Baleanu operator to solve same problem.
Journal of Function Spaces By taking the Elzaki transform, we get Now applying Elzaki perturbation transform technique to Eq. (47), we obtain On taking Elzaki inverse transform of Eq. (48), we get In Eq. (38), υ E ð:Þ denotes the nonlinear terms given in Eq. (23). By repeating the same process for nonlinear terms, we obtain the following terms: c cos 2 υ 4 ,

Results and Discussion
In this article, a detailed investigation of error analysis between exact and approximate solutions, as stated by Tables 1 and 2, has been conducted with greater accuracy. In table, calculating the absolute error at various fractional-orders demonstrates the simplicity and accuracy of the provided method. The error analysis between the exact and approximate solutions is shown in Tables 1 and 2, indicating that the series solution quickly converges to a small value. Also, in Tables 3 and 4, we show the numerical simulation of the proposed method solution. As a result, we will only use the third order of the series solution throughout the numerical evolution. The correctness of the error analytical result will be increased by inserting more terms of approximation solution. Figures 1 and 2

depict the
Journal of Function Spaces  9 Journal of Function Spaces behaviour of the exact and proposed approach solutions and describe the properties of the approximate solution. We also present the proposed approach solution at different fractionalorders for a better understanding of the problems characteristics. We concluded that the recommended technique solution was in good agreement with the exact solution based on the tables and graphs.

Conclusion
The main goal of this study is to use an efficient technique to determine the solution to the fractional Harry Dym equation and fractional Rosenau-Hyman equation. The proposed method is used in addition to two fractional derivatives: Caputo fractional derivative and Atangana-Baleanu fractional derivative. Tables and figures are used to specify the results of the comparative solution. The tables and figures show that the suggested technique solution and the exact result have a better understanding. From the derived results, it shows the reliability of the algorithm, and it is greatly suitable for nonlinear fractional partial differential equation.

Data Availability
The numerical data used to support the findings of this study are included within the article.