New Local Fractional Mohand–Adomian Decomposition Method for Solving Nonlinear Fractional Burger’s Type Equation

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Introduction
Fractional diferential equations are widely used in many felds of science and engineering to describe physical phenomena that exhibit nonlocal and nonlinear behavior.Burger's type equation is one such equation that describes the dynamics of a wide range of phenomena including gas dynamics, fuid mechanics, and population dynamics [1].However, the solution of Burger's type equation becomes challenging due to its nonlinearity and fractional derivatives [2].To address this challenge, various numerical methods have been proposed, and one such method is the local fractional Mohand-Adomian decomposition method (LFMADM).In general, fractional partial diferential equations do not have exact solutions, and only approximate and numerical methods can be employed to obtain solutions.
Burger's equation: is a fundamental nonlinear partial diferential equation in fuid mechanics.Burger's equation is a crucial model utilized in various felds of applied mathematics, including gas dynamics, heat conduction, trafc fow, and acoustic waves.
Its importance was initially highlighted by Bateman in 1915 [3] who identifed its steady solutions as worthy of exploration.Later, Burger proposed it as one of the equations describing mathematical models of turbulence in 1948 [4].Benton and Platzman [5] conducted a comprehensive survey of one-dimensional Burger's equation in 1972, examining its exact solutions.Te pursuit of numerical or analytical solutions to such equations is of signifcant importance in applied mathematics, leading to numerous studies by scientists to determine these solutions [6,7].Te area of fractional calculus fnds extensive applications across a wide range of engineering and scientifc disciplines.Tese include viscoelasticity, fuid mechanics, biological population modeling, electrochemistry, and optics.Fractional calculus is particularly valuable when it comes to modeling physical and engineering systems that are most accurately described by fractional diferential equations.Tese models are employed to provide precise representations of systems with specifc damping requirements.Recently, various numerical and analytical methods have been introduced in these domains, often applied to tackle novel and challenging problems.Mathematical modeling frequently leads to the formulation of fractional diferential equations and a range of problems that encompass special functions of mathematics, along with their generalizations in more variables.Furthermore, many physical phenomena in felds such as quantum mechanics, fuid dynamics, electricity, and ecological systems are governed by fractional-order partial diferential equations (PDEs) within their applicable scope.Consequently, it is of growing signifcance to have a comprehensive understanding of both traditional and newly devised techniques for solving fractional PDEs and the practical applications of these methods [8][9][10].
Numerous investigations have been conducted by researchers to acquire both numerical and analytical solutions for Burger's equation.Ozis and Ozdes [11] employed a direct variational approach, whereas Aksan and Ozdes [12] introduced a variational method based on time discretization.Numerous wavelet-based approaches have been meticulously examined to evaluate their precision and efciency in tackling both linear and nonlinear fractional diferential equations.Te authors have delved into the challenges faced by researchers in this particular feld and emphasized the importance of interdisciplinary collaboration to advance the exploration of various wavelets for solving diferential equations spanning diverse orders.Several wavelet methodologies, such as the cubic B-spline wavelet, Haar wavelet, Legendre wavelet, Legendre multiwavelet, and Chebyshev wavelet methods, have been subjected to scrutiny for their applicability in solving fractional diferential equations.Among these techniques, the Legendre multiwavelet method, when combined with the Galerkin method, has demonstrated efectiveness in providing approximate solutions to initial value problems associated with fractional nonlinear partial diferential equations.Tese wavelet-based approaches adeptly convert fractional diferential equations into systems of algebraic equations, facilitating their straightforward resolution through conventional methods.Kutluay et al. [13] derived a numerical solution for Burger's equation using fnite diference methods, while Varoḡlu and Liam Finn [14] utilized a weighted residue method.Caldwell et al. [15] employed fnite elements, and Evans and Abdullah [16] utilized the group explicit method.Mittal and Singhal [17] employed the Galerkin method to compute numerical solutions for Burger's equation.In 2005, Gorguis [18] presented a comparison between the Cole-Hopf transformation and the decomposition method for solving Burger's equation.Tus, it can be inferred that researchers have dedicated signifcant attention to obtaining solutions for the fractional Burger's equation.
Nonhomogenous fractional Burger's equation: Time-fractional Burger's Kdv equation: Fractional modifed Burger's equation: Considering 0 < α ≤ 1, the symbol U(y, t) signifes the velocity in the spatial dimension y and at time t.Due to its signifcant relevance, researchers have focused extensively on acquiring both precise and numerical solutions for equations resembling Burger's equation.
Te objective of this study is to create a dependable and efective numerical technique, utilizing the LFMADM, for solving nonlinear fractional Burger's equation.Te proposed method will undergo validation and analysis, assessing its convergence behavior, computational efciency, and sensitivity to parameters and boundary conditions.Te method will be applied to practical problems in fuid mechanics and population dynamics.Te results of this research are anticipated to enhance the development of more precise and efective numerical methods for solving nonlinear fractional diferential equations.
Furthermore, the proposed method will be applied to solve practical problems related to fuid mechanics or population dynamics, which will serve as a demonstration of its efectiveness in real-world scenarios.Trough the outcomes of this research, the development of more accurate and efcient numerical methods for solving nonlinear fractional diferential equations is expected to be advanced.Te successful development of this numerical technique will provide valuable insights into the behavior and solutions of nonlinear fractional Burger's equation, which can be utilized in various felds that require a precise understanding of complex mathematical models, such as engineering and physics.
Tis paper is organized as follows.In Section 2, we highlight some basic defnitions which are used.In Section 3, we develop the numerical method and present the graphical representation of the proposed methods.In Section 4, we draw the conclusion and give the future direction.

Definitions Defnition 1. Te CFD of a function
provided the right side is pointwise defned on (0, ∞), where n � [c] + 1 in case c is not an integer and n � c in case c is an integer.

Mohand Transform.
Integral transforms provide a systematic approach for resolving challenges within engineering sciences, such as heat conduction, radioactive decay, beam vibration, and population growth problems.Numerous researchers have employed a wide range of integral transforms, including Fourier, Laplace, Mahgoub, Kamal, Mohand, Sumudu, and Elzaki transforms, to provide solutions for various classes of equations such as delay differential equations, integral equations, partial diferential equations, and partial integrodiferential equations.One particularly powerful technique in this regard is the Mohand transform, which fnds its roots in the classical Fourier integral.Tis transformative method, pioneered by Mohand Mahgoub, simplifes the resolution of both partial and ordinary diferential equations within the time domain, leveraging its mathematical elegance and inherent properties.While Laplace, Fourier, Elzaki, Aboodh, Sumudu, and Kamal transforms are established mathematical tools for addressing diferential equations, the Mohand transform and its fundamental characteristics also ofer valuable resources for tackling such mathematical challenges.Te operator denoted as M(.), commonly referred to as the Mohand transform, is rigorously defned through integral equations as follows: and here, for t ≥ 0 and k1 ≤ s ≤ k2, the variable k1 ≤ s ≤ k2 plays a role in factoring the variable k1 ≤ s ≤ k2 within the argument of the function f.Tis transformation exhibits notable connections with the Laplace, Elzaki, and Aboodh transforms.

Local Fractional Mohand-Adomian Decomposition Method.
Te combination of the Mohand transform and Adomian decomposition method, known as the Mohand-Adomian decomposition method, ofers notable advantages in solving diferential equations.Tis approach not only provides algebraic values but also delivers closed-form solutions in the form of power series.It proves to be a straightforward technique for obtaining solutions to both linear and nonlinear fractional diferential equations, characterized by its efciency in terms of computational workload.An advantageous feature of this method lies in its capability to address nonlinear fractional diferential equations without the necessity of employing He's or Adomian's polynomials for handling nonlinear terms.Tis proposed methodology stands out due to its absence of restrictive assumptions and linearization requirements.As a result, it proves instrumental in analytically addressing a wide array of practical problems represented by fractional-order ordinary and partial diferential equations.Defnition 2. Te Mohand transform, as proposed by Mahgoub in 2017 [19], is defned as the transformation of the function f α (p * ) in the following manner: Additionally, the inverse Mohand transform is expressed as Defnition 3. Local fractional convolution: when dealing with the Mohand transform of functions, where f 1,α (p * ) * f 2,α (p * ) is defned as Defnition 4. Te Mittag-Lefer function, denoted as E α (y) for α > 0, is defned as follows: where α is a member of a complex number set.(i) Let f(t) � 1; then, by the defnition, we have

Mohand
(ii) Let f(t) � t; then: In the typical scenario, when n is a positive integer, then

Numerical Methods
In this section, we proposed a numerical Mohand-Adomian decomposition method for homogenous fractional Burger's equation.

Mohand-Adomian Decomposition
Method.To illustrate the method described above, we have chosen a homogenous fractional Burger's equation: where (0 < α < 1) and the initial condition is Upon applying the Mohand transform to (16), we obtain: Now using the defnition of the Mohand transform, we get and taking inverse Mohand transform, 4

Journal of Mathematics
Terefore, the initial condition will be utilized to determine the frst term of u(y, t).
Ultimately, we derive the recursive general relation in the following form: which gives u 0 is given, so we get u 1 easily.Similarly, we have Hence, the solution to (21) can be expressed as

Examples.
Next, we will employ the previously mentioned method in specifc scenarios.
Example 1.Consider the following homogenous fractional Burger's equation: where (0 < α < 1) and initial condition In conclusion, we arrive at the recursive general relation in the following form: In a similar manner, we obtain that sin(y).
(33) Also, 3!α 2 sin(y), Figure 1 represents the line graph plotting of homogenous fractional Burger equation (26), while Figure 2 illustrates the 3D surface plots of (26) using various fractional orders..In this section, we developed a numerical method based on the Mohand-Adomian decomposition method for nonhomogenous fractional Burger's equation.Let us now consider the following nonhomogenous fractional Burger's equation:
Applying the Mohand transform to (36), we get Now using the defnition of the Mohand transform, we get Taking inverse Mohand transform, we get Journal of Mathematics Finally, we obtain the recursive general relation form as

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Journal of Mathematics Terefore, from (41), we get Since u 0 � 0, we get Similarly, By adding all the terms, the series solution of (36) can be found as . (45) Figure 3 depicts a line graph illustrating the plotting of solutions derived from nonhomogenous fractional Burger equation (36).Tis visual representation ofers a clear insight into the behavior and trends depicted by these solutions.Moreover, in Figure 4, there are 3D surface plots showcasing (36) at work, demonstrating its behavior under diferent fractional orders.Tis visual exploration allows for a comprehensive understanding of how varying fractional orders impacts the overall solution landscape.Additionally, the exact solution of (36) is presented in Figure 5, providing a comparative perspective to the plotted solutions.Tis fgure serves as a reference point to understand the characteristics and deviations of the exact solution concerning the solutions portrayed in Figure 4.

Another Type of Fractional Burger's Equation.
In this section, we developed a numerical method based on the Mohand-Adomian decomposition method for another type of fractional Burger's equation.
We examine the subsequent fractional Burger's equation: with initial condition Applying the Mohand transform to (46), we get Now using the defnition of the Mohand transform, we get    (50) We derive the general recursive relation in the following form: Terefore, from (46), we get Similarly,

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Journal of Mathematics Summing the above terms yields In Figure 6, a precise line graph illustrates the solutions obtained from fractional type Burger equation (46).Tis graph ofers a clear and detailed view of the complex behavior and trends displayed by these solutions.Further, Figure 7 provides intricate 3D surface plots that demonstrate (46) under various fractional orders.Tese visuals allow a comprehensive understanding of how altering fractional orders intricately infuences the overall solution patterns.
Moreover, Figure 8 exhibits the exact solution derived from (46) for comparison purposes.Tis fgure acts as a standard reference, aiding in a nuanced examination of specifc traits and deviations when contrasted with the solutions depicted in Figure 7.

Conclusion
In this research, we have applied the Mohand transform and Adomian decomposition method to address fractional Burger's equations, fractional Burger's Kdv equations, and fractional modifed Burger's equations.Te objective of this investigation is to showcase the efectiveness and simplicity of the Mohand transform as a viable approach for obtaining precise and approximate solutions to these equations.Furthermore, we have illustrated several applications to underscore the versatility of this methodology.
By analyzing the graphs of the solutions, we have observed that the behavior of the solution varies signifcantly with diferent values of the fractional-order derivative.Tis indicates the importance of considering diferent fractionalorder values when studying these equations.Notably, when setting α � 1 in the given examples, we have obtained the exact solutions that have been previously investigated in references [20,21].
It is worth noting a few crucial points regarding the Mohand transform.Firstly, this method provides the solution in terms of easily computable components, making it highly practical for real-world applications.Te solutions obtained using the Mohand transform exhibit rapid convergence, which is benefcial for solving physical problems accurately.Te numerical results obtained from this approach have shown excellent agreement with their respective exact solutions, further validating its efectiveness.
Secondly, the methods employed in this study were applied directly, without resorting to linearization, perturbation, or restrictive assumptions.Tis direct approach demonstrates the broad applicability of the Mohand transform in solving various linear and nonlinear fractional problems encountered in applied science.
For the purpose of presenting graphical representations of the solutions, we have utilized MATLAB in this thesis.In future work, we can further explore the capabilities of the Mohand transform by employing it to solve nonlinear diferential equations and systems of linear equations and compare its outcomes with other existing methods to evaluate its efciency and performance.
Overall, the results obtained in this study highlight the efectiveness and practicality of the Mohand transform and Adomian decomposition method in solving fractional differential equations.Te potential of this approach for solving a wide range of linear and nonlinear fractional problems demonstrates its signifcance in applied science and paves the way for future research and applications.