Abstract
This paper investigates a modified analytical method called the Adomian decomposition transform method for solving fractional-order heat equations with the help of the Caputo-Fabrizio operator. The Laplace transform and the Adomian decomposition method are implemented to obtain the result of the given models. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the suggested method. Furthermore, due to the straightforward implementation, the proposed method can solve other nonlinear fractional-order problems.
1. Introduction
The fractional generalization of differential equations has proven to be an effective and precise tool for interpreting real-world problems. Several methods have been proposed in this direction, including Ross and Miller, Podlubny, Caputo, and Kilbas et al. Many scholars, including Das and Gupta, Srivastava et al., and Singh et al., investigated the vibration equation’s fractional extension using the Caputo fractional derivative. Due to the Caputo and other fractional derivatives’ singular kernel, mathematical models involving these fractional derivatives do not consider the entire memory. For describing nonlocal behavioral models, fractional calculus (FC) has become a common mathematical approach. Fractional derivatives have mathematically interpreted many physical problems in recent decades; these representations have produced excellent solutions in real-world modeling issues. Riesz, Coimbra, Hadamard, Riemann-Liouville, Grunwald-Letnikov, Weyl, Caputo, Caputo-Fabrizio, Atangana-Baleanu, among others, gave many basic definitions of fractional operators [1, 2]. A wide variety of nonlinear equations have been developed and commonly applied in numerous nonlinear physical sciences such as chemistry, biology, mathematics, and various branches of physics such as plasma physics, condensed matter physics, fluid mechanics, field theory, and nonlinear optics over the past few years. The exact result of nonlinear equations plays a crucial role in deciding the characteristics and behavior of physical processes, but as dealing with linear equations, it is impossible to find their exact results. Many effective techniques have been used to solved nonlinear FPDEs, for example, the homotopy perturbation transform technique [3, 4], the homotopy analysis transform technique [5, 6], reduced differential transform technique [7, 8], the finite element technique [9], the finite difference technique [10], q-homotopy analysis transform method [11], and so on.
The heat equation was created in 1822 by Joseph Fourier. The analysis of Brownian motion is related to the heat equation. Robert Brown treated this motion. The interaction of particles suspended in a liquid (fluid or gas) with the liquid’s fast-moving atoms causes the erratic movement of the particles. Heat refers to the complex energy of particles that are being exchanged. Faster moving atoms in the atmosphere clash with the scheme’s dividers as emotional vitality is transferred from an increased environment to a cooler system, exchanging some of this energy with the system’s atoms and enabling them to travel faster [12–14]. The wave model is a critical partial differential equation for explaining waves in conventional nuclear physics, such as mechanical waves. The electromagnetic radiation equation is a significant partial differential equation that occurs in areas such as acoustics, electrostatics, and quantum mechanics. Sound, light, gravity, and even matter are examples of them (in the Klein Gordon equation, relativistic quantum mechanics). Alembert and Euler discovered the one-dimensional and three-dimensional wave equations. The results of heat equations have been used of many researcher of mathematics, such as the Adomian decomposition technique [15], the optimal homotopy asymptotic technique [16], the differential transform technique [17], the variational iteration technique [18], Bernstein polynomials with the operational matrix [19], the homotopy perturbation method (HPM) [20], apply the variational iteration technique and Yang-Laplace transformation of time-fractional heat equations [21], Elzaki transformation and the differential transformation technique for nonlinear wave equation [22], and the Aboodh decomposition method [23]. The Laplace Adomian decomposition method (LADM) is a simple and efficient technique for solving linear and nonlinear FPDEs. LADM combines the Laplace transform and Adomian decomposition technique. It is observed that the proposed form, unlike RK4, does not require a predefined declaration size. In comparison to other analytical techniques, this method needs less parameters, no discretization, and no linearization [24]. Modified decomposition method is also compared to ADM to analyze the FPDE result provided in [25]. The result of Kundu Eckhaus equations is introduced in [26], using ADTM. Multistep ADTM are applied to investigate fractional partial differential equations in [27]. Laplace Adomian decomposition method is also applied for the result of fractional-order Smoke model [28]. In this article, we implemented LADM for the investigation of fractional-order heat equations. The current method has a very straightforward and simple method. Absolute error is used to measure accuracy.
2. Definitions and Preliminary Concepts
Definition 1. If , , , then Caputo-Fabrizio of fractional derivative is expressed as is the normalization function with . If ,
Definition 2. The fractional Caputo-Fabrizio integral due to is defined by
Definition 3. The Laplace transformation of relation with Caputo-Fabrizio as If
3. The General Methodology of the Present Method
In this section, the general result finds the FPDEs with the help of the Laplace Adomian decomposition method. where the Caputo operator , where is linear and nonlinear term, and the source term is .
With initial condition
Using the Laplace transformation to Equation (6), we get apply the Laplace differentiation transform property, we obtain as
The LADM result shows the infinite-series: the nonlinear term can be solve by Adomian polynomials putting Equation (11) and Equation (12) in Equation (10), we achieve as
Implementing of the Laplace transformation,
Usually, we can written as
Applying inverse transform in Equation (16), we get
4. Numerical Implementation
4.1. Problem 1
Consider the one-dimensional fractional heat equation (26): with initial condition
Using the Laplace Caputo-Fabrizio operator in Equation (18),
Applying inverse Laplace transformation,
Implement the ADM processes, we achieve
for
The subsequent terms are
The LADM solution for Problem 1 is when , then the LADM result is
The close form of the Problem 1 is
Figure 1 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 1. In Figure 2, the first 3D graph shows the different fractional-order of , and the second 2D figure shows the different fractional-order of . In Table 1, values of the error, however, are proved to be proportional to the divergence of exact solutions and corresponding to the uncertainty in the approximate solutions.
4.2. Problem 2
Consider the time-fractional two-dimensional heat equation (26): with the initial condition
Applying the Laplace transform of Equation (28), we get
Applying inverse Laplace transformation,
Applying the ADM processes, we achieve as
for
The subsequent terms are
The LADM solution for Problem 2 is
The closed form of the above equation is
Figure 3 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 2. In Figure 4, the first 3D graph shows the different fractional-order of, and the second 2D figure shows the different fractional-order of.
4.3. Problem 3
Consider the time-fractional three-dimensional heat equation (26): the initial condition is
Using the Laplace transformation of Equation (37),
Implementing inverse Laplace transformation
Apply the ADM processes, we achieve
for
The subsequent terms are
The LADM solution for Problem 3 is when , then the LADM solution is
This is the close form solution of the above Problem 3
Figure 5 shows the exact and approximate solution graphs with the help of Caputo-Fabrizio operator of example 3. In Figure 6, the 3D graph shows the different fractional-order of . In Figure 7, 2D figures show the different fractional-order of with respect to and .
5. Conclusions
In this paper, the fractional-order approximate results of heat equations are calculated via LADM with the Caputo-Fabrizio operator. The solutions have shown the highest agreement with the actual results for the examples. This method results for some problems have revealed the accuracy of the suggested technique. It is observed that the achieved analytical series solutions for the first few terms are very precise and converge very rapidly to the solutions of real physical problems. In the future, LADM can be applied to investigate the analytical result of other linear and nonlinear fractional partial differential equations, which are frequently implemented in engineering and science.
Data Availability
The numerical data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
One of the coauthors (A. M. Zidan) extends his appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through a research group program under grant number R.G.P.-2/142/42. This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program to support publication in the top journal (Grant no. 42-FTTJ-42).