Abstract
In this work, the solution of the linear, nonlinear, and coupled system fractional singular two-dimensional pseudoparabolic equation is examined by using a three-dimensional Laplace Adomian decomposition method (3-DLADM). Analysis of the method is discussed, and some demonstrative examples are mentioned to confirm the power and accuracy of the recommended method, and numerical analysis is applied to sketch the exact and approximate solution.
1. Introduction
The fractional derivative has been attracting much attention in physical and engineering problems, for instance, acoustics, viscoelasticity, and control. The sufficient condition for commutators of a fractional integral operator is discussed in [1]. The authors in [2] addressed the boundedness of commutators of the multidimensional Hardy-type operators with bounded mean oscillation coefficients. The fractional derivative of the Riemann zeta function was computed, using the Caputo derivative in the Ortigueira sense (for more details, see [3]). The author in [4] analyzed the fractional derivative of the Riemann function and discussed the functional equation with the distribution of prime numbers.
The parabolic equation occurred in several fields of applied mathematics, for example, the heat diffusion equation and fluid mechanics (for a model, see [5–8]). The solution fractional diffusion equation problems have been obtained via the Adomian decomposition method and series expansion method by the authors [9, 10]. Several articles have been found in previous studies, which are associated with qualities and applications of a fractional derivative [11–13]. The pseudoparabolic equation represents a diversity of physical operation. The author in [14] discussed the existence, uniqueness, and continuous dependence of powerful solutions of the one-dimensional pseudoparabolic equation. Overall, certainty of the nonlinear equations of real life is so far very hard to solve either theoretically or numerically. Currently, many researchers have suggested an exact solution to a one-dimensional coupled parabolic equation (for more details, see [15, 16]). The convergence of the Adomian method was discussed by many researchers (we refer the readers to see [17–20]). The author has modified the 2-D Laplace decomposition method to solve coupled pseudoparabolic equations in order to accelerate the convergence of the series solution [21].
Recently, the three-dimensional Laplace Adomian decomposition (3-DLADM) has been successfully applied to solve regular and singular coupled Burgers’ equations (see [22]). The objective of this research is to find the solution of singular 2-D fractional pseudoparabolic equations by applying a more successful technique, which is called 3-DLADM.
2. Some Basic Idea of the Double and Triple Laplace Transform and Caputo Fractional Derivative
In this unit, we offer basic definitions, properties of fractional calculus, Mittag-Leffler function, and Laplace transform theory which should be used in this work.
Definition 1 (see [23]). The left-sided Caputo fractional derivative of is given by
Definition 2 (see [24]). The Caputo fractional partial derivative of function with respect to is given by
Definition 3 (see [25]). Let be a continuous function of two variables ; then, the two-dimensional Laplace transform (2-DLT) of is given where , indicate 2-DLT, and are complex variables.
Definition 4 (see [26]). Let be a piecewise continuous function on the interval of exponential order. Consider for some . Under these conditions, 3-DLT is defined by where the symbol indicates 3-DLT and .
Definition 5. The inverse 3-DLT of the function is determined by where indicate inverse 3-DLT with respect to and .
Furthermore 3-DLT of the derivatives and are presented by
The 2-DLT formulas for the partial fractional Caputo derivatives are denoted by where (for more details, see [27]).
In the following, we provided one and two parameters; the classical Mittag-Leffler function is useful in this work.
2.1. Mittag-Leffler Function
The Mittag-Leffler function of one parameter is established by
The Mittag-Leffler function with two parameters is determined by (see [28, 29]). If we set in Equation (11), we get Equation (10). It appears from Equation (11) that hence, in general,
Differentiation of the Mittag-Leffler function is represented by (for more details, see [30]).
Next, we provide the Laplace transform (LT) of Mittag-Leffler functions helpful in this research:
In the same way, the single Laplace transform (LT) of two-parameter Mittag-Leffler functions
3. Multidimensional Laplace Transforms (-DLT)
Here, we deal with the multidimensional Laplace transform (-DLT) which is very useful to this work.
Definition 6. Let be a piecewise continuous function on the interval of exponential order. Consider for some Under these conditions, -DLT is defined by where , the symbol indicate -DLT, and and are complex variables.
Definition 7. The inverse -DLT of the function is determined by where and indicate inverse -DLT with respect to and .
3.1. Existence Condition for the Multi-Laplace Transform
If is said to be of exponential order (>0) and (>0) on , , if there exists a positive constant such that for all and : where can be written as at and Or, equivalently,
where the function is simply called an exponential order as , , and clearly, it does not grow faster than as , .
Theorem 8. If a function is a continuous function in every finite intervals and and of exponential order , then the double Laplace transform of exists for all and provided and .
Proof. We have
For and .
In the following theorem, we generalize Equations (8) and (9) to a multidimensional Laplace transform.
Theorem 9. Let be so that , for any (>0) and (>0): hold for any constant , then where is -DLT of
Proof. With the use of Theorem 9 and definition of multidimensional Laplace transform, we can deduce the proof easily.
Theorem 10. Let be piecewise continuous on -DLT of the partial derivatives of order th and are given by where
Proof. By employing definition of -DLT for , we get
by taking partial derivatives for both sides of Equation (31), we have
The integral inside bracket is determined by
where ; hence, we find that
by substituting Equations (33) and (34) into Equation (34) and using Theorem 9, we achieved
Similarly, we can obtain Equation (30).
In particular, at we have
4. Singular -D Fractional Pseudoparabolic Equation and 3-DLADM
The following is the procedure demonstrating two problems that are related to the linear and nonlinear singular -D pseudoparabolic equation:
Problem 11. The 3-DLADM is an effective technique for solving linear singular two-dimensional pseudoparabolic equation.
We consider let us consider a general fractional singular -D pseudoparabolic equation:
where the linear parts and are called Bessel’s operator and , are known functions. For the objective to solve Equation (38), we are implementing the following steps.
Step 1. By multiplying the two sides of Equation (38) by
Step 2. By implementing Equations (36), (37), and (7) for the equation in the first step and 2-DLT for condition, we get
Step 3. Operating the integral form of Equation (41), from to and to with respect to , respectively, we get
Step 4. The series solution of the singular -D pseudoparabolic equation is therefore entirely determined by
Step 5. Working with the 3-DLT for both sides of Equation (42) and applying Equation (43), we obtain in view of the first approximation, and the remaining components , are denoted by
Here, we consider that the inverse 3-DLT with respect to and of Equations (45) and (46) exists. To display the applicability of the method explained previously, we current the following example.
Example 12. Singular -D pseudoparabolic equation is given by By applying the above steps and Theorem 8 for Equation (47), all terms of the sequence are computed as follows: according to 3-DLADM, we get the following components: In the same way, we receive that by adding all the terms, we have therefore, the approximation solution of Equation (47) is denoted by By using and Equations (13) and (14), the approximation solution becomes
Figures 1(a) and 1(b) show the approximate and exact solutions of Equation (47); at and , we obtain the exact solution of Equation (47); by taking different values of such as , , and , we get the approximate solution. Figures 1(c) and 1(d) show the plot of function in three dimensions.
In the next problem, we apply the previous method.
Problem 13. Consider the next nonlinear singular -D pseudoparabolic equation: By applying the previous technique, the first approximation is given by and the rest of the terms are given by where nonlinear terms and are decomposed as The nonlinear terms and are denoted by
To demonstrate this technique for a nonlinear problem, we examine the following example.
Example 14. Consider the following nonlinear pseudoparabolic equation: By using the mentioned 3-DLADM and Theorem 8, we have where and are defined in Equations (65) and (69). The subsequent terms are introduced by Following in a similar manner, we have Hence, according to Equation (43), we have if we set and Equations (13) and (14), then the exact solutions of Equation (70) are presented by
Figures 2(a) and 2(b) show the approximate and exact solutions of Equation (70); at and , we get the exact solution of Equation (70). For the different values of such as , , and , we obtain the approximate solution. Figures 2(c) and 2(d) represent the surface of the function .
5. Singular 2-D Coupled Pseudoparabolic Equation and 3-DLADM
The aim of this part is devoted to establishing the solution of the coupled singular -D pseudoparabolic equation by applying 3-DLADM.
The coupled singular 2-D pseudoparabolic equation of fractional order is given by where , , and are given functions and is the coupling parameter. One can obtain the solution of Equation (79) by using 3-DLADM. This method contains the following steps. (1)Multiplying both sides of Equation (79) by leads to the following equation:(2)We apply 3-DLT and both sides of Equation (81) and 2-DLT for Equation (80); we get on using Theorem 8 and 2-DLT for condition, we obtain (3)By integrating Equation (83) from to and to with respect to and , respectively, we have
Finally, by using the inverse of 3-DLT, we can evaluate and as follows:
The 3-DLADM decomposes the unknown functions and by the infinite series of components as
By substituting Equation (88) into Equations (86) and (87), we get where
Our method recommends that the zeroth components and are determined by the initial conditions and nonhomogeneous parts:
The remainder terms are given by
In order to check the applicability of our method for solving the fractional coupled pseudoparabolic equation, the next example has been considered.
Example 15. Time fractional coupled pseudoparabolic equations are given by where with initial condition
Using the 3-DLADM procedure Equations (94), (95), and (96), we obtain the following components: at by the same way, at we have at , at , thus for remaining elements. Using Equation (88), therefore, the approximate solutions are determined by
We set , the fractional solution becomes
Hence, the exact solution becomes
Figures 3(a)–3(d) show the approximate and exact solutions of Equation (98); put and , we have the exact solution of Equation (98); by using different values of such as , , and , we get the approximate solution. The approximate solutions of the functions and are given by Figures 3(e)–3(h).
Conclusion 16. In the current study, a powerful method called the 3-DLADM was approved for finding approximate and exact solution of the time-fractional singular 2-D pseudoparabolic equation. The suggested method is easier in its precept and active in solving linear and nonlinear singular two-dimensional pseudoparabolic equation. Therefore, we conclude that the 3-DLADM is very effective and more precise for any fractional order partial differential equations.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The authors read and approved the final manuscript.
Acknowledgments
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research group No. (RG-1440-030).