Abstract
In this work, we combine conformable double Laplace transform and Adomian decomposition method and present a new approach for solving singular one-dimensional conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation. Furthermore, some examples are given to show the performance of the proposed method.
1. Introduction
Fractional partial differential equations have attracted much attention in applied sciences and engineering such as acoustics, control, and viscoelasticity. The parabolic equation appeared in different fields of applied mathematics, such as heat conduction and fluid mechanics (for instance, see [1–4]). The authors in [5, 6] studied the fractional diffusion equations problems by using the Adomian decomposition method and series expansion method. Many papers exist in the literature, which are related to conformable fractional derivative with its properties and applications [7, 8]. This new method was quickly generalized by Katugampola [9, 10]. The authors in [11] investigated existence and uniqueness theorems for sequential linear conformable fractional differential equations. The authors in [12] revisited the Grünwald Letnikov, Riemann–Liouville, and Caputo fractional derivatives and analysed under the light of the proposed criteria. The nonhomogeneous nonlocal theory has been presented based on conformable derivatives (CD) to study the critical point instability of micro/nanobeams under a distributed variable-pressure force (see [13]). The authors in [14] proposed a new fractional nonlocal model and its application in free vibration of Timoshenko and Euler–Bernoulli beams. Recently, several researchers applied the conformable Laplace transform method to solve different types of fractional differential equation (see [15, 16]). Many exact solutions in various wave forms for the nonlinear conformable time-fractional parabolic equation with exponential nonlinearity are formally constructed in [17]. The goal of this paper is to investigate the solution of singular conformable fractional pseudoparabolic equation and conformable coupled pseudoparabolic equation by conformable double Laplace transform decomposition methods (CDLTDMs). Moreover, we are able to prove some theorems related to this work.
1.1. Conformable Partial Derivatives
Definition 1. (see [18]). Given a function , the conformable space fractional partial derivative of order of the function is denoted by
Definition 2. (see [18]). Given a function , the conformable time partial derivative of order of the function is defined aswhere and are called the fractional derivatives of order and , respectively.
In Theorem 1, the connection between the conformable derivatives and the first derivative can be represented as follows.
Theorem 1. Let and be differentiable at a point .
Then,
Proof. By using definitions 1 and 2 and in equation (1), we haveSimilarly, we prove equation (2).
In the next example, we introduce the conformable derivative of specific functions, by using Theorem 1 as follows.
Example 1. Let and then(1)(2), (3) (4) (5) (6)
2. Some Properties of the Conformable Laplace Transform
Here, we work with the single conformable Laplace transform and conformable double Laplace transform (CDLT) which are defined, respectively as follows.
Definition 3. (see [7, 19, 20]). Let and . Then, the fractional Laplace transform of order is defined by
Definition 4. (see [21]). Let be a piecewise continuous function on the interval of exponential order. Consider for some sup , in these conditions (CDLT) is defined bywhere , and the integrals are by means of conformable integral with respect to t and x, respectively.
Theorem 2. If thenwhere is the Heaviside unit step function defined by when and and when and .
Proof. By applying the definition of double conformable Laplace transform,which is, by putting , we haveIn the next example, we reported that some conformable Laplace transforms of definite functions are important in this study.
Example 2. (1), where and are positive integers(2)(3)
Theorem 3. Let be piecewise continuous on ; the (CDLT) of the conformable partial derivatives of orders -th and -th, , and is given by
Proof. By using definition (CDLT) for , we haveBy applying Theorem 1, equation (13) becomesThe integral inside bracket given byBy substituting equation (15) into equation (14), we obtainIn the same manner, the (CDLT) of , and can be obtained.
Double Laplace transform of the function and are studied in the next theorem.
Theorem 4. If the (CDLT) of the conformable partial derivatives is given by equation (11), then double Laplace transform of and are given bywhere .
Proof. By applying the nth derivative with respect to p for both sides of equation (6), we get equation (17) as follows:We obtainSimilarly, we can prove equation (18).
3. Singular One-Dimensional Conformable Fractional Pseudoparabolic Equation
The conformable double Laplace decomposition methods (CDLTDMs) are an efficient technique which is used to obtain the solution linear and nonlinear singular pseudoparabolic equation.
Problem. We consider and as singular one-dimensional pseudoparabolic equations with initial conditions in the formsubject towhere, the term, is called conformable Bessel’s operator and and are known functions. In order to solve equation (21), we apply the following steps: Step 1: multiplying equation (21) by : Step 2: using Lemma 1 and equation (18) for equations in step 1 and single conformable Laplace transform for equation (22), we obtain where the symbol indicates (CDLT) with respect to . Step 3: applying the integral for both sides of equation (24), from to with respect to , we have Step 4: next, the (CDLTDM) consists of representing the solution of the singular pseudoparabolic equation as by the infinite series Step 5: working with the double Laplace transform on both sides of equation (25) and using equation (26), we receiveWe define the following recursive formula:The rest of the terms can be written as follows:where indicates double inverse Laplace transform with respect to and .
Here, we assume that double inverse Laplace transform with respect to p and s exists for each terms in equations (28) and (29). To confirm our method, we solve the next example.
Example 3. Consider the following nonhomogeneous form of a singular one-dimensional pseudoparabolic equation:with the conditionBy applying the above steps and Theorem 1, we obtainBased on the (CDLTDM), we obtainIn a similar manner, we obtain thatBy adding all the terms, we getThus, the exact solution is obtained as follows:By taking and , the fractional solution becomes
Problem. Consider the following nonlinear singular one-dimensional pseudoparabolic equation:subject toUsing our method, we getThe rest of the terms are given bywhere and are the so-called Adomian polynomials, given byThe nonlinear terms and are represented asTo illustrate this method for nonlinear problem, we consider the following example.
Example 4. Consider the following nonlinear pseudoparabolic equation:subject toBy applying the aforesaid conformable double Laplace decomposition method and Theorem 1, we haveProceeding in a similar manner, we haveSo according to equation (26), we havewhich is the exact solution of equation (44).
4. Conformable Double Laplace Transform Method and Singular Conformable Coupled Pseudoparabolic Equation
The purpose of this part is to examine the use of the (CDLTDM) for the linear one-dimensional conformable coupled pseudoparabolic equation. We consider the following conformable coupled pseudoparabolic equations:with conditionswhere , and are the known functions and is the coupling parameter. One can get the solution of equation (49), by using (CDLTDM); this method consists of the following steps:(1)Multiply both sides of equation (49) by leading to the following equation:(2)Applying (CDLT) on both sides of equation (51) and single conformable Laplace transform for equation (50), we get On using Theorem 1 and Theorem 2, we obtain(3)By integrating both sides of equation (53) from to with respect to , we have By applying double inverse Laplace transform for equation (54), we have The (CDLTDM) defines the solutions of conformable coupled pseudoparabolic equations as and by the infinite series: By substituting equation (57) into equations (55) and (56), we get(4)Working with the double Laplace transform on both sides of equation (25) and using equation (26), we receive
This technique suggests that the zeroth components and are identified by the initial conditions and from source terms as follows:
The rest of the terms are given by
In order to ensure the four techniques for solving the conformable fractional coupled pseudoparabolic equation, we will consider the following example.
Example 5. Consider the following homogeneous form of conformable coupled pseudoparabolic equations:wherewith conditionsBy applying the above method and Theorem 1 for equation (62) and (64) and using equations (59), (60), and (61), we obtainand similarly for the rest components. Using equation (57), the series solutions are therefore given byHence,The exact solution is obtained by taking and , as follows:
5. Numerical Result
In this section, we discuss the precision and efficiency of the (CDLTDM) by numerical results of for the exact solution when and approximate solutions at and taking different fractional values for conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation. The solutions of equation (30) are depicted in Figures 1 and 2, respectively.
In Figure 1, the approximate solutions of equation (30) at and , taking different fractional values, are compared and we found that the numerical solution becomes close to the exact solution when the fractional value increases:
Figure 2 indicates that the exact solution at of equation (30) and the approximate solution of equation (30) decrease at the fractional derivative values and . Similarly, the exact solution and approximate solution of equation (44) are demonstrated in Figures 3 and 4. Figure 3 give the plots of the behaviour of equation (44) when and with different fractional values taken in this case; the solution becomes close to the exact solution at close to one.
Figure 4 shows the approximate solution of equation (44) with , , and ; in such a case, the function gradually decreases. Finally, Figure 5 suggests that in the solutions of equation (62) at and , we find that the numerical solution becomes close to the exact solution when the fractional value increases.
Figure 6 demonstrates that the exact solution at of equation (62) and the approximate solution of equation (62) are concave upward at the fractional derivative increasing when and fixed.
6. Conclusion
In this work, singular one-dimensional conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation have been considered. Then, new conformable double Laplace transform decomposition methods have been applied to the problems. Finally, we gave three differential examples to show that this method is applicable and valid. The suggested method can also be applied for systems with more than two linear and nonlinear partial differential equations. In addition, if we let and in Examples 3 and 4, we get the solution which is considered in [22]. All figure results are obtained by using Matlab.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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 (RG-1440-030).