A Note on Conformable Double Laplace Transform and Singular Conformable Pseudoparabolic Equations

Fractional partial differential equations have attracted much attention in applied sciences and engineering such as acoustics, control, and viscoelasticity. 'e parabolic equation appeared in different fields of applied mathematics, such as heat conduction and fluid mechanics (for instance, see [1–4]). 'e 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]. 'is new method was quickly generalized by Katugampola [9, 10]. 'e authors in [11] investigated existence and uniqueness theorems for sequential linear conformable fractional differential equations.'e authors in [12] revisited the Grünwald Letnikov, Riemann–Liouville, and Caputo fractional derivatives and analysed under the light of the proposed criteria. 'e 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]). 'e 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 transformmethod 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]. 'e 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.


Introduction
Fractional partial differential equations have attracted much attention in applied sciences and engineering such as acoustics, control, and viscoelasticity. e parabolic equation appeared in different fields of applied mathematics, such as heat conduction and fluid mechanics (for instance, see [1][2][3][4]). e 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]. is new method was quickly generalized by Katugampola [9,10]. e authors in [11] investigated existence and uniqueness theorems for sequential linear conformable fractional differential equations. e authors in [12] revisited the Grünwald Letnikov, Riemann-Liouville, and Caputo fractional derivatives and analysed under the light of the proposed criteria. e 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]). e 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]. e 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.
Similarly, we prove equation (2). In the next example, we introduce the conformable derivative of specific functions, by using eorem 1 as follows.

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.
where H(x, t) is the Heaviside unit step function defined by Proof. By applying the definition of double conformable Laplace transform, In the next example, we reported that some conformable Laplace transforms of definite functions are important in this study.
Proof. By using definition (CDLT) for z α u/zx α , we have By applying e integral inside bracket given by By substituting equation (15) into equation (14), we obtain In the same manner, the (CDLT) of z β u/zt β , z 2α u/zx 2α , and z 2β u/zt 2β can be obtained.
Proof. By applying the nth derivative with respect to p for both sides of equation (6), we get equation (17) as follows: We obtain Similarly, we can prove equation (18). Problem. We consider 0 < α ≤ 1 and 0 < β ≤ 1 as singular one-dimensional pseudoparabolic equations with initial conditions in the form subject to where, the term, Journal of Function Spaces known functions. In order to solve equation (21), we apply the following steps: Step 1: multiplying equation (21) by x α /α: Step 2: using Lemma 1 and equation (18) for equations in step 1 and single conformable Laplace transform for equation (22), we obtain d dp where the symbol L α x L β t indicates (CDLT) with respect to x and t.
Step 3: applying the integral for both sides of equation (24), from 0 to p with respect to p, we have Step 4: next, the (CDLTDM) consists of representing the solution of the singular pseudoparabolic equation as u(x, t) by the infinite series Step 5: working with the double Laplace transform on both sides of equation (25) and using equation (26), we receive We define the following recursive formula: e rest of the terms can be written as follows: where L − 1 p L − 1 s indicates double inverse Laplace transform with respect to p and s.
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 condition By applying the above steps and eorem 1, we obtain Based on the (CDLTDM), we obtain In a similar manner, we obtain that By adding all the terms, we get us, the exact solution is obtained as follows: By taking α � 1 and β � 1, the fractional solution becomes Problem. Consider the following nonlinear singular onedimensional pseudoparabolic equation: subject to Using our method, we get e rest of the terms are given by where A n and B n are the so-called Adomian polynomials, given by (42) e nonlinear terms uu x and (u x ) 2 are represented as (43)

Journal of Function Spaces
To illustrate this method for nonlinear problem, we consider the following example.
Example 4. Consider the following nonlinear pseudoparabolic equation: subject to By applying the aforesaid conformable double Laplace decomposition method and eorem 1, we have Proceeding in a similar manner, we have So according to equation (26), we have which is the exact solution of equation (44).

Conformable Double Laplace Transform Method and Singular Conformable Coupled Pseudoparabolic Equation
e 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 conditions where f(x, t), g(x, t), f 1 (x), and g 1 (x) are the known functions and ζ 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 x α /α 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 eorem 1 and eorem 2, we obtain 6 Journal of Function Spaces d dp (3) By integrating both sides of equation (53) Figure 3 give the plots of the behaviour of equation (44) when (t � 1) and (α � β) with different fractional values taken in this case; the solution u(x, t) becomes close to the exact solution at (α � β) close to one. Figure 4 shows the approximate solution of equation (44) with (0 < α ≤ 1), (β � 0.99), and (t � 1); in such a case, the function u(x, t) gradually decreases. Finally, Figure 5 suggests that in the solutions of equation (62) at (t � 1) and (0 < α � β ≤ 1), 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 (α � β � 1) of equation (62) and the approximate solution of

Conclusion
In this work, singular one-dimensional conformable pseudoparabolic equation and conformable coupled pseudoparabolic equation have been considered. en, 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. e suggested method can also be applied for systems with more than two linear and nonlinear partial differential equations. In addition, if we let α � 1 and β � 1 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
e authors declare that they have no conflicts of interest.