A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method.


Introduction
In recent years, many mathematicians have been studying and discussing the linear and nonlinear fractional differential equations (FDEs) which arise in various fields of physical sciences, as well as in engineering. These types of equations play a significant role and also help to develop mathematical tools in order to understand fractional modelling.
However, there are many different methods to obtain exact and approximate solutions of these kinds of equations. In [1], the author point out a major flaw in the so-called conformable calculus. Recently, many researchers have also paid much attention to study the numerical and exact methods for finding the solution of conformable differential equations. In [2], the authors proposed so-called conformable derivatives. In [3], the conformable heat equation was studied. Similarly, in [4], the nonlinear conformable problems were also studied. The authors in [5] discussed the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. In a very short period of time, many mathematicians became interested and provided mathematical models related to conformable derivatives, for the details we refer reader to see [6][7][8][9]. In [10,11], the conformable derivatives were applied to some problems in mechanics, and in [12] total frational derivative and directional fractional derivative of functions of several variables were studied.
In order to solve the conformable derivatives, the single Laplace transform method was first introduced and used in [13]. In [14], the idea was extended to the conformable double Laplace transform. In [15], the modified Laplace transform was applied to solve some ordinary differential equations in the frame work of a certain type generalized fractional derivatives. The authors in [16] applied the double Laplace decomposition method to solve singular linear and nonlinear one-dimensional pseudohyperbolic equations.
In this present research, the main objective is to solve linear and nonlinear singular pseudohyperbolic equations by using the conformable double Laplace transform decomposition method, which is a combination of the conformable double Laplace transformation and decomposition method.

Properties of Conformable Derivative and Conformable Double Laplace Transform
In this part, we present some background about the nature of the conformable Laplace transform. In the following example, we present the conformable partial derivatives of certain functions as follows.
Example 1. Let µ, ν ∈ (0, 1] and a, b, m, n, λ, µ ∈ R, then the conformable derivative follows Next we recall the conformable single and double Laplace transforms, see [14,17], repectively. Definition 1. Let f : [0, ∞) → R be a real valued function. The conformable single Laplace transform of f is defined by Similarly, if we let u x µ µ , t ν ν be a piecewise continuous function on [0, ∞) × [0, ∞) of exponential order and for some a, b ∈ R, Then the conformable double Laplace transform is defined by where p, s ∈ C, 0 < µ, ν ≤ 1 and integrals by means of conformable integrals with respect to x and t, respectively.
Further, the first and second order partial derivatives of the conformable double Laplace transform with respect to x µ µ are given by Similarly, with respect to t ν ν they are given by In the following examples we state some conformable Laplace transforms of certain functions which are useful in this to Examples 3, 4, and 5.

Example 2.
In this example we calculate the conformable double Laplace for certain functions The next result generalizes the conformable double Laplace transform, see [14]. Theorem 1. Let 0 < µ, ν ≤ 1 and m, n ∈ N such that u x µ µ , t ν ν ∈ C l (R + × R + ), and l = max (m, n) . Further, we also let the conformable Laplace transforms of the functions be denoted by and where n = 1, 2, 3, ....

Proof.
Using the definition of conformable double Laplace transform for Equation (6), we get by taking the n th derivative with respect to p for both sides of Equation (8), we have and further we obtain Similarly, we can prove the Equation (7).

Conformable Derivatives Double Laplace Transform Decomposition Method Applied to Singular Pseudohyperbolic Equation
The main aim of this section is to discuss the applicability of the conformable double Laplace transform decomposition method (CDLDM) for the linear and nonlinear singular pseudohyperbolic equation. The pseudo-hyperbolic equations arise, for example, in the description of the electron diffusion processes in a plate, and they also arise in hydrodynamics in the study of fluid motion with an alternating viscosity. In this study we define the conformable double Laplace transform of the function u x µ µ , t ν ν by U µ,ν (p, s). To illustrate the idea of our method, let us suggest here two important problems.

The first problem:
Consider the linear pseudohyperbolic equations subject to the condition where are source term and initial conditions, respectively.

The method:
In order to obtain the solution of Equation (9) by using conformable double Laplace transform decomposition methods, we applying the following steps: Step 1: Multiplying both sides of Equation (9) by the term x µ µ , we have Step 2: Applying conformable double Laplace transform for Equation (11) we get Step 3: On using Equations (5)- (7), we obtain − d dp where the conformable Laplace transforms of u x µ µ , 0 and respectively, and using given initial condition Equation (13) becomes Step 4: By applying the integral for both sides of Equation (14), from 0 to p with respect to p, where p is transform of the variable x µ µ , we have where F µ,ν (p, s) , F 1 (p, 0), and F 2 (p, 0) are conformable Laplace transforms of the functions f x µ µ , t ν ν , f 1 x µ µ , and f 2 x µ µ , respectively.
Step 5: By taking the inverse conformable double Laplace transform for Equation (15), we can compute the solution u x µ µ , t ν ν as follows where L −1 p L −1 s indicates the double inverse conformable derivatives double Laplace transform. Here, we assume that the double inverse Laplace transform with respect to p and s exists for each term in the right hand side of Equation (16).
Step 6: The conformable double Laplace transform decomposition method (CDLDM) defines the solutions u( x µ µ , t ν ν ) with the help of infinite series as: By substituting Equation (17) into Equation (16), we obtain The zeroth component u 0 , as suggested by Adomian method, is always identified by the given initial condition and the source term L −1 p L −1 s 1 s 2 F µ,ν (p, s) , both of which are assumed to be known. Accordingly, we set The other components u k+1 , k ≥ 0 are given by using the relation the first few components from the last recursive relation are, at k = 0, etc. The important terms used in infinite series depend on the problems and may be three terms or four terms, etc.
In order to give a clear overview of this method, we present the following example: Example 3. Consider singular conformable derivatives in one dimensional pseudohyperbolic equations with the indicated initial condition and By using the aforesaid method subject to the initial condition, we have taking the integral for Equation (22), from 0 to p with respect to p, we get Employing the inverse conformable derivatives double Laplace transform to Equation (23), we get by substituting Equation (17) into Equation (24), we obtain: By applying the conformable double Laplace transform decomposition method, we obtain eventually, we have the general recursive relation, given by where k ≥ 0, therefore by using partial fractional and inverse Laplace transform with respect to s, we have In the view of the above equations, the series solution is given by

Hence, the exact solution of Equation (20) is given by
Second problem: Consider the following general form of the nonlinear singular pseudohyperbolic equations in one dimension of the form: where the functions a( x µ µ ) are arbitrary. In order to obtain the solution of Equation (25), we use the following steps: First step: By multiplying Equation (25) by x µ µ and taking conformable double Laplace transform, we have where conformable Laplace transform of u x µ µ , 0 and ∂ ν u x µ µ ,0 ∂t ν are given by Second step: Applying Equations (5)-(28) into Equation (27), one can get that where, Third step: By taking the integral for Equation (29), from 0 to p with respect to p, where p is a transform of x µ µ , we have Fourth step: Using CFDLDM, the solution can be written in the infinite series as in Equation (17). By using the inverse Laplace transformation to Equation (30), we obtain.
furthermore, the nonlinear terms u ∂ µ u ∂x µ and u 2 can be defined by: We have a few terms of the Adomian polynomials for A n and B n that are denoted by and where n = 0, 1, 2, .... By putting Equations (33)-(32) into Equation (31), we get the few components of the Adomian polynomials of Equations (33) and (34) are given as follows and Hence, the zeroth component u 0 from Equation (35) is given by and where k ≥ 0.

Example 4. Consider the nonlinear singular pseudohyperbolic equation in one dimensional is governed by
subject to the following initial conditions The conformable double Laplace transform decomposition method leads to the following scheme proceeding in a similar manner, we have so that the solution u x µ µ , t ν ν is given by and hence the conformable solution is given by By substituting µ = 1 and ν = 1 into Equation (42), the solution becomes Conformable double Laplace transform method and Singular conformable coupled pseudohyperbolic equation.
In this section, conformable double Laplace decomposition method is considered for the one-dimensional conformable derivatives coupled pseudohyperbolic equation since the method is much simpler and more efficient in the study of linear equations.
The thrid problem: Let us consider the conformable derivatives coupled pseudohyperbolic equations (2): We apply conformable double Laplace transform on both sides of Equation (45) and single conformable Laplace transform for Equation (44), we get on using theorem 1 and theorem 2, we obtain d dp U µ,ν (p, s) = 1 s d dp (3): By integrating both sides of Equation (47) from 0 to p with respect to p, we have where F 1 (p, 0) , F 2 (p, 0) , G 1 (p, 0) , and G 2 (p, 0) are conformable Laplace transform of the functions f 1 x µ µ , and g 2 x µ µ , respectively. By applying double inverse Laplace transform for Equation (48), we have and The conformable double Laplace decomposition methods represent the solutions of Equation (43), by the infinite series By substituting Equation (51) into Equations (49) and (50), we get and Our method suggests that the zeroth components u 0 and v 0 are identified by the initial conditions and from source terms as follows The remaining terms are given by and Here, we assume that the double inverse Laplace transform with respect to p and s exists for each term in the right hand side of the above equations.
To illustrate our method for solving the conformable derivatives coupled pseudohyperbolic equations, we will consider the following example: By taking µ = 1 and ν = 1, the conformable solution becomes

Numerical Result
In this section, we shall illustrate the accuracy and effciency of the double conformable Laplace transform method by numerical results of u(x, t) for the exact solution when µ = ν = 1, and approximate solutions when µ and ν taken different fractional values in Equations (20) and (40), which are depicted through Figures 1-4, respectively.
The three dimensional surface in Figure 1 shows the exact solution of Equation (20) in standard form of singular pseudohyperbolic equation at µ = ν = 1. Figure 2 compares the approximate solutions of Equation (20) when t = π 2 . In Figure 2a, the numerical solution at 0 < µ = ν ≤ 1, in this case u(x, t), increases hastily at fractional derivative decrease, Figure 2b shows the solution at µ = 0.99 and ν = 0.95, 0.90, 0.85 and we see u(x, t) increasing regularly when ν decreases, and in Figure 2c   Similarly, the exact solution and approximate solution of Equation (40) were demonstrated in Figures 3 and 4 when t = 1. In the case ν = µ = 1, we get the exact solution of a singular pseudohyperbolic equation, as seen in Figure 3. Figure 4 shows the approximate solution of Equation (40) with different values of µ and ν. Figure 4a gives plots of the behavior of Equation (40) when 0 < µ = ν ≤ 1, in this case the function u(x, t) increases quickly, and in Figure 4b we have obtained the solution for the values of µ = 0.99 and different values of 0 < ν ≤ 1, in this case the function u(x, t) increases gradually, and Figure 4c gives the behavior of Equation (40) at ν = 0.99 and different values of µ, in this case the function u(x, t) increasing tardily.
It is clear from the solutions of Equations (20) and (40) that the conformable double Laplace decomposition method has good agreement with the exact solutions of the problems. The fractional-order solution of these two problems and exact solution of integer order problems are equal at 0 < µ = ν ≤ 1, in this case we have no error.

Conclusions
In the present work we have studied singular linear and nonlinear pseudohyperbolic equations by employing the conformable double Laplace transform decomposition method (CDLDM), and we obtain analytic solutions when µ = ν = 1 and numerical solutions for different fractional values. Further, we also studied singular coupled pseudohyperbolic equations. It is clear that the solutions of Equations (20) and (40) were obtained as infinite series by using the conformable double Laplace decomposition method and they are in good agreement with the exact solutions of the problems. We have provided three different examples in order to demonstrate the efficiency, high accuracy, and the simplicity of the present method. Further, we plot the exact solutions, as well as the numerical solutions, in Figures 1-4, and we can easily see the efficieny of and agreement among the solutions.