A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential Equation

and Applied Analysis 3 which boil down to f (x, y, t)


Introduction
The topic of partial differential equations is one of the most important subjects in mathematics and other sciences.The behaviour of the solution very much depends essentially on the classification of PDEs therefore the problem of classification for partial differential equations is very natural and well known since the classification governs the sufficient number and the type of the conditions in order to determine whether the problem is well posed and has a unique solution.The Laplace transform has been intensively used to solve nonlinear and linear equations [1][2][3][4][5][6][7].The Laplace transform is used frequently in engineering and physics; the output of a linear time invariant system can be calculated by convolving its unit impulse response with the input signal.Performing this calculation in Laplace space turns the convolution into a multiplication; the latter is easier to solve because of its algebraic form.The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering [1][2][3][4][5][6][7].The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.The original differential equation can then be solved by applying the inverse Laplace transform.The English electrical engineer Oliver Heaviside first proposed a similar scheme, although without using the Laplace transform, and the resulting operational calculus is credited as the Heaviside calculus.Recently Kılıc ¸man et al. [8][9][10][11] extended the Laplace transform to the concept of double Laplace transform.This new operator has been intensively used to solve some kind of differential equation [11] and fractional differential equations.The aim of this work is to extend the Laplace transform to the triple Laplace transform.We will start with the definition of the triple Laplace transform.

Definitions and Theorems
Definition 1.Let  be a continuous function of three variables; then, the triple Laplace transform of (, , ) is defined by where, , ,  > 0 and , ,  are Laplace variables, and is the inverse triple Laplace transform.(3)

Uniqueness and Existence of the Triple Laplace Transform
In this section, we will study the uniqueness and existence of triple Laplace transform.First of all, let (, , ) be a continuous function on the interval [0,∞) which is of exponential order, that is, for some , ,  ∈ .Consider sup ,,>0 (, , ) Under the previous condition, the triple Laplace transform, exists for all  > ,  > , and  >  and is in actuality infinitely differentiable with respect to  > ,  >  and  > .
All functions in this study are assumed to be of exponential order.The following theorem shows that (, , ) can be uniquely obtained from (, , ).
for the triple inverse Laplace transform, can be used to obtain By hypothesis, we have that (, , ) = (, , ).then replacing this in the previous expression, we have the following: which boil down to and this proves the uniqueness of the triple Laplace transform.
Theorem 4. If, at the point (, , ), the integrals are convergent and in addition if is absolutely convergent, then, the following expression: (, , ) =  1 (, , )  2 (, , )  3 (, , ) is the Laplace transform of the function and the integral is convergent at the point (, , ); for the readers who are interested, they can see the proof in [11,12].
Theorem 5. A function (, , ) which is continuous on [0, ∞) and satisfies the growth condition (4) can be recovered from only (, , ) as Evidently, the main difficulty in using Theorem 5 for computing the inverse Laplace transform is the repeated symbolic differentiation of (, , ).
Let us see how Theorem 5 can be applicable.Let us consider the following functions: Naturally the triple Laplace transform of the previous function is given later as Now applying the high-order mixed derivative to the previous expression, we obtain the following: Applying Theorem 5 in the previous expression, we obtain the following result: Making a change of variable in the previous expression, we obtain the following simplified result: Using together, the application of logarithm and the L'Hôpital's rule on the previous expression, we arrive at the following result:

Some Properties of Triple Laplace Transform
In this section, we present some properties of the triple Laplace transform.Note that these properties follow from those of the double Laplace transform introduced by Kılıc ¸man and Eltayeb [8].The properties of the triple Laplace transform will enable us to find further transform pairs {(, , ), (, , )}: ( We will present the proof Note that the integral inside the bracket satisfies the properties of the double Laplace transform and is given as [11] (∬ ×  (, , )  ) =  (,  + ,  + ) .
and this completes the proof.
(ii) The following can also be observed: We will present the proof Note that the double integral inside the bracket satisfies the property of the double Laplace transform as [11] (∬ Thus and this completes the proof. (iii) The following property can also be observed: We will present the proof Then, Now making use of the convergence properties of the improper integral involved, we can interchange the operation of differentiation and integration and differentiate with respect to , , and  under the integral sign.Thus, we arrive at the following expression: Note that the expression in the bracket satisfies the property of the double Laplace transform as [11] And finally, we obtain and this completes the proof.Now using the previous three properties, we will show the proof of Theorem 5.
Proof of Theorem 5. Let us define the set of functions depending on parameters , , and V as It worth noting that the previous function is a kind of threedimensional density of probability, and it therefore follows that In addition of this, we will have that lim Here if one applies the third property, in particular by replacing  = /,  = /,  = V/ as follows: ++V [  ( (, , ))] Now let us put (, , ) =  −−− (, , ).Now if we make use of (38), we obtain the following Now taking into account properties (i) and (ii), (42) together with the function (, , ), we arrive at the following: Now observe that from (44) with the fact that (, , ) = (1, 1, 1) ++ , we arrive at the following: The previously mentioned is true for any , ,  in the complete space, in particular, for  = 0,  = 0,  = 0, and in this case Theorem 5 is covered.

Application to Third-Order Partial Differential Equation
In this section, we present the application of this operator for solving some kind of third-order partial differential equations.
Factorising the right side of equation ( 49), we obtain the following: Now applying the inverse triple Laplace transform on the previous equation we obtain the following solution: This is the exact solution for Mboctara equation.
subjected to the following initial and boundaries conditions: Now applying the triple Laplace transform on both sides of (52), we obtain the following: Factorising the right side of (54), we obtain the following: Now applying the inverse triple Laplace transform on the previous equation, we obtain the following solution This is the exact solution for nonhomogeneous Mboctara equation.(58) Factorising the right side of (59), we obtain the following: Abstract and Applied Analysis 9 Now applying the inverse triple Laplace transform on the previous equation, we obtain the following solution: This is the exact solution for nonhomogeneous Mboctara equation.

Triple Laplace Transform of Some Functions of Three Variables
In this section, we examine the triple Laplace transform of some functions in Table 1: (64)

Conclusion
This work presents the definition of the triple Laplace transform.Some triple Laplace transform is presented in Table 1.Some theorems and properties of this new relatively new operator are presented.Applications of the new operator, for solving some kind of third-order partial differential equations called Mboctara equation, are presented.Numerical solutions of the Mboctara equation are given.

Figure 1 :
Figure 1: Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations.

Table 1 :
Table of triple Laplace transform for some function of three variables.