On Multi-Laplace Transform for Solving Nonlinear Partial Differential Equations with Mixed Derivatives

A novel approach is proposed to deal with a class of nonlinear partial equations including integer and noninteger order derivative. This class of equations cannot be handled with any other commonly used analytical technique. The proposed method is based on the multi-Laplace transform. We solved as an example some complicated equations. Three illustrative examples are presented to confirm the applicability of the proposed method. We have presented in detail the stability, the convergence and the uniqueness analysis of some examples.


Introduction
In order to investigate the present and future behaviors of a physical problem, many scholars always convert the observed fact into mathematical formula.In the recent decade it was observed that many physical problems were described with partial differential equation with either integer order derivative or fractional order derivative.Some of these equations can be classified as ordinary (linear or nonlinear) partial (linear or nonlinear) differential equations.The partial differential equations, both fractional and integer-order, have been documented as an overriding modeling technique particularly in the last few decades [1][2][3][4][5].To accurately replicate the nonlocal, frequency-, and history-dependent properties of power law phenomena, some different modeling tools based on fractional operators have to be introduced.
However, as soon as this conversion is done, the next challenge is to find the solutions to these equations.Many scholars have developed method to show the existence and the uniqueness of the solution of these equations [6][7][8].But when we are dealing with real world fact, one needs to find approximate, numerical, or exact solution of these equations in order to predict and analyze the solution as function of time and space and this renders the existence concept useless [9][10][11].It is perhaps important to recall that finding the exact solution implies the existence of a solution and this is more convenient than just to provide a proof of the existence without presenting the solution.In order to be more practical, several methods have been proposed to find solutions to these equations.In the case of linear equations, some techniques using integral transform such as Mellin transform, Laplace transform, the Fourier transform, and the Sumudu transform, as well as other recent techniques, were proposed.In the case of nonlinear partial differential equations, asymptotic methods are dealing with equations with small parameters, perturbation methods are dealing with multilayers problems, and for some strong linearity, iterations methods such as homotopy perturbation, Adomian's decomposition, homotopy decomposition, variational iteration, and many others have been documented and proven efficient with limitations [12][13][14][15][16][17].In 2011, Khan showed that it was possible to make use of the Laplace transform to actually derive solution of nonlinear equations.In his method, he coupled the Laplace series with the Poincare series [16].Others adapted this method using the Sumudu transform and the idea of the Lagrange multiplier.
The question that remains is what will happen if a nonlinear partial differential is made up with mixed derivative only?Can these commonly used methods be suitable in finding the approximate or exact solution?The answer is perhaps no.One of the purposes of this paper is to present a novel or extended method that will be used to handle this class of partial derivative.The method makes use of the double Laplace transform and the Poincare series.Without loss of generality, the general form of this class of equation is given as follows: where , , . . .,  are integer and noninteger numbers,  and  are linear and nonlinear operators with only mixed derivative, respectively, and  is a known function.However, to be more practical we will only consider the dimension to be two or three.The rest of the paper is presented as follows.The methodology of the technique is presented in Section 2. Application of the method with some examples is presented in Section 3. The stability, convergence, and uniqueness analysis will be presented in Section 4, and finally a conclusion is reached in Section 5.

Methodology
We devote this section to the discussion underpinning the methodology of the technique for solving (1).Nevertheless, before we present this methodology we will first present some properties of Laplace transform for integer and noninteger order.
Definition 1.The Laplace transform is a widely used integral transform with many applications in physics and engineering.The Laplace transform of the function  is defined as follows: 2.1.Properties of Fractional Calculus.Two properties of the Laplace transform can be used to define the fractional integral operator as follows [18,19]: Now using the recursive method, we arrive at the following: From the above equation we can obtain It is well known from the convolution theorem of Laplace transform that Now from the above formula if we chose () =  −1 , with the information of (4), the fractional integral operator can be defined as follows: Let us observe that Laplace transform of the fractional derivative with both Riemann-Liouville and Caputo is as follows: Caputo use the usual initial conditions or values of the functions.On the other hand, we have the Riemann-Liouville; that is, The above make use of the unusual initial value or conditions of the function; therefore, it is not suitable for the real world problems [19].
With the above references we will present the methodology of the technique.The first step in this technique is to apply on both sides of (1) the multi-Laplace transform to obtain At this stage two iteration formulae can be developed, the first one using the idea of Lagrange multiplier and the second one using the homotopy idea.If we use the idea of Lagrange multiplier, we have that the Lagrange multiplier in Laplace space is given as follows: With the above Lagrange multiplier in hand, we proposed the general integration to be in the form of  0 (, , . . ., ) =  (, , . . ., ) ,  +1 (, , . . ., ) =   (, , . . ., ) and the approximate or special solution can be obtained as If we use the idea of homotopy, we will assume that the solution can be in the form of series as follows: With replacing this expression in (11) and after comparing terms of the same power of , as well as using the polynomial proposed in [15], we obtain with of course H  [(, , . . ., )] the polynomial proposed in [15].We will illustrate this method with some examples and this is done in the next section

Application
We present in this section the application of this extension by solving some nonlinear and linear partial differential equations with mixed derivative only.
Example 2. To illustrate these methods, let us consider the following simple linear equation: Making use of methodology 2 presented is Section 2, we have and the general iteration formula for this is given by  0 (, ) =  (, 0) +  (0, ) −  (0, 0) , Therefore, using the iteration formula, we obtain And then, the summation of the first 11 terms is given as Realize that if  is very large, then the solution of this equation is This is the exact solution of our equation.We will examine the solution for the fractional version.
Example 4. Let us consider the following nonlinear partial differential equation : The above equation is very complicated due to the strong nonlinearity; we will therefore present a special solution to it by applying methodology two presented in Section 2.
We will present in the next section the analysis of convergence and uniqueness of the especial solution of (29) for using method 2.

Convergence and Uniqueness Analysis
The reason of this part is to demonstrate in detail the convergence and the uniqueness of the nonlinear equation while using the proposed iteration method; we will therefore consider the following equation: Think about the Hilbert space H =  2 ((, ) × [0, ]) defined as Then the operator is of the form We can accordingly declare the consequential theorem for the sufficient condition for the convergence of (41).

Theorem 6.
Let us think about and think about the initial and boundary condition for (41); then, the proposed technique shows the way to a particular solution of (41).
We will present the proof of this theorem by just verifying Hypotheses 1 and 2.
Proof.Using the definition of our operator , we have the following: With the above reduction in hand, it is therefore possible for us to evaluate the following inner product: We will examine case after case and start with the high nonlinear part.Take for granted that , V are bounded; consequently, we can find a positive constant  such that (, ), (V, V) <  We will recall that We moreover have also that the Cauchy-Schwarz-Bunyakovsky inequality yields 2 .It follows by the use of Schwartz inequality that( 3  2 ( 3 − V 3 ) ,  − V) ≤       3  2 ( 3 − V 3 )      ‖ − V‖ .(43)However, we can find a positive constant 1 such that ‖ 2  2 ( 3 − V 3 )‖ ≤  1 ‖ 3  2 ( 3 − V 3 )‖ it follows from equation (43) that ( 2  2 ( 3 − V 3 ) ,  − V) ≤  1  2  3       3 − V 3      ‖ − V‖ .(44)