MIXED OF ELZAKI TRANSFORM AND PROJECTED DIFFERENTIAL TRANSFORM METHOD FOR A NONLINEAR WAVE-LIKE EQUATIONS WITH VARIABLE COEFFICIENTS

In this work, a mixture of Elzaki transform and projected di¤erential transform method is applied to solve a nonlinear wave-like equations with variable coe¢ cients. Nonlinear terms can be easily manipulated by using the projected di¤erential transformation method. The method gives the results show that the proposed method is very e¢ cient, simple and can be applied to other applications. 1. Introduction The integral transforms play a signicant role in many elds of science and in the literature, its are largely used in mathematical physics, optics, mathematical engineering and in some others in order to solve the di¤erential equations such as Laplace, Fourier, Mellin, Hankel and Sumudu. Recently, Tarig Elzaki [5] introduced a new integral transform, called Elzaki transform, which is applied to solve an ordinary and partial di¤erential equations. By applying the Adomian decomposition method (ADM), M. Ghoreishi solved some types of nonlinear wave-like equation [6], V.G. Gupta and S. Gupta worked out by using homotopy perturbation transform method ( HPTM) these types of equation tool [7], furthermore, A. Aslanov [3], F. Yin and et al [12] and A. Atangana and et al [2] researched for solving nonlinear heat and wave-like equation by using homotopy perturbation, variational iteration and homotopy decomposition methods respectively. Moreover, various techniques, such as homotopy analysis, perturbations, decompositions, iterations, di¤erential and Laplace transformation techniques have been used to handle similar types of these wave-like and also heat-like problems numerically and analytically as in references [1],[7],[10],[11]. In this work, we will present the mixed of Elzaki transform and projected di¤erential transform method, in order to solve a nonlinear wave-like equations. This method called Elzaki projected di¤erential transform method (EPDTM). 2010 Mathematics Subject Classication. Primary 35L05, 35L10; Secondary 35A22, 35A35.


Introduction
The integral transforms play a signi…cant role in many …elds of science and in the literature, it's are largely used in mathematical physics, optics, mathematical engineering and in some others in order to solve the di¤erential equations such as Laplace, Fourier, Mellin, Hankel and Sumudu.
Recently, Tarig Elzaki [5] introduced a new integral transform, called Elzaki transform, which is applied to solve an ordinary and partial di¤erential equations.
By applying the Adomian decomposition method (ADM), M. Ghoreishi solved some types of nonlinear wave-like equation [6], V.G.Gupta and S. Gupta worked out by using homotopy perturbation transform method ( HPTM) these types of equation tool [7], furthermore, A. Aslanov [3], F. Yin and et al [12] and A. Atangana and et al [2] researched for solving nonlinear heat and wave-like equation by using homotopy perturbation, variational iteration and homotopy decomposition methods respectively.Moreover, various techniques, such as homotopy analysis, perturbations, decompositions, iterations, di¤erential and Laplace transformation techniques have been used to handle similar types of these wave-like and also heat-like problems numerically and analytically as in references [1], [7], [10], [11].
In this work, we will present the mixed of Elzaki transform and projected di¤erential transform method, in order to solve a nonlinear wave-like equations.This method called Elzaki projected di¤erential transform method (EPDTM).
These types of equations are of considerable signi…cance in various …elds of applied sciences, mathematical physics, nonlinear hydrodynamics, engineering physics, biophysics, human movement sciences, astrophysics and plasma physics.These equations describe the evolution of erratic motions of small particles that are immersed in ‡uids, ‡uctuations of the intensity of laser light, velocity distributions of ‡uid particles in turbulent ‡ows.

Elzaki transform
De…nition 2.1.A new integral transform called Elzaki transform [4], [5] de…ned for functions of exponential order, is proclaimed.We consider functions in the set A de…ned by, where v is the factor of variable t.
T. M. Elzaki and S. M. Elzaki in [4], showed the modi…ed of Sumudu transform [9] or Elzaki transform was applied to partial di¤erential equations, ordinary di¤erential equations, system of ordinary and partial di¤erential equations and integral equations.
Proposition 2.2.To obtain Elzaki transform of partial derivative we use integration by parts and then we have Proof.
We assume that f is piecewise continuous and it is of exponential order.Now using the Leibniz rule to …nd By the same method we …nd and Let @f @t = g; then we have We can easily extend this result to the n th partial derivative by using mathematical induction, we have The proof is complete.
Properties of Elzaki transform can be found in [4], [5] we mention only the following

Projected di¤erential transform method
In this section, we introduce the basic idea of modi…ed version of the di¤erential transform method (DTM), the projected di¤erential transform method (PDTM) [8].The DTM is based on the Taylor series for all variables.Here, we consider the Talyor series of the function u with respect to the speci…c variable.Assume that the speci…c variable is the variable t.
De…nition 3.1.The projected di¤ erential transform U (X; k) of u (X; t) with respect to the variable t at t 0 is de…ned by where X = (x 1 ; x 2 ; :::; x n ); u (X; t) is the original function and U (X; k) is the transformed function of u(X; t).
De…nition 3.2.The projected di¤ erential inverse transform of U (X; k) with respect to the variable t at t 0 is de…ned by Combining Eqs.(3.1) and (3.2), we have the Taylor series expansion of the function u at t = t 0 as follows From the above de…nitions, the fundamental operations of the PDTM are given by the following theorems Theorem 3.3.Let U (X; k); W (X; k) and Z(X; k) be the projected di¤ erential transform of the functions u(X; t); w(X; t) and z(X; t) respectively, where X = (x 1 ; x 2 ; ::: where and are constants.then ::: f1; 2; :::g : f1; 2; :::; ng ; n 2 f1; 2; :::g : x a 1 1 x a 2 2 :::x an n ; k m = a m 0; otherwise :
Then by EPDTM we have the solution of Eqs.(4.1) with initial condition (4.2) in the form of in…nite series which converge rapidly to the exact solution as follows where U (X; k) is projected transform function of u(X; t): Proof.In order to to achieve our goal, we consider the following nonlinear wave-like Eqs.(4.1) with the initial conditions (4.2).
First, we take the Elzaki transform on both sides of (4.1) subject to initial conditions (4.2), we get Using the di¤erentiation property of Elzaki transforms 2.2 and above initial conditions, we have Applying the inverse Elzaki transform on both sides of Eq. (4.4), to …nd represents the term arising from the source term and the prescribed initial conditions.Now, we apply the projected di¤erential transform method.
Then, the solution of Eqs.(4.1) and (4.2) is given as follows The proof is complete.
To illustrate the capability and simplicity of the method, some examples for nonlinear partial di¤erential equations will be discussed in particular nonlinear wave-like equations.

Numerical Application
In this section, we apply mixture of Elzaki transform and the projected di¤erential transform method for solving various types of nonlinear wavelike equations with variable coe¢ cients and we compare the approximate analytical solution obtained for our nonlinear wave-like problems with known exact solutions .
Example 2 Consider the following nonlinear wave-like equation with variable coe¢ cients with initial conditions (5.7) u(x; 0) = e x ; u t (x; 0) = e x ; where u = u(x; t) is a …eld function, (x; t) 2 ]0; 1[ R + : The exact solution to (5.6) with initial conditions (5.7) is given by u(x; t) = e x+t : By taking Elzaki transform on both sides of (5.6) subject to initial conditions (5.7), and using the proposition 2.2, we obtain By applying the inverse Elzaki transform for (5.6), we get Applying projected di¤ erential transform method to obtain with (5.9) U (x; 0) = e x + te x ; where A(x; k); B(x; k) and C(x; k) are the projected di¤ erential transformed of the nonlinear terms The few nonlinear terms are as follows ] :::; and so on.
From the relationship in (5.8),(5.9),we obtain 5! e x ; :::; and so on.Which in closed form gives exact solution which is an exactly the same solution obtained by Adomian decomposition method [6] and Homotopy perturbation transform method [7] for the same test problem.Example 3 Consider the following one dimensional nonlinear wave-like equation with variable coe¢ cients (5.10) @ 2 u @t 2 = x 2 @ @x (u x u xx ) x 2 (u 2 xx ) u; with initial conditions (5.11) u(x; 0) = 0; u t (x; 0) = x 2 ; where u = u(x; t) is a …eld function, (x; t) 2 ]0; 1[ R + : The exact solution to (5.10) with initial conditions (5.11) is given by u(x; t) = x 2 sin t: By taking Elzaki transform on both sides of (5.10) subject to initial conditions (5.11), and using the proposition 2.2, we obtain By applying the inverse Elzaki transform for (5.12), we get Applying projected di¤ erential transform method to obtain where A(k) and B(k) are the projected di¤ erential transformed of the nonlinear terms u x u xx and u 2 xx having the value The few nonlinear terms are as follows and so on.
Which in closed form gives exact solution which is an exactly the same solution obtained by Adomian decomposition method [6] and Homotopy perturbation transform method [7] for the same test problem.

Discussion of Results
We present in this section to discuss our obtained results in comparison with their associated exact forms.We de…ne E n (X; t) to be the absolute error between the exact solution u(X; t) and n-term approximate solution by EPDTM as follows E n (X; t) = ju(X; t) ' n (X; t)j :

Conclusion
In this work, we used a mixed Elzaki transform and the projected differential transformation method, the advantage is providing an analytical approximation of the solution, usually an exact solution, in a fast convergent sequence for nonlinear wave-like equations with variable coe¢ cients.EPDTM can be performed very easily is more e¢ cient and reliable compared to the most known techniques (Adomian decomposition and homotopy perturbation) as in [6], [7].In addition, EPDTM is faster than ADM-HPM to solve this type of equation and can be applied to other nonlinear partial di¤erential equations.

Figure 1 .
Figure 1.The behavior of the exact solution and the 3terms approximate solution for Example 1 when x = y = 0:5.

Figure 2 .
Figure 2. The behavior of the exact solution and the 3terms approximate solution for Example 2 when x = 0:5.

Figure 3 .
Figure 3.The behavior of the exact solution and the 3terms approximate solution for Example 3 when x = 0:5.

Table 1 .
Comparison of the absolute errors for the obtained results and the exact solution for Example 1, where n=3

Table 2 .
Comparison of the absolute errors for Example 2,

Table 3 .
Comparison of the absolute errors for Example 3,