Efficient Approach for Solving (2+1) D- Differential Equations

: In this article, a new efficient approach is presented to solve a type of partial differential equations, such (2+1)-dimensional differential equations non-linear, and nonhomogeneous. The procedure of the new approach is suggested to solve important types of differential equations and get accurate analytic solutions i.e., exact solutions. The effectiveness of the suggested approach based on its properties compared with other approaches has been used to solve this type of differential equations such as the Adomain decomposition method, homotopy perturbation method, homotopy analysis method, and variation iteration method. The advantage of the present method has been illustrated by some examples.


Introduction:
Differential equations especially partial differential equations (PDEs) play an important role in everyday life, they have become a part of modern life 1 . Therefore, it has become necessary to have many and varied ways to solve such equations, which in turn solve life problems associated with them 2 .
They are used to describe many life models such as exponential growth, population growth of species or the change in investment return over time 3 , cooling and heating problems, bank interest, radioactive decay problems even flow problems in solving continuous compound interest problems, orthogonal trajectories 4 and also involving fluid mechanics problems, population or conservation biology 5 , circuit design, heat transfer, seismic waves 6 . They are used in specific fields such as, in the field of medicine, where modeling cancer growth or the spread of disease may be described as differential equations 7 .
The (1+1)-dimensional PDEs is applied to simulate the propagation of waves in a line. Actual atmospheric and oceanic motions do not occur on lines but planes. Accordingly, it is necessary to study higher-dimensional PDEs to describe the propagation of Rossby solitary waves. Gottwald first derived the (2+1) dimensional Zakharov kuznetsov (ZK) equation for nonlinear Rossby solitary waves in barotropic fluids 8 . In recent years, numerous scholars have obtained higherdimensional PDEs for Rossby solitary waves to explain the wave phenomenon in large-scale atmospheres and oceans. Yang et al 9  Many methods for solving (2+1)D-PDEs such as variable separation approach 12 , hyperbola function method 13 , expanded ( 2 ⁄ ) expansion method 14 , extended F-expansion method 15 , and complex method 16,17 , a Darboux Transformation 18,19 . In this paper, the researchers will use a stunner method to solve partial differential equations with (2+1)-dimension and obtain distinct and accurate analytical results. The next section explains the steps of the proposed method.
This paper has been arranged as follows: In section 2, the basic ideas of the suggested method will be given. In section 3, solving some examples of (2+1)D, such as cubic Klein-Gordon equation, Kadomtsev-Petviashvili equation, and Boussinesq equations by using the suggested method will be given. The convergence of the suggested techniques will be illustrated in section 4. Finally, the conclusion is given in section 5.
In the suggested method the unknown dependent function ( , , ) can be construed as infinite series of the form: ( , , ) = 0 ( , ) In the next step calculate the terms ( = 0, 1, 2 … ).

Convergence Analysis for Series Solution
The analysis of convergence for the series solution of the (2+1) D-PDEs is discussed. The sufficient requirement for convergence of the suggested approach is addressed. That is the series solution for (2+1) D-PDEs will appear to be close to the exact solution. Theorem 1. Let n presented as u 0 + …+ u n be an operator from a Hilbert space H to H. The series solution Theorem 1, is a specific case from the Banach's fixed point theorem which is a sufficient condition to study the convergence of the proposed method.
convergent, then this series will consider the exact solution of the present nonlinear problem. Now the following theorem shows the series "If χ and Ƴ are Banach spaces and ℵ: → Ƴ is a contractive nonlinear mapping, that is ∀ , * ∈ ; ∥ ℵ( ) − ℵ( * ) ∥≤ ∥ − * ∥ ,0 < < 1 Then according to Banach's fixed point theorem, ℵ has a unique fixed point , consider the exact solution of the present non-linear problem.

Illustrative Examples
In this section, some illustrative examples for solving (2+1) D-PDEs by using the suggested method are presented.

Example1
The suggested method is used to solve the (2+1)-dimensional cubic Klein-Gordon equation. This equation prescribes many problems in classical (quantum) mechanics, solitons, and condensed matter physics. For example, it models the dislocations in crystals and the motion of rigid pendula attached to a stretched wire. 20 Consider (2+1) D-cubic Klein-Gordon equation It is clear that ( ) = Comparing the results presented in this paper with other results shows that the suggested method is powerful, efficient, and adequate.
The Riccati-Bernoulli sub-ODE method was used to construct solitary wave solutions for the (2+1)-dimensional cubic nonlinear Klein-Gordon (cKG) equation and obtain a new infinite sequence of solutions by using a Bäcklund transformation. The Riccati-Bernoulli sub-ODE gives infinite solutions. Indeed, all presented solutions have so important contributions for the explanation of some practical physical phenomena and further nonlinear problems 20 .
Wang et al. 21 have presented only five solutions for the cKG equation, using the multifunction expansion method. Whereas Khan et al. 22 gave eight solutions, using the modified simple equation (MSE) method. Comparing these results with the presented result in this paper, one can deduce that the suggested method gives a unique exact traveling wave solution. Thus, the suggested method is more effective in providing an exact solution than these two methods.

Example 2
Kadomtsev and Petviashivili in 1970 first introduced this equation to describe slowly varying nonlinear waves in a dispersive medium and study weakly nonlinear dispersive waves in plasmas and also in the modulation of weakly nonlinear long water waves which travel nearly in one dimension that is, nearly in a vertical plane. The solitons are stable 23 .

Example 3
In this example, we solve the (2+1)dimensional Boussinesq equation which contains the second-order partial derivative u tt in addition to other partial derivatives. This family of nonlinear equations gained its importance because it appears in many scientific applications and physical phenomena 24  ( + − 2 )). This is the exact solution.
The ( ′/ )-expansion method is used to solve example 3, with Maple and getting solutions are in more general forms 24 .
In exp(Φ(η))-expansion method is applied to find exact traveling wave solutions to the (2+1)dimensional Boussinesq equation with the aid of Maple 25 .
Zheng studied the exact traveling wave solutions of the (2+1)-dimensional Boussinesq equation by using the (Gʹ/G)-expansion method and achieved three analytical solutions 26 .
Ajeel et al 27 were discussed the related existing theorem.
In this article, the new effective method for treating non-linear, (2+1)D -PDEs is implemented. A new decomposition technique has been introduced to compute exact analytic solutions for the non-linear (2+1) D-model equations such as (2+1) D-cubic Klein-Gordon equation, (2+1) D-Kadomtsev-Petviashvili model equation, and (2+1)D-Boussinesq equations. Series formulation is used throughout the entire procedure, which leads to a series solution being made use within the new procedure. The method is generally based on the well selected base functions and produces an exact solution. Illustrated examples showed that the proposed method has better accuracy with easy implementation. Furthermore, the results showed that when the number of iterations increases, the series solution becomes closer to the exact value as well. The suggested method can be used in the future to solve (3+1)D-PDEs.