Analytical and approximate solution of two-dimensional convection-diﬀusion problems

In this work, we have used reduced diﬀerential transform method (RDTM) to compute an approximate solution of the Two-Dimensional Convection-Diﬀusion equations (TDCDE). This method provides the solution quickly in the form of a convergent series. Also, by using RDTM the approximate solution of two-dimensional convection-diﬀusion equation is obtained. Further, we have computed exact solution of non-homogeneous CDE by using the same method. To the best of my knowledge, the research work carried out in the present paper has not been done, and is new. Examples are provided to support our work.


Introduction
We consider two-dimensional convection-diffusion equation as follows: ∂u(a,b,t) ∂t + β a ∂u(a,b,t) ∂a in Ω × (0, T ] , u(a, b, t) = g(a, b, t), (a, b) ∈ ∂Ω, t∈(0, T ], u(a, b, 0) = h(a, b), where β a and β b are progressive velocity components in the direction of a and b respectively, and α a > 0 and α b > 0 are the coefficients of diffusivity in the a and b directions, respectively. And α a > 0 and α b > 0 are g(a, b, c) and h(a, b) are smooth functions and Ω is a subset of R 2 and (0, T ] is the time interval, and ∂Ω is the boundary of Ω. This equation is frequently used in applied sciences and engineering especially in modeling and simulations of various complex phenomena in science and engineering. This paper first describes RDTM and then uses it to solve the Convectiondiffusion equation. In recent years, studies conducted on findings new analytical solutions of differential equations have attracted attention of scientists from all over the world (see [1]- [9]).
And some numerical solutions have been developed to solve these types of convection-diffusion problems. likes: Higher-Order ADI method [10] or rational high-order compact ADI method [11], the alternating direction implicit method [12], the finite element method [13], fourth-order compact finite difference method [14], decomposition Method [15], the finite difference method [16], restrictive taylors approximation [17], The fundamental solution [18], finite difference method [19], combined compact difference scheme and alternating direction implicit method [20], higher order compact schemes method [21], the finite volume method [22], the finite difference and legendre spectral method [23] and even the Monte *Corresponding Author Carlo method [24]. Keskin in [25] proposed the RDTM to solve various PDE and fractional nonlinear partial differential equations.
This method is a repetitive procedure for the solution of a Taylor series differential equations. This technique reduces the size of the computational work and can be easily applied to numerous physical problems. We organize the paper as follows. In section RDTM is used to four types of TDCDP, and section 4 concludes the paper.

Analysis of the RDTM
We have a function with three variables u(a, b, t), and presume that it can be shown as an invention of multiple of two functions u(a, b, t) = v(a, b)w(t). u(a, b, t) can be denoted as where U k (a, b) is called t-dimensional spectrum function of u(a, b, t). The three-dimensional RDTM are introduced are defined in [26] as follows: By inserting equation (3) in (4), we obtain Some basic properties of RDTM are presented in Table1 below. Original function Transformed function

Applications
We used the basic definitions (in Section 2) of the three-dimensional RDTM for solving four examples of Convection-diffusion equations (CDE).
By using Eq. (22) in Eq. (21), we obtain U k (a, b) values for k = {0, 1, 2, 3, · · · } as follows: by using the differential inverse reduced transform of U k (a, b),we get by using the closed form in the solution of (24), we obtain the following exact solution u(a, b, t) = e −t sin(πa)cos(πb).

Conclusion
In this study, we used RDTM to solve convectiondiffusion problems and showed that RDTM is an effective and appropriate technique for finding exact solutions of the TDCDP which we have investigated here. On the other hand the results are quite reliable for solving this problem. The exact closed form solution was obtained for all the examples presented in this paper. RDTM offers an excellent opportunity for future research.