NUMERICAL APPROXIMATION OF 1D AND 2D REACTION DIFFUSION SYSTEM WITH MODIFIED CUBIC UAH TENSION B-SPLINE DQM

In this paper, a new numerical approach “Modified cubic UAH tension B-spline DQM” is projected to find the numerical approximation of 1D and 2D Reaction-Diffusion system. The modified cubic UAH tension B-spline is used in space to discretize the partial derivatives. The obtained system of ODE is dealt with SSP-RK43 scheme. To check the adaptability and efficiency of the proposed scheme, five numerical examples are discussed. The present method is easy to implement and economical as compared to the existing approaches available in literature for different types of linear and non-linear PDEs.


1D Reaction Diffusion System:
1D non-linear Reaction Diffusion system of equations is as follows: with Dirichlet or Neumann Boundary conditions in the computational domain [a, b]. Where u(x, t) NUMERICAL APPROXIMATION OF 1D AND 2D REACTION DIFFUSION SYSTEM and v(x, t) are the real valued functions, 1 and 2 are the arbitrary constants.
Model of reaction Diffusion systems are the mathematical models related to several physical processes. Reaction Diffusion systems have a wide number of applications in different areas alike, geology, biology, physics, ecology and many others. Reaction Diffusion systems can represent several models like Semi linear partial differential equations including Brusselator model [1], Gray Scott model [2], Isothermal model [3], Schnakenberg model [4] and many more. Many researchers have solved Reaction-Diffusion systems numerically. Sahin [5] implemented FE method for getting numerical approximation. By different methods [6][7][8] represents the numerical approximation of Reaction Diffusion systems.

2D Reaction Diffusion System:
2D Reaction Diffusion Brusselator system of the non-linear system of partial differential equations is as follows: and Neumann Boundary conditions on boundary are defined by the lines = 0, = , = 0 = .
Because of such importance, these equations are very important from the numerical point of view.
Such problems have been solved by eminent researchers. Dehghan et al. [9][10][11][12][13][14]  proposed the numerical approximation as well as the stability analysis for the Brusselator system given in equation (3) and equation (4). Adomian [15] and Wazwaz [16] gave the decomposition method. Twizell et a. [17] proposed 2D FD scheme for the solution of Brusselator RD system. Ang [18] gave the dual-Reciprocity Boundary element approximation for the solution of Brusselator system numerically. Rohila [25] implemented the concept of the modified cubic B-spline to solve Reaction-Diffusion systems. Tamsir et al. [26] implemented exponential modified cubic B-spline to solve the nonlinear Burgers' equation. Arora and Joshi [27] implemented the notion of modified trigonometric cubic B-spline for the solution of 1D and 2D Burgers' equation.

Differential Quadrature Method:
DQM has attained the noticeable attention over some previous decades. The initial knowledge was based upon the work of Bellman and Casti [28]. This regime owes it's higher popularity due to its simplification and higher accuracy and efficiency. It is included in several applications of engineering and sciences. A comprehensive review of DQM was proposed by Bert and Malik [29].
DQM is actually a numerical discretization technique, in which several test functions can be used to get the weighting coefficients for the approximation of derivatives [30][31][32]. A lot of work has been reported in literature related to DQM. Korkmaz  After exploring the literature in detail, it is noticed that UAH tension B-spline has never been used to get the numerical approximation of 1D and 2D system of Reaction-Diffusion equations. As per author's knowledge, this scheme will open some new dimensions in the research of the numerical approximation of complex non-linear partial differential equations. This paper is organized into different sections. In Section 2, complete detail of the numerical scheme is provided. In Section 3, five test problems are provided. In Section 4, the crux of this research is given as conclusion.

[Modified Cubic UAH tension B-spline DQM]
Uniform algebraic hyperbolic tension B-spline of order 4 is defined as follows: By using following set of equations improvised values can be obtained [39].

NUMERICAL EXPERIMENTS AND DISCUSSION
In  Table 4, Numerical approximation of U(x, y, t) and V(x, y, t) are given at time levels t = 1.  Table 5, Numerical U and V are given at t = 0.1 and 1.0 respectively for 1 = 0.5, 1 = 1 and α = 0.002. In Figures 9 and 10, Numerical approximations of U and V are given graphically at t = 1, 2, 3 and 4 respectively for 1 = 1, 1 = 2 and α = 0.002. In Table 6, Numerical U and V are evaluated at t = 5 and t = 10 respectively.

E)ample 1:
Brusselator model [42] was proposed by Brussels school of Prigogine. Present model represents the Hypo-theoretical tri-molecular natured reaction, having very important traits in the chemical science area.

CONCLUSION
In present paper, modified cubic UAH tension B-spline based DQM is developed to solve the linear and non-linear partial differential equations. Solving such complex non-linear partial differential equations analytically is not always possible. That is why, it is a major need of time to develop some efficient and accurate numerical regimes. The obtained ODE system is dealt by SSP-RK43 scheme. Five numerical examples are discussed in this paper. Numerical approximation of 1D and 2D Reaction-Diffusion system is obtained. This scheme will help researchers in their future work to solve some other complex partial differential equations numerically, mainly where analytical solution is not available.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.