Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular approximation techniques. In this research, we introduce spectral collocation method based on Lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4 order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1 kind give lowest accuracy in the proposed method.


Introduction
Theory of Fractional calculus is almost of same age as that of the classical calculus. The concept of a fractional derivative initiates the theory of fractional calculus when this concept first appeared in a letter of L'Hospital to Leibnitz in 1695 [1]. But in 1819, Lacroix first mentioned the derivative of arbitrary order in a text [2]. Later from 1832 Liouville [3] starts to provide the initial foundations of the fractional calculus. A brief history and fundamental theory of fractional calculus can be found in [4]. Though many important physical processes in almost all branches of sciences can be described using various tools of classical calculus, there are also many complex phenomena that cannot be modeled using classical calculus. Being one of the most important generalizations of classical calculus, fractional calculus has been used to describe many such complex processes recently. In last few decades, scientists and engineers have found fractional order derivative very powerful to describe problems in anomalous diffusion, signal processing, control processing, fractional stochastic system and viscoelasticity.
In space fractional diffusion equation, spatial derivative of order 1 2 α < < replaces the classical second order spatial derivative in classical diffusion equation and results into super diffusive flow model. Unlike classical order derivative, fractional derivative has several kinds of definitions such as Caputo, Riemann-Liouville, Grünwald-Letnikov and Riesz. Detailed discussion about various definitions of fractional derivative can be found in [5] [6].
Spectral method is very effective and efficient and hence popular among scientists and engineers for numerical approximations of both classical and fractional differential and integral equations. In spectral method unknown solution is first approximated by the trial solution as a finite sum of a set of known basis functions. Then expansion coefficients in trial solution can be determined by several techniques like collocation, Galerkin, Petrov Galerkin, tau etc. In all of these techniques expansion coefficients are determined by minimizing the difference between exact and trial solutions. For spectral approximations of fractional differential equations collocation technique is the most efficient and widely used technique. In collocation technique expansion coefficients are determined by forcing the trial solution to satisfy the differential equation at some suitably chosen points from the domain known as collocation points. For a p parameter solution p collocation points are required within the domain and which results into p residual equations. In recent times, researchers showed immense interests to try out various types of polynomials as basis sets in spectral approximations and different kinds of collocation points. For numerical approximations of 1D space fractional diffusion equations Khader [7] used Chebyshev polynomials of space variable as the basis set in spectral approximations and roots of shifted Chebyshev polynomials as the collocation points to determine the expansion coefficients. Chebyshev polynomials are also used for both time and space domain as basis set in spectral approximations of 1D fractional diffusion equation by Azizi and Loghmani [8] and Xie et al. [9]. But Azizi [11] also chosen evenly spaced collocation points but expressed the trial solution in terms of Hermite polynomials for approximations of fractional order system of differential equations. Huang and Zheng [12] used Jacobi polynomials and Lagrange's basis polynomials for spectral approximations of 1D space fractional diffusion equations and used Gauss-Lobatto nodes for collocation. Doha et al. [13] presented spectral tau method for 1D space fractional diffusion equation where they expand the solution by Jacobi polynomials. For spectral expansion of trial solution for time fractional diffusion equations Lagrange interpolation polynomials are used in both 1D space and time domain by Huang [14]. For collocation purpose he chose Jacobi-Gauss nodes for time domain and for space domain he chosen Gauss-Lobatto nodes. Zayernouri and Karniadakis [15] introduced fractional Lagrange interpolants and developed a new fractional spectral collocation method for 1D fractional differential equations. Krishnaveni et al. [16]  In this research, we consider two dimensional space fractional diffusion equation with an initial condition and non-homogeneous Dirichlet's boundary conditions where spatial fractional derivatives are described in Riemann-Liouville sense. Variety of spectral collocation methods are available for 1D fractional diffusion equations but spectral collocation method haven't been tried with lots of variations in 2D problems. To the best of our knowledge Lagrange's basis polynomials haven't been used in spectral collocation method for 2D space frac-

Preliminaries
In this section, we present a brief introduction of Legendre, Jacobi and Chebyshev polynomials of 1 st and 2 nd kinds. Also short description of Lagrange's basis polynomials and Caputo's fractional order derivative are given.
Jacobi Polynomials: The nth degree Jacobi polynomials Riemann-Liouville Fractional Derivative: Riemann-Liouville fractional derivative operator of order α is denoted by D α and defined by: Then for Like classical integer order derivative, Riemann-Liouville fractional order derivative is also a linear operator.

Lagrange's Spectral Collocation Method
Lagrange's Spectral Collocation method for numerical approximations of 2D space fractional diffusion equation will be presented in this section. In this research, we consider following form of diffusion equation subject to the initial condition and non-homogeneous Dirichlet's boundary conditions 0 u y t g y t u a y t g y t 6 7 , 0, , , Here Lagrange's basis polynomials  First by setting the residual at Equation (16) to be zero at the interior points of the spatial domain, we have the equation Thus Equation (20) gives us the following system of ODE for time variable t Rest of the values of expansion coefficients can be obtained from Equation (32) and (

Numerical Results
In this section, we consider two examples to apply the proposed method with four different types of nodes to investigate the accuracy of the method and nodes.
System of ordinary differential equations in Equation (26)   , , , The exact solution of this problem is given by ( ) 3 3.6 , , e t u x y t x y − = .
Applying the proposed Lagrange's spectral collocation technique for         The exact solution of this problem is given by ( )

Conclusion
Spectral collocation method with Lagrange's basis polynomials for approximations of 2D space fractional diffusion equations is introduced in this research.
We used four different types of nodes generated from roots of Legendre, Jacobi,