Abstract
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 ap-proximation 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 4th 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 1st kind give lowest accuracy in the proposed method.
Highlights
Theory of Fractional calculus is almost of same age as that of the classical calculus
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
To approximate the solution of 2D space fractional diffusion equation by spectral collocation method Bhrawy [17] used shifted Legendre polynomials and Gauss-Lobatto nodes for 2D space domain
Summary
Theory of Fractional calculus is almost of same age as that of the classical calculus. Krishnaveni et al [16] proposed a hybrid method for 1D space fractional diffusion equation where trial solution is approximated by spectral expansion of fractional shifted Legendre polynomials. To approximate the solution of 2D space fractional diffusion equation by spectral collocation method Bhrawy [17] used shifted Legendre polynomials and Gauss-Lobatto nodes for 2D space domain. Other numerical methods such as ADI and Finite-Difference methods [18] [19] [20] [21] [22] are very efficient for 2D problems.
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