Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

A. K. Mittal; L. K. Balyan; A. K. Mittal; L. K. Balyan

doi:10.3934/math.2020111

AIMS Mathematics

2020, Volume 5, Issue 3: 1642-1662. doi: 10.3934/math.2020111

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Research article

Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

A. K. Mittal ^,,
L. K. Balyan

Discipline of Mathematics, IIITDM Jabalpur, Madhya Pradesh 482005, India

Received: 14 November 2019 Accepted: 03 February 2020 Published: 10 February 2020
MSC : 35C07, 35C11, 35R11

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order α and β. Moreover, numerical solutions are demonstrated to justify the theoretical results.
Citation: A. K. Mittal, L. K. Balyan. Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain[J]. AIMS Mathematics, 2020, 5(3): 1642-1662. doi: 10.3934/math.2020111

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Abstract

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order α and β. Moreover, numerical solutions are demonstrated to justify the theoretical results.

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