Wavelet based algorithm for numerical study of $(1+2)$-dimensional time fractional diffusion problems

Abstract

Abstract An effective and robust scheme is developed for solutions of two-dimensional time fractional heat flow problems. The proposed scheme is based on two-dimensional Haar wavelets coupled with finite differences. The time fractional derivative is approximated by an $L_{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -formula while spatial part is approximated by two-dimensional Haar wavelets. The proposed methodology first converts the problem to a discrete form and then with collocation approach to a system of linear equations which is easily solvable. To check the efficiency of the scheme, two error norms, $E_{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> an $E_{\mathrm{rms}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>rms</mml:mi> </mml:msub> </mml:math> , have been computed. The stability of the scheme has been discussed which is an important part of the manuscript. It is also observed that the spectral radius of the amplification matrix satisfies a stability condition. From computation it is clear that computed results are comparable with the exact solution.

Locations

• Advances in Difference Equations - View - PDF
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