《TABLE VI COMPARISON BETWEEN HAAR SOLUTIONS (J=1, m=4, ν=0.9) AND HAM[5]OF PROBLEM 4》

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《An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrdinger and Coupled Schrdinger Systems》

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In this paper，we extended the capability of the Haar wavelet collocation method（HWCM）for the solution of timefractional coupled system of partial differential equations.The main advantage of HWCM is the ability to achieve a good solution and rapid convergence with small number of collocation points.The presence of maximum zeros in the Haar matrices reduces the number of unknown wavelet coefficients that is to be determined，which as a result diminishes the computation time as well.The scheme is tested on some examples of time fractional Schr¨odinger equations.The presented procedure may very well be extended to solve two dimensional Schr¨odinger equation and other similar nonlinear problems of partial differential equations of fractional order.The problem discussed here is just for showing the applicability of the proposed computational technique to handle the complex system of differential equation in fractionalorder problems in a straight forward way.Also，the Haar wavelet method proves to be capable to efficiently handle the nonlinearity of partial differential equations of fractional order.The main advantages of the proposed algorithm are，its simple application and no requirement of residual or product operational matrix.Numerical solutions for different order of fractional time derivative by Haar wavelet are shown in Tables and Figures.The increasing values ofνshow that the solutions are valuable in understanding their respective exact solutions forν=1.Comparisons between our approximate solutions of the problems with their actual solutions and with the approximate solutions achieved by a homotopy analysis method[5]confirm the validity and accuracy of our scheme.