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Application of Bernstein Polynomial Multiwavelets for Solving Non Linear Variational Problems with Moving and Fixed Boundaries

[ Vol. 14 , Issue. 4 ]


Sandeep Dixit, Shweta Pandey and S.R. Verma*   Pages 441 - 458 ( 18 )


Background: In this article, an efficient direct method has been proposed in order to solve physically significant variational problems. The proposed technique finds its basis in Bernstein polynomials multiwavelets (BPMWs). The mechanism of the proposed method is to transform the variational problem into an algebraic equation system through the use of BPMWs.

Objective: Since the necessary condition of extremization consists of a differential equation that cannot be easily integrated into complex cases, an approximated numerical solution becomes a necessity. Our primary objective was to establish a wavelet-based method for solving variational problems of physical interest. Besides being computationally more effective, the proposed approach yields relatively more accurate results than other comparable methods. The approach employs fewer basis elements, which in turn increase the simplicity, decrease the calculation time, and furnishes better results.

Methods: An operational matrix of integration, which is based on the BPMWs, is presented. We substituted the approximated values of x , unknown function ξ (x) and their derivative functions ξ ʹ′(x),ξ ʹ′(x),.....ξ n (x) with BPMWs operational matrix of integration and BPMWs. On substituting the respective values in the given variational problem, it gets converted into a system of algebraic equations. The obtained system is further solved using the Lagrange multiplier.

Results: The results obtained yield a greater degree of convergence as compared to other existing numerical methods. Numerical illustrations based on physical variational problems and the comparisons of outcomes with exact solutions demonstrate that the proposed method yields better efficiency, applicability, and accuracy.

Conclusion: The proposed method gives better results than other comparable methods, even with the use of a fewer number of basis elements. The large order of matrices, such as 32, 64, and 512, obtained by using other available methods is far too high to achieve accuracy in results in comparison to the ones we obtained by using matrices of relatively lower orders, such as 7, 8 and 13, in the proposed method. This method can also be used for extremization functional occurring in electrical circuits and mechanical physical problems.


Wavelet analysis, Bernstein polynomials multiwavelets, Operational matrix of integration, variational problem, extremal, Haar wavelet method.


Department of Mathematics, University of Petroleum and Energy Studies, Dehradun 248007, Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar 249404, Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar 249404

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