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 in complex cases, an approximated numerical solution becomes a necessity. Our primary objective is 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 increases the simplicity, decreases the calculation time, and furnishes better results.
Methods:
An operational matrix of integration, which is based on the BPMWs, is presented. We substitute the approximated values of , unknown function and their derivative functions 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 obtain 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.
Bernstein polynomials method and it’s error analysis for solving nonlinear problems in the calculus of variations: Convergence analysis via residual function