Abstract
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 ′ x ξ ′′ 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.Keywords: Wavelet analysis, bernstein polynomials multiwavelets, operational matrix of integration, variational problem, extremal, haar wavelet method.
Graphical Abstract
Call for Papers in Thematic Issues
Advancements in Brain Tumor Detection and Treatment using Machine Learning, Deep Learning, and Blockchain Technology
Brain tumors are among the most challenging diseases to diagnose and treat, requiring specialized expertise and advanced technology for accurate detection and effective treatment. In recent years, machine learning and deep learning algorithms have shown promise in improving the accuracy and efficiency of brain tumor detection through medical imaging analysis. ...read more
Advances of Biometric Data Analysis to Boost Intelligent Applications
With the rapid development of artificial intelligence (AI) technology, more and more intelligent devices and applications are appearing in our daily lives, such as smart home, smart agriculture, health diagnosis, educational support, and environmental monitoring. Interaction with these devices or applications has become a pressing issue that can greatly improve ...read more
Attainment of SDGs through the Advancement in Solar Energy Systems
With less than a decade until we reach 2030, it is crucial to address the deep inequalities affecting not only our health but also our quality of life, and the economy of countries worldwide. Few of the UN's Sustainable Development Goals (SDGs) can be directly and indirectly achieved through the ...read more
Emerging Intelligent Computing Techniques and their Applications Using Machine Learning
This special issue will aim to discuss computational intelligence approaches, initiatives and applications in engineering and science fields (including Machine Intelligence, Mining Engineering, Modeling and Simulation, Computer, Communication, Networking and Information Engineering, Systems Engineering, Innovative Computing Systems, Adaptive Technologies for Sustainable Growth, and Theoretical and Applied Sciences). This collection should ...read more
Related Journals
Recent Patents on Engineering
7