Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

Amin, Rohul; Shah, Kamal; Khan, Imran; Asif, Muhammad; Salimi, Mehdi; Ahmadian, Ali

doi:10.3390/math8111874

Cited by 6 publications

(2 citation statements)

References 24 publications

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“…For example, Amin et al successfully obtained effective approximate solutions for 8th and 6th order differential equations using only first-order weak derivative Haar wavelets as trial functions. [155,156] It is worth mentioning that these two points have been a part of the reasons that have long plagued the development of traditional methods for solving high-order differential equations.…”

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

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Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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“…Recently, an emerging class of machine learning (ML) models, such as artificial neural networks (ANNs), random forest (RF), adaptive neuro-fuzzy inference-based system (ANFIS), gene expression programming (GEP), group method of data handling (GMDH), support vector machine (SVM), and ensemble ML models were proposed and successfully applied in the literature for surface water and groundwater quality prediction [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The ANNs are the computational network models based on the biological neural network that forms the structure of human brain.…”

Section: Introductionmentioning

confidence: 99%

Comparative Assessment of Individual and Ensemble Machine Learning Models for Efficient Analysis of River Water Quality

et al. 2022

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The prediction accuracies of machine learning (ML) models may not only be dependent on the input parameters and training dataset, but also on whether an ensemble or individual learning model is selected. The present study is based on the comparison of individual supervised ML models, such as gene expression programming (GEP) and artificial neural network (ANN), with that of an ensemble learning model, i.e., random forest (RF), for predicting river water salinity in terms of electrical conductivity (EC) and dissolved solids (TDS) in the Upper Indus River basin, Pakistan. The projected models were trained and tested by using a dataset of seven input parameters chosen on the basis of significant correlation. Optimization of the ensemble RF model was achieved by producing 20 sub-models in order to choose the accurate one. The goodness-of-fit of the models was assessed through well-known statistical indicators, such as the coefficient of determination (R2), mean absolute error (MAE), root mean squared error (RMSE), and Nash–Sutcliffe efficiency (NSE). The results demonstrated a strong association between inputs and modeling outputs, where R2 value was found to be 0.96, 0.98, and 0.92 for the GEP, RF, and ANN models, respectively. The comparative performance of the proposed methods showed the relative superiority of the RF compared to GEP and ANN. Among the 20 RF sub-models, the most accurate model yielded the R2 equal to 0.941 and 0.938, with 70 and 160 numbers of corresponding estimators. The lowest RMSE values of 1.37 and 3.1 were yielded by the ensemble RF model on training and testing data, respectively. The results of the sensitivity analysis demonstrated that HCO3− is the most effective variable followed by Cl− and SO42− for both the EC and TDS. The assessment of the models on external criteria ensured the generalized results of all the aforementioned techniques. Conclusively, the outcome of the present research indicated that the RF model with selected key parameters could be prioritized for water quality assessment and management.

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