In this paper, using orthogonally of Tchebychev polynomials, we present an orthonormal WAVELET BASIS for L2[0, 1]. We use this BASIS for solving Neumann problems with Galerkin method. The property of this BASIS is that a variety of integral operators is represented in this BASIS as sparse matrices, to high precision. Some examples are solved to illustrate the efficiency and accuracy of this method.

WAVELETs and radial BASIS functions (RBF) have ubiquitously proved very successful to solve different forms of partial differential equations (PDE) using shifted BASIS functions, and as with the other meshless methods, they have been extensively used in scattered data interpolation. The current paper proposes a framework that successfully reconciles RBF and adaptive WAVELET method to solve the Perona-Malik equation in terms of locally shifted functions. We take advantage of the scaling functions that span multiresolution subspaces to provide resilient grid comprising centers. At the next step, the derivatives are computed and summed over these local feature collocations to generate the solution. We discuss the stability of the solution and depict how convergence could be granted in this context. Finally, the numerical results are provided to illustrate the accuracy and efficiency of the proposed method.

Extended AbstractIntroductionDrought monitoring entails the simulation of indices which are categorized into single and combined types. Historically, simulations have predominantly relied on single indices, including Standardized Precipitation Index (SPI), Standardized Runoff Index (SRI), resulting in limited research on drought simulation using combined indices (i.e. MSPI and SPTI), particularly in conjunction with combined models. Over the years, several single models have been developed for simulating individual drought indices. For instance, the Autoregressive Integrated Moving Average (ARIMA) model has been applied to simulate drought indices like Standardized Precipitation Index (SPI) and Standard Index of Annual Precipitation (SIAP). Additionally, models such as Artificial Neural Network (ANN) and Long Short Term Memory (LSTM) have been used for simulating indices like SPI, DI, SIAP, and SHDI. Recent studies suggest that combined models outperform single models. WAVELET ARIMA ANN (W-2A) and WAVELET ANFIS combined models to simulate the single drought index SPEI. Other researchers have developed combined models such as ARIMA-LSTM, WAVELET-ARIMA-LSTM, WAVELET-ARIMA-ANN and LSTM-CM to simulate single drought indices SPI, DI, SIAP. Despite the progress in developing drought simulation models, including single models and particularly combined models, their application has primarily focused on individual indices. Historically, simulations have predominantly relied on single indices, resulting in limited research on drought simulation using combined indices, particularly in conjunction with combined models. This study has combined the strengths of the WAVELET transformation, Autoregressive Integrated Moving Average (ARIMA), Artificial Neural Network (ANN) and Long Short Term Memory (LSTM) to test new methods of hybrid models for their ability to drought simulations based on the new combined index SRGI, employing the combined models W-AL and W-2A.Materials and MethodsDrought simulated in the Alashtar sub-basin between 48, 15 east longitude and 33, 54 north latitudes, covering an area of 811 square kilometers from 1991 to 2020, utilizing individual indices such as SPI, SRI, SGI, and the combined index SRGI. The study area encompasses the Karkheh River basin. Both single models (ARIMA, LSTM, ANN) and combined models (W-AL and W-2A) were employed for this purpose. Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Mean Error (ME) were used to evaluate the performance of the models. Also, relative frequency and error distribution charts were used to evaluate and compare the results of the models.Individual indices were calculated based on fitting the best cumulative probability function to monthly precipitation, monthly discharge, and monthly water table data, respectively for indices SPI, SRI, and SGI, and then inversely transforming to a N (0,1). The SRGI index is a combination of two drought indices, SGI and SRI (Feng et al., 2020). For this purpose, the copula function is used to obtain the best joint probability distribution function governing precipitation and water table data. The selection of the best copula function was done through the Kolmogorov-Smirnov K-S test at a significant level of 5%. In the current research, four copula functions of Frank, Clayton, Gamble and Joe were used.The process of building the combined models includes the analysis of the time series of the studied drought index, using DWT and decompose into two series named approximate and partial. Then, the approximate and detail series modeled by ARIMA and ANN respectively, in W-2A model and ARIMA and LSTM, respectively, in W-AL model. Results and DiscussionThe results demonstrate that the combined models W-AL and W-2A exhibit higher accuracy across all indices, both individual and combined, compared to single models ARIMA, LSTM, and ANN. The RMSE ranges for the combined models were 0.44 to 0.71, while for single models, they ranged from 0.47 to 1.54. Specifically, model W- AL displayed superior accuracy across all individual indices, with RMSEs of 0.44, 0.62, and 0.59, in contrast to model W-2A, which yielded RMSEs of 0.49, 0.71, and 0.63. However, W-AL's performance lagged behind W-2A for the combined SRGI index, with respective RMSEs of 0.64 and 0.61. Thus, the simpler model yielded more acceptable results in simulating the composite index.ConclusionAmong all the combined and individual models, the combined models perform better in simulating drought, based on all indices, compared to the individual models. Therefore, it can be said that combined models are more suitable for simulating and monitoring drought compared to individual models. However, the performance of the two combined models, W-2A and W-AL, in simulating the combined SRGI index is different. The performance of the simpler W-2A model is better than the more complex W-AL model, with RMSE values of 0.61 and 0.64, respectively. Therefore, in combined indices, despite the complexity of their computational process, there is not necessarily a need to use a more complex combined model. Overall, the use of combined models is recommended for monitoring various types of indices, especially drought based on combined indices such as SRGI. The major objectives of this study are: (1) to use hybrid models WAVELET-ARIMA-LSTM (W-AL) and WAVELET-ARIMA-ANN (W-2A) methods to predict monthly drought. (2) To analyze drought characteristics in Alashtar basin based on the new combined drought index, SRGI. It is expected that the research results will help to provide decision support which in turn will help in planning adaptative measures to reduce drought impacts and provide decision support for disaster prevention.

Introduction: Increasing water demand and water pollution due to the development of agricultural, urban and industrial activities have caused environmental problems all over the world. The significant increase in water pollution and the diversity of various urban, agricultural and industrial pollutants made the qualitative management of water resources inevitable. Short-term and long-term accurate forecasts of river quality parameters are essential for designing hydraulic structures, irrigation planning, optimal utilization of reservoirs and environmental planning. Given the stochastic characteristics of the hydrological events, forecasting the future status of surface waters is always associated with uncertainties. The purpose of the present study was to investigate the performance of two types of artificial neural networks, namely MLP and GMDH, combined with discrete WAVELET transform (DWT), to forecast two important quality parameters, electrical conductivity (EC) and sodium adsorption ratio (SAR) at Zayandeh-Rood River in 1, 2 and 3 months ahead. Material and methods: In this study, water quality data (EC and SAR) of Zayandeh-Rood River at Zaman Khan Station was used from 1363 to 1384. From 21 years of data, 15 years (approximately 70%) were used for training and 7 years (30%) were used to test the developed models. Two types of mother WAVELET dmey and db4 were evaluated. Statistical parameters such as RMSE and R2 were used to evaluate the performance of the models. Results and discussion: The results showed that the use of discrete WAVELET transform improves the performance of the models. Various combinations of input data (various delays) and two types of mother WAVELETs were evaluated. The results showed that WAVELET-MLP and WAVELET-GMDH hybrid models outperform single MLP and single GMDH models at all forecasting intervals. The results of the single MLP and GMDH models were only effective in forecasting SAR one month ahead but practically could not forecast two and three months later. In the EC parameter, the MLP and GMDH models performed better. Also, the results showed that the use of annual time lags does not increase the accuracy and in some cases even reduces it. The study of the types of mother WAVELETs also showed that the dmey WAVELET is the most suitable WAVELET type to forecast EC and SAR qualitative parameters. The comparison between WAVELET-MLP and WAVELET-GMDH models showed the relative superiority of the former model. By increasing the forecast period from one month to three months ahead, the accuracy of the models decreased. This decrease in precision was higher in forecasting SAR parameter, e. g. in the one month forecast, R2 was 0. 936 and in the 3 months ahead forecasts it was reduced to 0. 516. In the EC parameter, the R2 fell to 0. 641 in 3 months ahead forecasting. Conclusion: The results of this study can be used as a BASIS for future planning for water quality. It is suggested that the model presented in this study should be considered in other rivers. Also, the combination of other artificial intelligent models such as ANFIS and SVM with WAVELET transform can be evaluated.

Rainfall-runoff process is one of the most important and complex phenomena in the hydrological cycle and therefore different views have been presented for modeling the phenomenon. Obviously, the recognition of the behavior of the catchment can play an important role in selecting the appropriate model as well as saving time on the simulation. Previous studies have shown that the multi-linear models have an acceptable performance in the case of watersheds which usually have a regular rainfall pattern. In this study, the multilinear WAVELET-M5 model was introduced and the rainfallrunoff process in the Aji Chay catchment was investigated. At first, the main rainfall and runoff time series were decomposed to several sub-time series by the WAVELET transform to overcome its non-stationarity. Then the obtained sub-time series were imposed as input data to M5 model tree to forecast the runoff values and also the results were compared to the other models (i. e. ANN, M5 and WANN) by the root mean squared error and determination coefficient criteria. The results showed that the performance of the proposed hybrid WAVELET-M5 model increased up to 69% compared to the sole M5 model tree for the Aji Chay catchment.

This study attempts to examine the factor structure of Basic Adlerian Scales for Interpersonal Success Adult form (BASIS-A), and providing a reliable instrument to evaluate lifestyle in the Iranian population. Factor analysis was applied to study the inventory 513 subjects (%46.4=male, %53.6=female; %65.5=married, and %34.1=single) aging 18-40 years old (m=29.4; SD=7) were recruited and studied. Findings, showing the Iranian norm for BASIS-A, also displayed the same major factors as the American original version consisting 5 major factors as Wanting Recognition, Going Along, Taking Charge, Being Cautious, and Belonging-Social Interest. The correlation of the minor factors has been analyzed. This study concludes that BASIS-A could be used by the clinicians to get a better understanding about the Iranian clients lifestyle.

In this paper we will present an adaptive WAVELET scheme to solve the generalized Stokes problem. Using divergence free WAVELETs, the problem is transformed into an equivalent matrix vector system, that leads to a positive definite system of reduced size for the velocity. This system is solved iteratively, where the application of the infinite stiffness matrix, that is sufficiently compressible, is replaced by an adaptive approximation. Finally we prove that this adaptive method has optimal computational complexity, that is it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding WAVELET-best N-term approximation.