We have checked the Volterra integral equations of the second kind with an integral of the form of a convolution by using the Elzaki transform.

In this article, we have proposed the solution of Legendre's equation expressed by the initial value by using the Laplace transform, and the proposed method should be applied to another equations. Additionally, we have checked representations of the equation by using the Sumudu transform and Elzaki's.

Accurate demand forecasting is crucial for industries to make strategic decisions and maintain their competitive edge. However, existing demand forecasting methods have prodigious problems, especially when it comes to handling the uncertainty, complexity, and nonlinearity of demand forecasting. In addition, the lack of historical data and data biases can create unreliable sources, which discourages the utilization of demand forecasting at a higher level of implementation in businesses. In addition, lack of historical data and data biases can create unreliable sources, which discourages utilization of demand forecasting at higher level of implementation in businesses. The proposed hybrid model aims to improve demand forecasting performance by combining the strengths of existing methods such as K-means clustering, LASSO regression, and LSTM deep learning. By leveraging these techniques, the model can overcome the limitations of each method and improve the accuracy of demand forecasting in various industries. K-means clustering helps to group similar data points, LASSO regression helps to select the most relevant features, and LSTM deep learning helps to capture the temporal dependencies in the data. The combination of these techniques can result in a more accurate and robust demand forecasting model. The model was tested on 2,548 retail products, and outperformed three benchmarking models using the mMAPE, RMSE, and MAE indicators. The proposed model can be used in the retail industry to improve management performance and decision-making, and its ability to optimize variables for each cluster can improve resource allocation.