The boundary value problem, BVP, for the PDE heat equation is studied and explained in this article. The problem declaration comprises two intervals; the (0, T) is the first interval and labels the heating of the inside burning chamber, and the second (T, ∞) interval defines the normal cooling of the chamber wall when the chamber temperature concurs with the ambient temperature. It is necessary to prove the boundary function of this problem has its place in the space $H^1\left[0,\infty \right]$ in order to successfully apply the Fourier transform method. The applicability of the Fourier transform for time to this problem is verified. The method of projection regularization is used to solve the inverse boundary value problem for the heat equation and to obtain an evaluation for the error between the approximate and the real solution. These results are new and of practical interest as shown in the numerical case study.

In this paper, we introduce an approach via regularization and Homotopy way for resolving the inverse Cauchy problem of the Laplace of system partial differenti

The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP).

In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution.

In this study, Climate Research Unit (CRU) data crossing from 1901 to 2022 was utilized to analyze and predict evaporation rates in Almaty, Kazakhstan. The data was prepared utilizing Python, leveraging the Random Forest (RF) machine learning Techniques for its capability to handle complex and non-linear information. The essential measurements for assessing the model's execution were Cruel Supreme Blunder (MAE), Root Cruel Square Mistake (RMSE), and Cruel Squared Mistake (MSE). The Random Forest model was utilized in prediction and data examination. This model is recognized by its capacity to process non-linear and complex data by making a set of choice trees and joining their comes about to get exact and steady predictions. The model is based on arbitrary testing with bootstrap testing, which makes a difference decrease change and increment the precision of the demonstrate.