Addressing Earth’s Atmosphere under Fuzzy Schlömilch’s Integral Equation using Regularization-Successive Approximation

Schlömilch’s integral equation (SIE) can be used to model the interaction between the Sun and the Earth’s atmosphere. The problem occurs when the atmospheric information involves uncertainty that cannot be solved using standard SIE. To address these uncertain conditions, we will need to use the fuzzy concept. This work presents the solution of various kinds of SIE combined with the fuzzy concept named Fuzzy Schlömilch’s integral equation (FSIE). With fuzzy numbers in their parametric form, we were able to derive two distinct integral equations of Schlömilch-type. To solve the suggested model of equations, we combine the successive approximation method with the regularization method. The regularization method converts SIE from its first kind to its approximated second kind. Then the successive approximation method is used to derive the solution for the unknown functions. To demonstrate the reliability, efficiency, and accuracy of the suggested Regularization-Successive Approximation (R-SA) method, we solve examples for different types of FSIEs. This is a new iteration-based method that accurately converges to the exact solution. Graphs are created to show the visual results at different levels of uncertainty. Results of every case are carefully reviewed, and the report ends with a short summary of the study and possible future research paths. The uncertain interaction between the Sun and the Earth’s atmosphere can be solved by using the FSIEs.