تقدير معلمات المعادلات التفاضلية الجزئية مع تطبيق عملي

Estimating Parameters for Partial Differential Equations with Practical Application

Abstract
The Study Seeks to estimate the parameter of the partial differential equation, as some of its types and classifications were studied and its parameters were estimated by three methods of estimation (the Maximum likelihood Method, Gaussian process, the Standard Bayes Method) .For the purpose of comparing the methods of estimating the parameters of the partial differential equation, the Monte Carlo Simulation method was employed to conduct several experiments at nine different sizes (Small, Medium, large) and at levels of white noise(0.01,0.05,0.09) and through the four statistical criteria (Bais, Squara roots of average squared errors, Standard deviations, Coverage probabilities) .The Results showed the superiority of the standard Bayes method in calculating the estimations of the partial different equation at the small sample sizes according to the different level of noise and the superiority of the maximum likelihood method ,followed by the gaussian process for medium and large samples at the white noise level (0.01) , while at the white white noise level (0.05 & 0.09) the gaussian process method was superior to medium and large sample sizes . The best method in the simulation results , which is the standard Bayes method ,was applied to real data with a view (30) representing the amount of lymphocytes and their relationship to the (Covid -19 virus) . The advantage of the Bayes method has been proven ,as the standard Bayes method matched in representing the data accurately and correctly . we found that the rate of recrumitment of lymphocytes to the site of injury due to the inflammatory response in the lung per day is(11.86712) for all patients with (Covid -19) and that the average number of lymphocytes and infected lung cells the virus has reached (3556) and for all patients.