Estimating Parameters for Partial Differential Equations with Practical Application

A Thesis Submitted to

Council of  The Administration and Economics/ Karbala University as Partial fulfillment of the Requirements for the Degree of Master of Science in Statistics

Presented by researcher

Zahraa Mohsin Neamah AL-Araji

Supervised By

Ass. Prof. Dr. Mushtaq Kareem Abd Al-Rahem

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.