Application of statistical spline model for solving inverse source parabolic problems

Inverse parabolic problems are the most famous ill-posed problems‎. ‎Thus using stable inexact numerical methods for approximating these problems‎, ‎causes a large perturbation‎. ‎In this paper‎, ‎we study the problem of determining the source term of the inverse source problem‎ [{partial _t}T(x,t) = kappa ,{nabla ^2}T(x,t) + g(t)delta (x - {x^*}),x in {( circ ,,1)^d},t in ( circ ,,{t_f}),] ‎from the measured data given in the form of‎ [T({x_{measure}},{t_i}) = {y_i}, ,i = 1,2, ldots ,I,] ‎(additional condition) where d = 1,2 ‎and [delta ] is the Dirac delta function and (T,g) are the unknown functions‎. ‎Then‎, ‎using the statistical spline model and applying Levenberg-Marquardt method‎, ‎we obtain an approximate solution for quasi solution g=g(t)‎‎. ‎Finally‎, ‎to show the priority and accuracy of the introduced method some numerical examples are given‎