Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 1 2 2016 12 01 Numerical Solution of Optimal Heating of Temperature Field in Uncertain Environment Modelled by the use of Boundary Control حل عددی گرمادهی بهینه میدان دمایی در محیط تصادفی مدل‌سازی شده با استفاده از کنترل مرزی 23 38 3395 EN Ali Nehrani faculty of mathematical sciences, university of guilan, rasht, iran Mohammad Keyanpour Faculty of mathematical sciences, University of Guilan, Rasht, Iran Journal Article 2016 01 12 ‎In the present paper‎, ‎optimal heating of temperature field which is modelled as a boundary optimal control problem‎, ‎is investigated in the uncertain environments and then it is solved numerically‎. ‎In physical modelling‎, ‎a partial differential equation with stochastic input and stochastic parameter are applied as the constraint of the optimal control problem‎. ‎Controls are implemented as Dirichlet boundary conditions and representing the heating elements on the boundary of the field‎. ‎In numerical quantification‎, ‎stochastic input and parameter are approximated via Karhunen-Lo\'eve expansion and inserted to the problem‎. ‎In fact‎, ‎for numerical discretization of the problem stochastic Galerkin method is applied to generalize polynomial chaos‎. ‎Numerical optimization is performed via gradient method‎. ‎The problem is fully implemented and in order to show the applicability of the method‎, ‎numerical examples are solved and numerical results are represented through figures. https://mathco.journals.pnu.ac.ir/article_3395_2c4e86ec7879dc9ee757e6be0008d18b.pdf