Non homogeneous Heat Equation : Identi cation
نویسندگان
چکیده
We study the nonhogeneous heat equation under the form: u t ? u xx = '(t)f(x), where the unknown is the pair of functions (u; f). Under various assumptions about the the function ' and the nal value in t = 1 i.e. g(x), we propose diierent regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For ' 6 6 0 the solution is unique. I. Introduction Consider the problem: Find a pair of functions (u; f) satisfying the following equation and boundary and initial values: ? @u @t + u = '(t)f(x); (t; x) 2 (0; 1) (0; 1) u(1; t) = 0; u x (0; t) = u x (1; t) = 0 u(x; 0) = 0; u(x; 1) = g(x) (1) Where ' and g are two given functions. The previous problem is equivalent to nd a function f satisfying an integral equation of the 1st. kind of Volterra type:
منابع مشابه
Investigation of the Effects of Non-Linear and Non-Homogeneous Non-Fourier Heat Conduction Equations on Temperature Distribution in a Semi-Infinite Body
In this paper, the non-Fourier heat conduction in a semi-infinite body was examined. The heat wave non-Fourier heat conduction model was used for thermal analysis. Thermal conductivity was assumed temperature-dependent which resulted in a non-linear equation. The heat source was also considered temperature-dependent which resulted in a non-homogeneous equation. The Mac-Cormack predictor-correct...
متن کاملSECURING INTERPRETABILITY OF FUZZY MODELS FOR MODELING NONLINEAR MIMO SYSTEMS USING A HYBRID OF EVOLUTIONARY ALGORITHMS
In this study, a Multi-Objective Genetic Algorithm (MOGA) is utilized to extract interpretable and compact fuzzy rule bases for modeling nonlinear Multi-input Multi-output (MIMO) systems. In the process of non- linear system identi cation, structure selection, parameter estimation, model performance and model validation are important objectives. Furthermore, se- curing low-level and high-level ...
متن کاملNon-linear system identi®cation using Hammerstein and non-linear feedback models with piecewise linear static maps
We consider the identi®cation of Hammerstein/non-linear feedback models by approximating internal non-linearities using piecewise linear static maps. The resulting method utilizes a point-slope parameterization that leads to a computationally tractable optimization problem. The computational appeal of this technique is derived from the fact that the method only requires a matrix inverse and sin...
متن کاملNon-Fourier heat conduction equation in a sphere; comparison of variational method and inverse Laplace transformation with exact solution
Small scale thermal devices, such as micro heater, have led researchers to consider more accurate models of heat in thermal systems. Moreover, biological applications of heat transfer such as simulation of temperature field in laser surgery is another pathway which urges us to re-examine thermal systems with modern ones. Non-Fourier heat transfer overcomes some shortcomings of Fourier heat tran...
متن کاملSemi-Quantitative System Identi cation
System identi cation takes a space of possible models and a stream of observational data of a physical system, and attempts to identify the element of the model space that best describes the observed system. In traditional approaches, the model space is speci ed by a parameterized di erential equation, and identi cation selects numerical parameter values so that simulation of the model best mat...
متن کامل