### Integral Transform Methods for Inverse Problem of Heat Conduction with Known Boundary of SemiInfinite Hollow Cylinder and Its Stresses

Abstract: Three dimensional inverse transient thermoelastic problem of a semi-infinite hollow cylinder is considered within the context of the theory of generalized thermoelasticity. The lower surface, upper surface and inner surface of the semiinfinite hollow cylinder occupying the space D={(x,y,z)E R^{3}: a≤(x^{2}+y^{2})^{1/2} ≤b, 0≤z≤∞} known boundary conditions. Finite Marchi-Zgrablich transform and Fourier sine transform techniques are used to determine the unknown temperature gradient, temperature distribution, displacement and thermal stresses on outer curved surface of a cylinder. The distribution of the considered physical variables are obtained and represented graphically.

Keywords: Thermoelastic problem, semi-infinite hollow cylinder, Thermal Stresses, inverse problem, Marchi-Zgrablich transform and Fourier sine transform.

I. INTRODUCTION

Khobragade et al. [1, 5-11] have investigated temperature distribution, displacement function, and stresses of a thin as well as thick hollow cylinder and Khobragade et al. [2] have established displacement function , temperature distribution and stresses of a semi-infinite cylinder. Yoon Hwan Choi et. al. [16] discussed the temperature distributions of the heated plate investigated with the condition that the line heating process was automatic. The temperature variations were also investigated with the changes of those three variables. The numerical results showed that the peak temperature decreased as the moving velocity of the heating source increased. It also revealed that the peak temperatures changed linearly with the changes of the heating source. Xijing Li, Hongtan Wu, Jingwei Zhou and Qun He [15] studied one-dimensional linear inverse heat problem. This ill-posed problem is replaced by the perturbed problem with a non localized boundary condition. After the derivation of its closed-form analytical solution, the calculation error can be determined by the comparison between the numerical and exact solutions.

Michael J. Cialkowski and Andrzej Frąckowiak [12] presented analysis of a solution of Laplace equation with the use of FEM harmonic basic functions. The essence of the problem is aimed at presenting an approximate solution based on possibly large finite element. Introduction of harmonic functions allows reducing the order of numerical integration as compared to a classical Finite Element Method. Numerical calculations confirm good efficiency of the use of basic harmonic functions for resolving direct and inverse problems of stationary heat conduction.