Thermoelastic phenomena in the majority of practical engineering applications are adequately simulated with the classical Fourier heat conduction equation, there is still few important body of problems that require due consideration of thermoelastic treatment. It is well known that ‘Changes in temperatures causes thermal effects on materials’. Some of these thermal effects include thermal stress, strain, and deformation. The first effect ‘thermal stresses’ was considered during our study & serious attention has been paid to the thermoelasticity theories in solving thermoelastic problems. To understand this book one must have knowledge of Residue Theory, Differential Equation, Analysis, Integral Transform and Thermoelasticity. The investigations in the present volume are based on the research work of Roychoudhury, Tanigawa, Noda, Deshmukh, Wankhede and Nowacki where they have studied the direct and inverse problems of thermoelasticity on homogeneous materials.At last, it may be mentioned that the results obtained here are mathematical and vary from both the physical and practical point of view.