ABSTRA CT

ln this thesis, the extended application of separation of variables method is discussed for the large class of "Partial Differential Equations" (PDEs) containing mixed derivatives. The applicability of the method can be possible with some modifications in classical separation of variables method by differentiating the non-separable equation either by the time coordinate t or by the spatial coordinate x . In view of the particular application, applicability of the method has been extended to a linear homogeneous string-like equation that describes the vibration of a conveyor belt system. The exact analytical solutions of such problems have been a challenging subject. The classical separation of variables method cannot exhibit a solution as it is incapable to deal the mixed derivative term s. For example, the term Coriol is acceleration 2V uxt is a mixed-derivative term presented in governing equation of motion under consideration, or the material damping OUxxt or OUxxxxt in case of a string or a beam, respectively. The general initial and the fixed boundary conditions have been considered for a proposed string-like equation. From a physical view point, the problem represents a simple mathematical model to describe vibrations (transversal) of a conveyor belt system. The axial velocity of the string is assumed to be positive, constant and small compared to velocity of wave. The exact analytical solution is obtained by employing modified or extended version of separation of variables method. Furthermore, with proposed modification in separation of variables method, applicability of method may be extended to the large class of PDEs. It has been shown that the eigenfunctions are complicated corresponding to complete valued eigenvalues. To some numerical error these eigenfunctions are orthogonal to one another. The obtained results have been shown for given modes for certain given values of the parameters where it can easily be seen that the solutions are stable in the initial time evolution, as the time increases there is arising truncation error in the
solution curves for fixed modes.