In this thesis, the conjugate gradient method will be extended to solve symmetric and non-symmetric nonlinear equations. Several variations of the conjugate gradient method are introduced and investigated. To extend the conjugate gradient method for nonlinear problems, there are four methods discussed; the pre-conditioning conjugate gradient method, the pre-conditioning conjugate gradient method for better initial guess, the fixed point conjugate gradient method and the fixed point conjugate gradient method for better initial guess. The four methods are discussed in detail and tested on various numerical examples. The linear numerical examples include a two dimensional structural frame problem, a Hilbert matrix system and an aero-acoustic application. The nonlinear examples include an artificially constructed set of nonlinear equations and analysis of a nonlinear transient heat transfer problem with a moving heat source. The artificially constructed nonlinear examples include symmetric as well as non-symmetric examples. The heat transfer analysis problem is non-symmetric. The direct extension of the conjugant gradient method is working properly for the nonlinear problems. However, none of the variations proposed provide convergence to all of the nonlinear example problems. Out of all the proposed methods, the fixed point conjugate gradient method holds the most potential which converged to correct solutions for all the examples except the heat transfer analysis problem. If converged, the method was also found to be more efficient than the standard conjugate gradient method. Further investigation of the fixed point conjugate gradient method is needed in order to make the method more robust
