Optimal System Of The Zero-coupon The Bond Pricing Model And The Symmetry Reduction

Lie group theory is applied to differential equations occurring as a mathematical model in financial problems. The Zero-coupon bond pricing (ZCB for short) model is studied. Its one-paramctcr Lie point symmetries and corresponding group of adjoint representations are obtained. An optimal system of one-dimensional subalgebra is derived and used to construct distinct families of special closed-form solutions of the equation. The method above is also used to study the (2+1)-dimensional nonlinear Sine-Gordon equation, obtaining the corresponding optimal system and some symmetry reductions.An outline of the paper is as follows.In Chapter 2 we determine the Lie symmetries admitted by the ZCB model occurring as a mathematical model in financial problems :and give the communication relations of the corresponding Lie symmetries.In Chapter 3 we construct the adjoint representations of the ZCB Lie group and use it to perform the classification of one-dimensional subalgebras of the ZCB Lie algebra.The optimal system of the obtained symmetry groups is also discussed and many interesting group-invariant solutions are obtained.In Chapter 4 the same method discussed above is used to study (2+1)-dimensional nonlinear sine-Gordon equation,getting some lower dimensional differential equations.Finally in Chapter 5 we conclude.