Reduction for Ordinary and Partial Differential Equations by Using Lie Group

In this publication, We have done Lie group theory is applied to reduce the order of ordinary differential equations (ODEs) with 1-parameter and reduce a PDEs to ODEs . Also set up algorithm to solve ODEs and PDEs to obtain the exact solution .


Introduction
The Lie symmetries and their various generalizations have become an inseparable part of the modern physical description of wide range of phenomena of nature from quantum physics to hydrodynamics.Such success of a purely mathematical theory of continuous groups developed by Lie and Engel in 19th century [8] is explained by the remarkable fact that the overwhelming majority of mathematical models of physical, chemical, and biological processes possess nontrivial Lie symmetry.Symmetries can be applied to reduce the order of an ODE .[9] proposed a generalization of Lie's method called the non-classical method of group-invariant solutions.It is well known that the classical Lie symmetry method of point transformations is often used for reducing the number of independent variables in partial differential equation to obtain ordinary one can obtain partial solutions of the equation under study [1], [2], [3].For partial differential equations , Lie symmetries are used to reduce the equation to an ODE via appropriate reduction variables.These can lead to the group-invariant solutions [4] of PDEs which are so important today [5], [6].

2.Reduction of Order for ODE,[1]
Consider the n th order ODE as:

3.Applications for ODEs
In this part, we give algorithm to established some problems for ODEs how computed reduction order by using symmetries,

1-Algorithm
Step1: Write the vector field of the form: Step2: Find the n th prolongation for equation (2.1) in style: ...
Step3: Applying the prolongation X [n] on equation (2.1) found in Step2 as:
Step5: Replacing Step7: Reduce order of ODE by using new vector field found in Step6.Step8: Appling the n th prolongation for the new vector field .
Step9: Find invariance using [] 0 n X I  whose characteristic system is : We need the 2-prolongation of (3.2) then: Applying equation (3.3) in equation (3.1) given as: and  from definition: Now , substituting 2 2 y y x y y by y we find: By separation of the coefficient , , , , , y y y y y y y y y and y y The 2 nd prolongation of X 1 is : [ We find invariance using  3.21) we result two invariance of X [2] given by: First write the vector field given as: ( , ) ( , ) To compute equation (3.32) we need the 2 nd prolongation given by the following: XX y y and   from definition are: .39) By separation of the coefficient of the above system: The generator given as : The 2 nd prolongation of X 1 is : Solving the characteristic system [2] 0 X I  is given the differential invariants .x y we get : Multiplying equation (3.57) by u 2 we find: Then the equation (3.59) reduced 1 s t order ODE .

4.Reduction of Order for PDEs
In this part, established reduction order for PDEs the technique is basically the same as for ODEs Now, consider the second order of PDE is : The point symmetries of the form: * If and only if is the 2 th extension of the infinitesimal generator X introduced in the following: ( Then G is some function of the invariants write as: 12 ( , ), ( , , ),..., ( , , , , , , , ) Which implies that: 0 XI ... (4.5)This condition (4.5) can be expressed by using the characteristic of the group which is: From condition (4.5) , the surface u = u(t,x) is invariant provided that: 0 Q  when u = u(t,x) ... (4.7)In other words from [7], every invariant solution satisfies the invariant surface condition write as: Now , suppose that  and  are not both zero then the invariant surface condition is a 1-order quasilinear PDE can be solved by the method of characteristic equation are:

5.Applications for PDEs
In the following, we confirmed algorithm for reducing order of PDE by using symmetry .Moreover calculated some examples about it.