Abstract
This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically.
Highlights
The Riccati differential equation is named after the Italian nobleman Count Jacopo Francesco Riccati (1676-1754)
An analytic solution of the non-linear Riccati equation reached see [2] using A domain Decomposition Method. [5] used differential Transform Method(DTM) to solve Riccati differential equations with variable co-efficient and the results are compared with the numerical results by (RK4) method.[14] solved Riccati differential equations by using (ADM) method and the numerical results are compared with the exact solutions
We will consider in our test only a first order differential equation and the numerical solution to second-order and higher-order differential equations is formulated in the same way as it is for first-order equations
Summary
The general solution y (x ) of equation (1), we suppose s ( x ) is a particular solution for (1) and assume the general solution is y It's clear equation (5) is linear equation from first order after solution we obtain y (x ) and by choosing particular s ( x ) via (Trial and Error) we obtain the general solution y. A symmetry group of differential equation is termed a group admitted by this equation. The generator X of the group admitted by differential equation is termed an operator admitted by this equation. An essential feature of a symmetry group G is that it conserves the set of solutions of the differential equation admitting this group. It may happen that some of the integral curves are individually unaltered under group G. Such integral curves are termed invariant solutions
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