# Taylor Series Method

**Taylor series Method : **Taylor series method is the fundamental numerical method for the solution of the initial value.

Let us consider the initial value problem

...................1.1

Expanding y(x) in Taylor series about any point x_{i} ,with the Lagrange form of remainder,

we obtain

................1.2

where 0 < θ < 1, x ∈ [x_{0}, b] and b is the point up to which the solution is required.

We denote the numerical solution and the exact solution at x_{i} by yi and y(x_{i}) respectively.

Now, consider the interval [x_{i}, x_{i 1}]. The length of the interval is h = x_{i 1} – x_{i}

Substituting x = x_{i 1} in (1.2), we obtain

Neglecting the error term, we obtain the Taylor series method as

......................1.3

Note that Taylor series method is an explicit single step method.

The truncation error of the method is given by

......................1.4

we say that the Taylor series method (1.3) is of order p.

For p = 1, we obtain the first order Taylor series method as

..............1.5

This method is also called the Euler method. The truncation error of the Euler’s method is

..................1.6

Sometimes, we write this truncation error as

..................1.7

Since,

...................1.8