1.1      Definition

 Truncation errors are those that result from using an approximation in place of an exact mathematical procedure.

 Example           Approximation of the derivative

 One of the most important methods used in numerical methods to approximate mathematical functions is: Taylor series 

1.2      Using Taylor series to estimate truncation errors

Let us see how Taylor series may be used to estimate truncation errors.

How to determine the error introduced by using this formulation to compute the derivative instead of using the real mathematical definition?

Using a Taylor series truncated at the first-order:

Therefore,

We have now an estimate of the truncation error: 

Truncation error =

So the order of the error due to the formulation used to compute the derivative is h.

2.1      Definition

An equation that consists of derivatives is called a differential equation.  Differential equations have applications in all areas of science and engineering.  Mathematical formulation of most of the physical and engineering problems lead to differential equations.  So, it is important for engineers and scientists to know how to set up differential equations and solve them.

2.2      Euler’s Method:

Numerically approximate values for the solution of the initial-value problem 𝑦 ′ = 𝐹(𝑥, 𝑦), 𝑦 𝑥0 = 𝑦0, with step size ℎ, at 𝑥_𝑛 = 𝑥_𝑛−1 + ℎ, are

𝑦_𝑛 = 𝑦_𝑛−1 + ℎ ∙ 𝐹(𝑥_𝑛−1, 𝑦_𝑛−1)

Example:

2.3      Runge-Kutta 2^nd order:

The Runge-Kutta 2nd order method is a numerical technique used to solve an ordinary differential equation of the

Only first order ordinary differential equations can be solved by using the Runge-Kutta 2nd order method. In other sections, we will discuss how the Euler and Runge-Kutta methods are used to solve higher order ordinary differential equations or coupled (simultaneous) differential equations.

3.1     Newton-Raphson method:

Let x₀ be an initial guess to the root α of f(x) = 0. Let h is the correction i.e. α = x₀ + h. Then f(α) = 0 implies f(x₀ + h) = 0. Now assuming h small and f twice continuously differentiable, we find

3.2      TRAPEZOIDAL RULE

Trapezoidal rule is based on the Newton-Cotes formula that if we approximate the integrand by an n^th order polynomial, then the integral of the function is approximated by the integral of that n^th order polynomial.

The trapezoidal rule works by approximating the region under the graph of the function f(x){\displaystyle f(x)} as a trapezoid and calculating its area. It follows that

{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right]}

3.3      SIMPSON’S 1/3^RD RULE

Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration.  Simpson’s 1/3rd rule is an extension of Trapezoidal rule where the integrand is approximated by a second order polynomial.

Since for Simpson’s 1/3^rd Rule, the interval  is broken into 2 segments, the segment width is

Hence the Simpson’s 1/3^rd rule is given by

Since the above form has 1/3 in its formula, it is called Simpson’s 1/3^rd Rule.

4.1      False position method

A shortcoming of the bisection method is that in dividing the interval from x_l to x_u into equal halves, no account is taken of the magnitude of f(x_i) and f(x_u). Indeed, if f(x_i) is close to zero, the root is more close to x_l than x₀.

The false position method uses this property:

A straight line joins f(x_i) and f(x_u). The intersection of this line with the x-axis represents an improved estimate of the root. This new root can be computed as:

Giving                         This is called the false-position formula

4.2 Secant method

Here we don’t insist on bracketing of roots. Given two initial guess. Given two approximation x_n−1, x_n, we take the next approximation x_n+1 as the intersection of line joining (x_n−1, f(x_n−1)) and (x_n, f(x_n)) with the x-axis. Thus x_n+1 need not lie in the interval [x_n−1, x_n]. If the root is α and α is a simple zero, then it can be proved that the method converges for initial guess in sufficiently small neighborhood of α.

Thanks,

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