Numerical Methods | Engineering Mathematics - Civil Engineering (CE) PDF Download

Truncation errors and the taylor series

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.
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.

Numerical solution of ordinary differential equations (ODE)

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. 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
   𝑦𝑛 = 𝑦_{𝑛 − 1} + ℎ ∙ 𝐹(𝑥_{𝑛 − 1}, 𝑦_{𝑛 − 1})
   Example:
   
3. Runge-Kutta 2nd order
   The Runge-Kutta 2nd order method is a numerical technique used to solve an ordinary differential equation of the
   dy / dx = f(x, y), y(0) = y₀
   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.
   Example: Rewrite
   dy / dx + 2y = 1.3e^-x, y(0) = 5
   in
   dy / dx = f(x, y, y(0) = y₀ form.
   Solution: dy / dx + 2y = 1.3e^-x, y(0) = 5
   dy / dx = 1.3e^-x -2y, y(0) = 5
   In this case
   f(x, y) = 1.3e^-x - 2y

Numerical Integration

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
   
2. Trapezoidal rule
   Trapezoidal rule is based on the Newton-Cotes formula that if we approximate the integrand by an nth order polynomial, then the integral of the function is approximated by the integral of that nth order polynomial.
   The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that
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 1/3^rd 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.

Roots of Equations

1. False-Position Formula
   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
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.