Numerical Method Study Notes for Civil Engineering

•  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

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)

• 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.

Euler’s Method

• Numerically approximate values for the solution of the initial-value problem y′ = F(x,y), yx_o = y_o , 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:

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.

Numerical Integration

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

Trapezoidal Method

• 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]}

Simpsons 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 [a,b] is broken into 2 segments, the segment width is h=b-a/2
• 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 Equation

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:

Here

• This is called the false-position formula

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 the 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 the sufficiently small neighborhood of α.

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