Numerical Methods for Engineers (saylor.org)

Numerical Methods for Engineers (saylor.org)
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Numerical Methods for Engineers (saylor.org)
Numerical methods have been used to solve mathematical expressions of engineering and scientific problems for at least 4000 years. Such methods apply numerical approximation in order to convert continuous mathematical problems (for example, determining the mechanical stress throughout a loaded truss) into systems of discrete equations that can be solved with sufficient accuracy by machine. This course will provide you with an introduction to several of those numerical methods which you may then find opportunity to practice later in the curriculum.

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Please note: this legacy course does not offer a certificate and may contain broken links and outdated information. Although archived, it is open for learning without registration or enrollment.

Let us consider a few examples in which numerical methods might offer a net benefit to the engineer.

1. A new problem results in a numerical expression involving logarithmic, polynomial, and trigonometric terms. Not only do you wish to find the zeros of this particular expression, but you need to find the zeros of 100,000 similar expressions per day. Hence, you need an automated procedure for doing such that can be implemented on a computer.

2. Consider the manufacture of an oddly shaped part out of very expensive materials. The manufacturing process involves cutting, heating, cooling, and bending of the part. In order to optimize the manufacturing process, you wish to understand the details of the heating, cooling, and shaping processes. Modeling of these processes requires consideration of transient heat flow in three dimensions and transient deformation in three dimensions; further, the part is an oddly-shaped domain. One approach to improve understanding of the process is to numerically simulate the heating, cooling, and deformation; this simulation involves the numerical solution of systems of partial differential equations.

3. You have an empirical model of a process that predicts temperature and oxygen content as a function of time. The empirical model involves four parameters that need to be estimated from measurements under a variety of conditions. There are several nonlinear regression or optimization algorithms which might be suitable for this task.

As you progress through the curriculum, you will encounter many more problems which might benefit from analysis by numerical methods. Indeed, you may gain much insight by applying numerical methods to a variety of problems that you encounter later in the curriculum.

This course will consist of ten units: an introduction to the numerical properties of machine computations; numerical differentiation; solution of non-linear equations; linear algebra (or the solution of systems of linear equations); interpolation; regression and optimization; numerical integration; numerical solution of ordinary differential equations; numerical solution of partial differential equations; and some miscellaneous numerical tools. Each unit is accompanied by lectures, readings, and exercises.

Upon successful completion of this course, the student will be able to:

- Quantify absolute and relative errors.

- Distinguish between round-off and truncation errors.

- Interconvert binary and base-10 number representations.

- Define and use floating-point representations.

- Quantify how errors propagate through arithmetic operations.

- Derive difference equations for first and second order derivatives.

- Evaluate first and second order derivatives from numerical evaluations of continuous functions or table lookup of discrete data.

- Describe situations in which numerical solutions to nonlinear equations are needed

- Implement the bisection method for solving equations.

- List advantages and disadvantages of the bisection method

- Implement both Newton-Raphson and secant methods.

- Describe the difference between Newton-Raphson and secant methods.

- Demonstrate the relative performance of bisection, Newton-Raphson, and secant methods.

- Define and identify special types of matrices.

- Perform basic matrix operations.

- Define and perform Gaussian elimination to solve a linear system.

- Identify pitfalls of Gaussian elimination.

- Define and perform Gauss-Seidel method for solving a linear system.

- Use LU decomposition to find the inverse of a matrix.

- Define and perform singular value decomposition; explain the significance of singular value decomposition.

- Define interpolation.

- Define and use direct interpolation to approximate data and find derivatives.

- Define and use Newton’s divided difference method of interpolation.

- Define and use Lagrange and spline interpolation.

- Define regression.

- Perform linear least-squares regression and nonlinear regression.

- Derive and apply the trapezoidal rule and Simpson’s rule of integration.

- Distinguish Simpson’s method from the trapezoidal rule.

- Estimate errors in trapezoidal and Simpson integration.

- Derive and apply Romberg and Gaussian quadrature for integration.

- Define and distinguish between ordinary and partial differential equations.

- Implement Euler’s methods for solving ordinary differential equations.

- Investigate how step size affects accuracy in Euler’s method.

- Implement and use the Runge-Kutta 2nd order method for solving ordinary differential equations.

- Apply the shooting method to solve boundary-value problems.

- Define Fourier series and the Fourier transform.

- Find Fourier coefficients for a given data set or function and domain.

- Describe the finite element method for one-dimensional problems.



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Free Course

MOOC List is learner-supported. When you buy through links on our site, we may earn an affiliate commission.