# Chapter one

Mathematical skills

## Section six: Inverse Laplace Transforms

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

This section focuses on inverse Laplace transforms. We give a pragmatic approach for an engineer who needs to use the tool and gain insight rather than giving a rigorous development.

In practice, we recommend you use a look-up table once you have sufficient understanding and/or use computer tools to do any numerical computations.

### 1. Introduction

Introduces the concept of the look-up table as a simple short cut method. Gives a number of concise illustrations which are elaborated in the resources to follow.

A talk through video is on YouTube. Read the notes (PDF, 584KB).

### 2. Partial fractions and expansion methods

Partial fractions are needed to express the transform in the forms in the look-up table. This is to solve the underlying signal. Some simple illustrations are given.

A talk through video is on YouTube. Read the notes (PDF, 458KB).

### 3. Partial fractions and cover-up rule

The cover-up rule allows the residues of simple partial fractions to be computed very fast, enabling insight.

A talk through video is on YouTube. Read the notes (PDF, 517KB).

### 4. Partial fractions with quadratic factors

This section is for completeness and understanding of the topic. Handling quadratic factors would rarely be a paper and pen exercise in practice.

A talk through video is on YouTube. Read the notes (PDF, 792KB).

### 5. Solving ODEs

These resources demonstrate the solutions of ODEs using Laplace methods. The second video also extends to more authentic scenarios such as feedback.

Two talk through videos are on YouTube (video one, video two). Read the notes (PDF, 813KB).

### 6. Using MATLAB

These resources demonstrate how the MATLAB tool can be used to find inverse Laplace solutions in many alternative forms.

**Analytic solutions**

A talk through video is available on YouTube. Read the notes (PDF, 863KB).

**Numerical solutions**

A talk through video is available on YouTube. Read the notes (PDF, 856KB).

**Partial fractions**

A talk through video is available on YouTube. Read the notes (PDF, 732KB).

### 7. Videos with worked solutions

This section poses questions and suggests viewer tries themselves before continuing to see the solution.

**Laplace transforms**

A talk through video is on YouTube.

**Inverse Laplace**

A talk through video is on YouTube.

**Final value theorem**

A talk through video is on YouTube.

**Behaviour**

A talk through video is on YouTube.

### 8. Tutorial sheets

Introductory tutorial sheet with outline solutions.

Download the tutorial sheet (PDF, 540KB).