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What Is Applied Mathematics
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What Is Applied Mathematics. Applied mathematics is a broad subject area dealing with those problems that come from the real world. Applied mathematics deals with all the stages for solving these problems, namely:

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  • October 4, 2021
  • 76
  • 2021/2022
  • Class notes
  • Jordi-lluis figueras
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Applied Mathematics.

Jordi-Lluı́s Figueras

October 9, 2014

,ii

,Contents

Some words v

1 What is Applied Mathematics. 1


I Mathematical modelling 3
2 Dimensional analysis and Scaling. 7
2.1 Dimensions and units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Laws and unit free laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Pi theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Example 1: Atomic bomb. . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Example 2: Heat transfer problem. . . . . . . . . . . . . . . . . . . . 13
2.4 Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


II Analytical methods. 15
3 Perturbation methods. 17
3.1 Regular perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Poincaré-Lindstedt method. . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Big O and little o notation. . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Singular perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Boundary layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 The WKB approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Calculus of variations. 27
4.1 Variational problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Necessary conditions for extrema. . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Normed linear spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Derivatives of functionals. . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 The simplest problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Higher derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iii

, iv CONTENTS

4.4.2 Several functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.3 Natural boundary conditions. . . . . . . . . . . . . . . . . . . . . . . 33
4.5 More problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Dynamical systems. 35
5.1 Discrete dynamical systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.1 Equilibria and stability. . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Continuous dynamical systems. . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.1 Vector fields and phase space portraits. . . . . . . . . . . . . . . . . . 38
5.2.2 Stationary orbits and stability. . . . . . . . . . . . . . . . . . . . . . . 39
5.2.3 Periodic orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Chaotic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Introduction to partial differential equations. 43
6.1 Some examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Linearity and superposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Laplace’s equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 Evolution problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.6 Eigenfunction expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Sturm-Liouville problems. 51

8 Theory of transforms. 53
8.1 Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 Fourier transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.3 Other transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

9 Integral equations. 59
9.1 Volterra equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9.2 Fredholm equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.2.1 Fredholm equations with degenerate kernel. . . . . . . . . . . . . . . 62
9.2.2 Symmetric kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.3 Perturbation methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Appendices 67

A Solving some ODEs. 69
A.1 First order linear ODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.2 Second order linear ODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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