Source Of Complexity

Lesson

Introduction
Matter of scale
Variability
Non-linearity
Introduction to differential equations
- What is a differential equation
- Classification of PDEs
- Existence and Uniqueness
Governing equations
Different approaches
Exercises
To go further
Conclusion

What is a differential equation

The field of differential equations is very rich and contains a variety of different species. However, there is a basic feature to all problems defined by differential equation: the equation relates a function to its derivatives in such a way that the function itself can be determined.

The differential equations can be subdivided into Partial Differential Equations (PDEs) and Ordinary differential Equations (ODE). In PDEs we deal with partial derivatives, whereas ODEs involve only derivatives with respect to one variables. We can give some typical ordinary differential equations:

$u^{'}(t) = u(t) \quad \quad \quad \quad \quad \quad (a)\\ u^{'}(t) = u^{2}(t) \quad \quad\quad \quad \quad \quad (b)\\ u^{''}(t) = u(t) + \sin(t) \cos(t) \quad (c)\\ u^{''}(x) + u^{'}(x) = x^{2} \quad \quad \quad \quad (d)\\ u^{''''}(x) = \sin(x) \quad \quad \quad \quad \quad (e)\\$

The above equation are ODEs of different order (first, second and fourth).

We can also give some typical partial differential equation:

$u_{t}(x,t) = u_{xx}(x,t) \quad \quad \quad \quad \quad\quad(f) \\ u_{xx}(x,y) + u_{yy}(x,y) = 0 \quad\quad\quad\quad(g) \\ u_{t}(x,t) + (x^2 + t^2)u_{x}(x,t) = 0 \quad \quad(h) \\ \quad\\ \text{Where} \\ u_{t} = \frac{\partial u}{\partial t}, \quad u_{t} = \frac{\partial^2 u}{\partial x^2}, \quad \text{and so on}$

For the most environmental complex systems, their dynamics is described using PDEs.

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