Partial Differential Equations

Here we be shown some types of partial differential equations such as transport equation, heat equation, wave equation, etc. The illustration of each equations is presented in one- and two-dimensional form. This page is created for Calculus class Week 10 Partial Derivatives, Informatics, Telkom University.

1. Transport Equations:

One-dimensional transport equation:

$https://latex.codecogs.com/svg.image?\huge \frac{\partial u(t,x)}{\partial t} + a \frac{\partial u(t,x)}{\partial x} =0$

Illustration transport equation (source: [1])

Two-dimensional transport equation:

2. Heat Equations

One-dimensional heat equation

$https://latex.codecogs.com/svg.image?\huge \frac{\partial u(t,x)}{\partial t}= D \frac{\partial^2 u(t,x)}{\partial x^2}$

Two-dimensional heat equation

3. The Wave Equation

One-dimensional wave equation

Two-dimensional wave equation

Laplace’s Equation

References:

[1] https://en.wikipedia.org/wiki/Upwind_scheme

[2] https://upload.wikimedia.org/wikipedia/commons/6/6f/GaussianUpwind2D.gif?20161127062043

[3] https://commons.wikimedia.org/wiki/File:Heat_equation_numerical_solution.gif

[4] https://gfycat.com/liquidsociableladybug-heat-equation-dirchlet-matlab

[5] https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/02%3A_The_Classical_Wave_Equation/2.01%3A_The_One-Dimensional_Wave_Equation

[6] https://commons.wikimedia.org/wiki/File:2D_Wave.gif

[7] https://www.physics.uoguelph.ca/chapter-10-laplaces-equation

1. Transport Equations:

2. Heat Equations

3. The Wave Equation

Laplace’s Equation

References:

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