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https://github.com/dayyass/calculus-of-variations

Project on solving the Calculus of variations problems using symbolic mathematics (2018).
https://github.com/dayyass/calculus-of-variations

calculus calculus-of-variations derivatives euler-lagrange-equation minimize-functionals optimization

Last synced: 2 months ago
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Project on solving the Calculus of variations problems using symbolic mathematics (2018).

Host: GitHub
URL: https://github.com/dayyass/calculus-of-variations
Owner: dayyass
License: mit
Created: 2019-02-23T19:09:26.000Z (almost 6 years ago)
Default Branch: master
Last Pushed: 2024-02-02T18:14:19.000Z (11 months ago)
Last Synced: 2024-09-18T02:58:16.296Z (4 months ago)
Topics: calculus, calculus-of-variations, derivatives, euler-lagrange-equation, minimize-functionals, optimization
Language: Python
Homepage: https://it-mm.rea.ru/uploads/arhiv/2019/sertificat/299.pdf
Size: 1 MB
Stars: 16
Watchers: 1
Forks: 5
Open Issues: 3
Metadata Files:
- Readme: README.md
- License: LICENSE

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README

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### About

My bachelor project on solving the **Calculus of variations** problems using symbolic mathematics (2018).

I [participated](https://it-mm.rea.ru/uploads/arhiv/2019/sertificat/299.pdf) with this project at the IX International Scientific and Practical Conference named after A.I. Kitov "Information Technologies and Mathematical Methods in Economics and Management" ([link](https://it-mm.rea.ru/eng)).


My [presentation](presentation.pdf) about this project.

### What is Calculus of variations

According to [Wikipedia](https://en.wikipedia.org/wiki/Calculus_of_variations):


The **calculus of variations** is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.


### Simplest problem

The definition above might seem quite difficult to understand, so let's consider the simplest problem:










This is the task of finding an extrema in continuously differentiable functions space



, where:

 - functional to maximize/minimize,


 - fixed closed line segment,


 - continuously differentiable function,


 - function derivative,


 - boundary conditions.


#### Example










### Euler–Lagrange equation

Functions that maximize or minimize functionals may be found using the [*Euler–Lagrange equation*](https://en.wikipedia.org/wiki/Euler–Lagrange_equation) of the calculus of variations:

, where

 - partial derivative of  w.r.t. ,


 - partial derivative of  w.r.t. .


Arbitrary constants arising when solving this differential equation, find them from the given boundary conditions.

### List of supported problems

- [Simplest problem](https://github.com/dayyass/calculus-of-variations/wiki/Simplest-problem)

- [Boltz problem](https://github.com/dayyass/calculus-of-variations/wiki/Boltz-problem)

- [Isoperimetric problem](https://github.com/dayyass/calculus-of-variations/wiki/Isoperimetric-problem)

- [Higher derivatives problem](https://github.com/dayyass/calculus-of-variations/wiki/Higher-derivatives-problem)

- [Multidimensional problem](https://github.com/dayyass/calculus-of-variations/wiki/Multidimensional-problem)

More about each task in [project wiki](https://github.com/dayyass/calculus-of-variations/wiki).

### Usage

First, install the package:

```

pip install calculus-of-variations

```

Usage for example above:

```python3

import calculus_of_variations

solver = calculus_of_variations.SimplestSolver(

    L="x_diff ** 2 + t * x",

    t0="0", t1="1",

    x0="0", x1="0",

)

solver.solve()

# integral from 0 to 1 of (x_diff ** 2 + t * x)dt -> extr

# x(0) = 0

# x(1) = 0

# general_solution: C1 + C2*t + t**3/12

# coefficients: {C1: 0, C2: -1/12}

# particular_solution: t**3/12 - t/12

# extrema_value: -1/180

```

Other cases:

```python3

# Simplest problem

solver = calculus_of_variations.SimplestSolver(

    L="x_diff ** 2",

    t0="0", t1="1",

    x0="0", x1="1",

)

solver.solve()

# Boltz problem

solver = calculus_of_variations.BoltzSolver(

    L="x_diff ** 2 + 2 * x",

    l="x_t0 ** 2",

    t0="0", t1="1",

)

solver.solve()

# Isoperimetric problem

solver = calculus_of_variations.IsoperimetricSolver(

    f0="x_diff ** 2",

    t0="0", t1="1",

    x0="0", x1="1",

    f_list="x",

    alpha_list="0",

)

solver.solve()

# Higher derivatives problem

solver = calculus_of_variations.HigherDerivativesSolver(

    n="2",

    L="x_diff_2 ** 2",

    t0="0", t1="1",

    x0="0", x1="0",

    x0_array="0",

    x1_array="1",

)

solver.solve()

# Multidimensional problem

solver = calculus_of_variations.MultidimensionalSolver(

    L="x1_diff**2 + x2_diff**2",

    t0="0", t1="1",

    x1_0="0", x1_1="1",

    x2_0="0", x2_1="1",

)

solver.solve()

```

For specific examples see [**examples.sh**](https://github.com/dayyass/calculus-of-variations/blob/master/examples.sh).


List of **allowed functions** that you can use as parameters: [*link*](https://github.com/dayyass/calculus-of-variations/wiki/Allowed-functions).

### Web-interface

The project supports simple web-interface for solving problems.

You can specify **host** (`--host`) and **port** (`--port`) (default values: `host: 127.0.0.1` and `port: 8050`):

```

# Simplest problem

python -m web_interface.simplest_problem_dash --host "127.0.0.1" --port 8050

# Boltz problem

python -m web_interface.boltz_problem_dash --host "127.0.0.1" --port 8050

# Isoperimetric problem

python -m web_interface.isoperimetric_problem_dash --host "127.0.0.1" --port 8050

# Higher derivatives problem

python -m web_interface.higher_derivatives_problem_dash --host "127.0.0.1" --port 8050

# Multidimensional problem

python -m web_interface.multidimensional_problem_dash --host "127.0.0.1" --port 8050

```

You can also launch web-interface using docker.


To build docker image run:

```

docker image build -t calculus_of_variations .

```

You can also pull image from [Docker Hub](https://hub.docker.com/r/dayyass/calculus_of_variations):

```

docker pull dayyass/calculus_of_variations

```

To start docker container run (example for **simplest_problem_dash**):

```

docker container run -d -p 8050:8050 --name calculus_of_variations calculus_of_variations python -m web_interface.simplest_problem_dash --host 0.0.0.0 --port 8050

```

To access web-interface go to `http://localhost:8050`

### Tests

To launch [**tests**](https://github.com/dayyass/calculus-of-variations/tree/master/tests) run the following commands:


`python -m unittest discover`

To use [**pre-commit**](https://pre-commit.com) hooks run:


`pre-commit install`

To measure [**code coverage**](https://coverage.readthedocs.io) run the following commands:


`coverage run -m unittest discover && coverage report -m`

## Requirements

Python >= 3.6

### Citation

If you use **calculus-of-variations** in a scientific publication, we would appreciate references to the following BibTex entry:

```bibtex

@misc{dayyass2018variations,

    author       = {El-Ayyass, Dani},

    title        = {Calculus of Variations problems solving using symbolic mathematics},

    howpublished = {\url{https://github.com/dayyass/calculus-of-variations}},

    year         = {2018}

}

```