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# <span style="color:rgb(213,80,0)">Numerical Methods with Applications</span>


[![View on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://www.mathworks.com/matlabcentral/fileexchange/111490-numerical-methods-with-applications) or [![Open in MATLAB Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=README.mlx)

![MATLAB Versions Tested](https://img.shields.io/endpoint?url=https%3A%2F%2Fraw.githubusercontent.com%2FMathWorks-Teaching-Resources%2FNumerical-Methods-with-Applications%2Frelease%2FImages%2FTestedWith.json)

**Curriculum Module**

_Created with R2021b. Compatible with R2021b and later releases._


[![View on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://www.mathworks.com/matlabcentral/fileexchange/111490-numerical-methods-with-applications) or [![Open in MATLAB Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=README.mlx)

![MATLAB Versions Tested](https://img.shields.io/endpoint?url=https%3A%2F%2Fraw.githubusercontent.com%2FMathWorks-Teaching-Resources%2FNumerical-Methods-with-Applications%2Frelease%2FImages%2FTestedWith.json)

**Curriculum Module**

_Created with R2021b. Compatible with R2021b and later releases._

# Information

This curriculum module contains interactive [MATLAB® live scripts](https://www.mathworks.com/products/matlab/live-editor.html) that teach fundamental concepts and basic terminology related to designing and implementing numerical methods related to interpolation, numerical integration and differentiation, and numerical solutions to ordinary and partial differential equations.
Expand Down Expand Up @@ -53,30 +53,30 @@ MATLAB® is used throughout. Tools from the Symbolic Math Toolbox™ are used fr

# Scripts
## [**Interpolation.mlx**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=Interpolation.mlx)
| <img src="Images/hand.png" width="171" alt="hand.png"> <br> | **In this script, students will...** <br> <br>- define interpolation and explain how it is different from regression and extrapolation <br> <br>- demonstrate multiple solutions to an interpolation problem, even when interpolating by polynomials <br> <br>- implement three different interpolation functions <br> | **Applications** <br> <br>- Drawing a smooth rendition of their hand <br> <br>- Tracking the path of a cyclone <br> **Scaffolded Template Scripts** <br> [<samp>linInterp.m</samp>](./FunctionLibrary/linInterp.m) <br> [<samp>pcHInterp.m</samp>](./FunctionLibrary/pcHInterp.m) <br> [<samp>pcSpline.m</samp>](./FunctionLibrary/pcSpline.m) <br> |
| <img src="Images/hand.png" width="171" alt="hand.png"> <br> | **In this script, students will...** <br> <br>- define interpolation and explain how it is different from regression and extrapolation <br>- demonstrate multiple solutions to an interpolation problem, even when interpolating by polynomials <br>- implement three different interpolation functions <br> | **Applications** <br>- Drawing a smooth rendition of their hand <br>- Tracking the path of a cyclone <br><br> **Scaffolded Template Scripts** <br> [<samp>linInterp.m</samp>](./FunctionLibrary/linInterp.m) <br> [<samp>pcHInterp.m</samp>](./FunctionLibrary/pcHInterp.m) <br> [<samp>pcSpline.m</samp>](./FunctionLibrary/pcSpline.m) <br> |
| :-- | :-- | :-- |

## Supporting Scripts
- [<samp>Hand.mlx</samp>](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=Hand.mlx)
- [<samp>TrackStorms.mlx</samp>](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=TrackStorms.mlx)
## [**NumericalDerivatives.mlx**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=NumericalDerivatives.mlx)
| <img src="Images/NoisyDerivative.png" width="171" alt="NoisyDerivative.png"> <br> | **In this script, students will...** <br> <br>- determine numerical derivative approximations formulas <br> <br>- use Taylor's theorem to calculate the order of the error for a numerical approximation to a derivative <br> <br>- demonstrate how numerical derivatives can magnify approximation errors <br> | **Applications** <br> <br>- Numerical solutions to differential equations <br> |
| <img src="Images/NoisyDerivative.png" width="171" alt="NoisyDerivative.png"> <br> | **In this script, students will...** <br> <br> - determine numerical derivative approximations formulas <br>- use Taylor's theorem to calculate the order of the error for a numerical approximation to a derivative <br> - demonstrate how numerical derivatives can magnify approximation errors <br> | **Applications** <br>- Numerical solutions to differential equations <br> |
| :-- | :-- | :-- |

## [**NumericalIntegration.mlx**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=NumericalIntegration.mlx)
| <img src="Images/BakerLakeSR.png" width="171" alt="BakerLakeSR.png"> <br> | **In this script, students will...** <br> <br>- implement Euler's method, Gaussian 2\-point approximations, and Simpson's rule for numerical integration <br> <br>- explain why higher\-order approximations may not be appropriate in applications <br> | **Applications** <br> <br>- Measure the area of a lake <br> **Scaffolded Template Scripts** <br> [<samp>eulerMethod.m</samp>](./FunctionLibrary/eulerMethod.m) <br> [<samp>gauss2pt.m</samp>](./FunctionLibrary/gauss2pt.m) <br> [<samp>simpsonsRule.m</samp>](./FunctionLibrary/simpsonsRule.m) <br> |
| <img src="Images/BakerLakeSR.png" width="171" alt="BakerLakeSR.png"> <br> | **In this script, students will...** <br> <br> - implement Euler's method, Gaussian 2\-point approximations, and Simpson's rule for numerical integration <br> - explain why higher\-order approximations may not be appropriate in applications <br> | **Applications** <br> - Measure the area of a lake <br><br> **Scaffolded Template Scripts** <br> [<samp>eulerMethod.m</samp>](./FunctionLibrary/eulerMethod.m) <br> [<samp>gauss2pt.m</samp>](./FunctionLibrary/gauss2pt.m) <br> [<samp>simpsonsRule.m</samp>](./FunctionLibrary/simpsonsRule.m) <br> |
| :-- | :-- | :-- |

## Supporting Scripts
- [<samp>MeasureLakeArea.mlx</samp>](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=MeasureLakeArea.mlx)
## [**NumericalODEs.mlx**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=NumericalODEs.mlx)
| <img src="Images/image_6.png" width="171" alt="image_6.png"> <br> | **In this script, students will...** <br> <br>- implement Euler's method for first\-order initial value problems <br> <br>- calculate the error of their numerical solution <br> <br>- implement a trapezoidal method <br> <br>- implement a four\-step Runge\-Kutta method <br> <br>- compare results with the built\-in MATLAB solver <samp>ode45</samp> <br> | **Applications** <br> <br>- Model a pendulum with increasingly realistic assumptions <br> **Scaffolded Template Scripts** <br> [<samp>eulerMethodDE.m</samp>](./FunctionLibrary/eulerMethodDE.m) <br> [<samp>rk4.m</samp>](./FunctionLibrary/rk4.m) <br> |
| <img src="Images/image_6.png" width="171" alt="image_6.png"> <br> | **In this script, students will...** <br> <br> - implement Euler's method for first\-order initial value problems <br> - calculate the error of their numerical solution <br> - implement a trapezoidal method <br>- implement a four\-step Runge\-Kutta method <br> - compare results with the built\-in MATLAB solver <samp>ode45</samp> <br> | **Applications** <br> - Model a pendulum with increasingly realistic assumptions <br><br> **Scaffolded Template Scripts** <br> [<samp>eulerMethodDE.m</samp>](./FunctionLibrary/eulerMethodDE.m) <br> [<samp>rk4.m</samp>](./FunctionLibrary/rk4.m) <br> |
| :-- | :-- | :-- |

## Supporting Scripts
- [<samp>PendulumModels.mlx</samp>](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=PendulumModels.mlx)
## [**NumericalPDEs.mlx**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Numerical-Methods-with-Applications&project=NumericalMethods.prj&file=NumericalPDEs.mlx)
| <img src="Images/heatSoln.gif" width="171" alt="heatSoln.gif"> <br> | **In this script, students will...** <br> <br>- identify errors from discretizing the problem and from discretizing the method and choose appropriate parameters to minimize overall error <br> <br>- explain the importance of stability when choosing a numerical method <br> <br>- implement explicit, implicit, and Crank\-Nicolson methods to solve a 1\-D heat equation <br> | **Applications** <br> <br>- Solve a heat equation <br> **Scaffolded Template Scripts** <br> [<samp>explicitPDE.m</samp>](./FunctionLibrary/explicitPDE.m) <br> [<samp>implicitPDE.m</samp>](./FunctionLibrary/implicitPDE.m) <br> [<samp>cnPDE.m</samp>](./FunctionLibrary/cnPDE.m) <br> |
| <img src="Images/heatSoln.gif" width="171" alt="heatSoln.gif"> <br> | **In this script, students will...** <br> <br> - identify errors from discretizing the problem and from discretizing the method and choose appropriate parameters to minimize overall error <br> - explain the importance of stability when choosing a numerical method <br> - implement explicit, implicit, and Crank\-Nicolson methods to solve a 1\-D heat equation <br> | **Applications** <br> - Solve a heat equation <br><br> **Scaffolded Template Scripts** <br> [<samp>explicitPDE.m</samp>](./FunctionLibrary/explicitPDE.m) <br> [<samp>implicitPDE.m</samp>](./FunctionLibrary/implicitPDE.m) <br> [<samp>cnPDE.m</samp>](./FunctionLibrary/cnPDE.m) <br> |
| :-- | :-- | :-- |

# License
Expand All @@ -102,4 +102,5 @@ Looking for more? Find an issue? Have a suggestion? Please contact the [MathWork

*©* Copyright 2024 The MathWorks™, Inc



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