1. 11/17/2015 http://numericalmethods.eng.usf.edu 1 Introduction to Partial Differential Equations http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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5. What is a Partial Differential Equation ? Ordinary Differential Equations have only one independent variable Partial Differential Equations have more than one independent variable subject to certain conditions: where u is the dependent variable, and x and y are the independent variables.

6. Example of an Ordinary Differential Equation Assumption: Ball is a lumped system. Number of Independent variables: One (t) Hot Water Spherical Ball

7. Example of an Partial Differential Equation Assumption: Ball is not a lumped system. Number of Independent variables: Four (r, θ, φ,t) Hot Water Spherical Ball

8. Classification of 2 nd Order Linear PDE’s where are functions of,and is a function of

9. Classification of 2 nd Order Linear PDE’s can be: Elliptic Parabolic Hyperbolic

10. Classification of 2 nd Order Linear PDE’s: Elliptic If,then equation is elliptic.

11. Classification of 2 nd Order Linear PDE’s: Elliptic Example: where, giving therefore the equation is elliptic.

12. Classification of 2 nd Order Linear PDE’s: Parabolic If,then the equation is parabolic.

13. Classification of 2 nd Order Linear PDE’s: Parabolic Example: where, giving therefore the equation is parabolic.

14. Classification of 2 nd Order Linear PDE’s: Hyperbolic If,then the equation is hyperbolic.

15. Classification of 2 nd Order Linear PDE’s: Hyperbolic Example: where, giving therefore the equation is hyperbolic.

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17. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate Acknowledgement

18. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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