Partial Derivatives

# Partial Derivatives

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## Partial Derivatives

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##### Presentation Transcript

1. Partial Derivatives Example: Example: Find grad(u) = gradient Find If If solution: solution: Partial Derivatives gradient

2. Partial Derivatives Example: Example: Find Laplacian of u Find divergence of = div(v) If If solution: solution: divergence laplacian

3. 7 1 2 4 Partial Differential Equations (PDE) Definition: a partial differential equation (PDE) is an equation that contains partial derivatives 3 5 6

4. 7 1 4 Partial Differential Equations (PDE) Order of a PDE: The order of a differential equation (ODE or PDE) is the order of the highest derivative in the equation. 3 6

5. Linear 2ed-Order PDE The general linear 2ed order PDE in two variables x, y. The discriminant of the equation = Definition:

6. Linear 2ed-Order PDE (Classification) The general linear 2ed order PDE in two variables x, y.

7. Linear 2ed-Order PDE

8. Solution of a Partial Differential Equation Definitions and Terminology Definition: Solution of PDE Any function which when substituted into a PDE reduces the equation to an identity, is said to be a solution of the equation. Can you think of another solution ??? A solution of a PDE is generally not unique

9. Boundary Condition Definitions and Terminology This PDE has an infinite number of solutions D BVP: Boundary Value Problem

10. Dirichlet Boundary Condition Definitions and Terminology D BVP: Boundary Value Problem D BVP: Boundary Value Problem Find a function which satisfy the PDE inside the domain and it assumes given values on the boundary

11. Dirichlet Boundary Condition Definitions and Terminology D BVP: Boundary Value Problem Find a function which satisfy the PDE inside the domain and it assumes given values on the boundary the boundary consists of two Vertical lines and two horizontal lines where

12. WHY PDE ?? Definitions and Terminology PDE in where BC on Analytic Solution: PDEs can be used to describe a wide variety of phenomena such as - sound - heat - electrostatics -……. - electrodynamics - fluid flow - elasticity These phenomena can be formalised in terms of PDEs

13. WHY PDE ?? Definitions and Terminology PDEs can be used to describe a wide variety of phenomena such as - sound - heat - electrostatics -……. - electrodynamics - fluid flow - elasticity These phenomena can be formalised in terms of PDEs Schrödinger equation Navier–Stokes equations Darcy law Biharmonic equation ------------- ------------- Heat equation the wave equation Laplace's equation Helmholtz equation

14. Numerical methods to solve PDEs Definitions and Terminology Analytical solution is not available (almost all) Numerical solution is almost the only method that can be used for getting information about the solution The three most widely used numerical methods to solve PDEs are • The finite element method (FEM), • The finite volume methods (FVM) • The finite difference methods (FDM).

15. Math-574 Topics (Part 1) In this course, we study the analysis, implementation and application of finite element methods. The following topics are studied in this course: