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

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slide1

Partial Derivatives

Example:

Example:

Find

grad(u) = gradient

Find

If

If

solution:

solution:

Partial Derivatives

gradient

slide2

Partial Derivatives

Example:

Example:

Find

Laplacian of u

Find divergence of

= div(v)

If

If

solution:

solution:

divergence

laplacian

slide3

7

1

2

4

Partial Differential Equations (PDE)

Definition:

a partial differential equation (PDE) is an equation that contains partial derivatives

3

5

6

slide4

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

slide5

Linear 2ed-Order PDE

The general linear 2ed order PDE in two variables x, y.

The discriminant of the equation =

Definition:

slide6

Linear 2ed-Order PDE (Classification)

The general linear 2ed order PDE in two variables x, y.

slide8

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

slide9

Boundary Condition

Definitions and Terminology

This PDE has an infinite number of solutions

D

BVP: Boundary Value Problem

slide10

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

slide11

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

slide12

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

slide13

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

slide14

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).
slide15

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: