1. Partial Derivatives bounded domain Its boundary denoted by
smooth or polygon
smooth or polyhedron
Closure of the domain
volume of the domain

2. Partial Derivatives 2D functions Partial Derivatives: Example:
Let be scalar functions
And be a vector-valued function
Example:
Partial Derivatives:
3D functions
Let be scalar functions
And be a vector-valued function
Example:

3. Partial Derivatives Operator2 2D functions Divergnce Example Example:
Let be scalar functions
And be a vector-valued function
Example:
Operator3
Laplacian (Laplace Operator)
Operator1
gradient
Example
Example
Remarks:
grad: scalar  vector
div: vector  scalar

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

5. 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. 1 3 4 7 6

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

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

8. Linear 2ed-Order PDE

9. 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

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

11. D D Dirichlet Boundary Condition Definitions and Terminology
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

12. D Dirichlet Boundary Condition Definitions and Terminology where
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

13. Definitions and Terminology
WHY PDE ??
Definitions and Terminology
PDE in
BC on
where
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

14. Definitions and Terminology
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

15. 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).

16. Partial Derivatives Normal Derivative Example 2D functions Example:
Let be scalar functions
And be a vector-valued function
Example:
Find
gradient
Example
Boundary Condition Variational Name Proper Name
u(x) = 0 essential Dirichlet
u(x) = 0 natural Neumann

17. 1D 2D Partial Derivatives
Boundary Condition Variational Name Proper Name
essential Dirichlet
natural Neumann 2D
Boundary Condition Variational Name Proper Name
essential Dirichlet
natural Neumann

18. 2D Boundary Conditions Inhomog Boundary Condition
mixed boundary conditions
Robin’s boundary condition