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

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:

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

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

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

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

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

Linear 2ed-Order PDE

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

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

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

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

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

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

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).

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

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

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