Partial Derivatives Example Find gradu gradient Find If

Partial Derivatives Example: Find grad(u) = gradient Find If If solution: Pa solution: rti al De riv gr at ad iv e s ie nt

Partial Derivatives Example: Find divergence of Find Laplacian of u = div(v) If If solution: di ve solution: rg en ce la pl a ci an

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

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 1 4 7 6

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

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

Linear 2 ed-Order PDE

Solution of aand Partial Differential Equation Definitions 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

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

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

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 consists of two Vertical lines and two horizontal lines where

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 - electrodynamics - heat - fluid flow - electrostatics - elasticity These phenomena can be formalised in terms of PDEs -…….

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

Numerical to solve PDEs Definitionsmethods 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).

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: No classes 2 TOPICS Introduction and fundamental concepts. Classification of second -order linear PDE • Quick introduction to PDE toolbox The Finite Element Method for Poisson’s Equation 5 • Triangulations • Data Storage Structures • Mesh Generation • The Space of Piecewise Linear • Quadrature and Numerical Integration • Green’s Formula • Variational Formulation • Finite Element Approximation • Derivation of a Linear System of Equations • Properties of the Stiffness Matrix • Computer Implementation • Assembly of the Stiffness Matrix • Assembling the Boundary Conditions • A Finite Element Solver for Poisson’s Equation • A Priori Error Estimates The Finite Element Method for Time-dependent Problems (The Heat Equation) 4 • Finite Difference Methods for Systems of ODE • The Heat Equation • Variational Formulation • Spatial Discretization • Time Discretization • Computer Implementation • Stability Estimates • A Priori Error Estimates The Finite Element Method for Time-dependent Problems (The Wave Equation) 3 • The Wave Equation • Variational Formulation • Spatial Discretization • Time Discretization • Computer Implementation • Stability Estimates • A Priori Error Estimates Iterative Methods for Large Sparse Linear Systems 1 • Direct Methods - Iterative Methods • Conjugate Gradient Method (CG) • Preconditioning • MINRES - GMRES • Jacobi’s Method - The Gauss-Seidel Method