Brief introduction to Partial Differential Equations (PDEs)

Slides:

Advertisements

Similar presentations

Finite Difference Discretization of Hyperbolic Equations: Linear Problems Lectures 8, 9 and 10.

Advertisements

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Part Eight Partial Differential Equations.

PART 7 Ordinary Differential Equations ODEs

ECE602 BME I Partial Differential Equations in Biomedical Engineering.

ECE602 BME I Partial Differential Equations in Biomedical Engineering (Cont’d)

Partial Differential Equations

PDEs: General classification

PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones.

Numerical Methods for Partial Differential Equations

COMPUTATIONAL MODELING FOR ENGINEERING MECN 6040 Professor: Dr. Omar E. Meza Castillo Department.

CISE301: Numerical Methods Topic 9 Partial Differential Equations (PDEs) Lectures KFUPM (Term 101) Section 04 Read & CISE301_Topic9.

Numerical methods for PDEs PDEs are mathematical models for –Physical Phenomena Heat transfer Wave motion.

Matmol Metoda linii Method of lines

1 Numerical Methods and Software for Partial Differential Equations Lecturer:Dr Yvonne Fryer Time:Mondays 10am-1pm Location:QA210 / KW116 (Black)

Classical Waves Calculus/Differential Equations Refresher.

Hyperbolic PDEs Numerical Methods for PDEs Spring 2007 Jim E. Jones.

CIS888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering Partial Differential Equations - Background Physical problems are governed.

Partial Differential Equations Introduction –Adam Zornes, Deng Li Discretization Methods –Chunfang Chen, Danny Thorne, Adam Zornes.

Math 3120 Differential Equations with Boundary Value Problems

Solving an elliptic PDE using finite differences Numerical Methods for PDEs Spring 2007 Jim E. Jones.

1 EEE 431 Computational Methods in Electrodynamics Lecture 3 By Dr. Rasime Uyguroglu.

D. R. Wilton ECE Dept. ECE 6382 Second Order Linear Differential Equations.

Introduction to PDE classification Numerical Methods for PDEs Spring 2007 Jim E. Jones References: Partial Differential Equations of Applied Mathematics,

Introduction to Numerical Methods for ODEs and PDEs Lectures 1 and 2: Some representative problems from engineering science and classification of equation.

Engineering Analysis – Computational Fluid Dynamics –

Differential Equations Brannan Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 09: Partial Differential Equations and Fourier.

Numerical methods 1 An Introduction to Numerical Methods For Weather Prediction by Mariano Hortal office 122.

Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.

Partial Derivatives bounded domain Its boundary denoted by

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

Partial Differential Equations Introduction – Deng Li Discretization Methods –Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006.

Boundary-Value Problems in Rectangular Coordinates

Chapter 1 Partial Differential Equations

Differential Equations Linear Equations with Variable Coefficients.

Differential Equations

Solving Partial Differential Equation Numerically Pertemuan 13 Matakuliah: S0262-Analisis Numerik Tahun: 2010.

Department of Mathematics Numerical Solutions to Partial Differential Equations Ch 12. Applied mathematics. Korea University.

Second Order Linear Differential Equations ECE 6382 Notes are from D. R. Wilton, Dept. of ECE David R. Jackson 1.

Ordinary Differential Equations (ODEs). Objectives of Topic  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods.

Ch. 12 Partial Differential Equations

Differential Equations

Introduction to Differential Equations

Part 8 - Chapter 29.

I- Computational Fluid Dynamics (CFD-I)

EEE 431 Computational Methods in Electrodynamics

Linear Equations Constant Coefficients

DIFFERENTIAL EQUATIONS

Introduction to Differential Equations

Boundary-Value Problems in Rectangular Coordinates

Christopher Crawford PHY

Lecture 13 Contents Partial Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Introduction to Partial Differential Equations

Ch 1.3: Classification of Differential Equations

Christopher Crawford PHY

Finite Difference Method

ME/AE 339 Computational Fluid Dynamics K. M. Isaac Topic2_PDE

Chapter 1:First order Partial Differential Equations

State Space Method.

Partial Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Second Order-Partial Differential Equations

RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE

PARTIAL DIFFERENTIAL EQUATIONS

AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac

Lecture 13 Contents Partial Differential Equations

AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac

Akram Bitar and Larry Manevitz Department of Computer Science

Chapter 1:First order Partial Differential Equations

Presentation transcript:

Brief introduction to Partial Differential Equations (PDEs) G P Raja Sekhar Department of Mathematics IIT Kharagpur rajas@iitkgp.ac.in

First-Order PDEs Propagation of wave with speed c First-order linear wave equation (advection eq.) Propagation of wave with speed c Advection of passive scalar with speed c

Second-Order PDEs Advection-diffusion equation (linear)

Popular PDEs Laplace and Poisson’s equations Helmholtz equation Time-dependent harmonic waves Propagation of acoustic waves

Popular PDEs Wave equation Fourier equation (Heat equation)

Classification Linear second-order PDE in two independent variables (x,y), (x,t), etc. A, B, C, …, G are constant coefficients (may be generalized) (discriminant)

Well-Posed Problem (1) the solution exists (existence) (2) the solution is unique (uniqueness) (3) the solution depends continuously on the auxiliary data Mathematical - Computational

Boundary and Initial Conditions Initial conditions: starting point for propagation problems Boundary conditions: specified on domain boundaries to provide the interior solution in computational domain R R s n

Hyperbolic PDEs Two real roots, two characteristic directions Two propagation (marching) directions Domain of dependence Domain of influence (ux,uy,vx,vy) are not uniquely defined along the characteristic lines, discontinuity may occur Boundary conditions must be specified according to the characteristics

Parabolic PDEs One real (double) root, one characteristic direction (typically time is const) The solution is marching in time (or spatially) with given initial conditions The solution will be modified by the boundary conditions (time-dependent, in general) during the propagation Any change in boundary conditions at t1 will not affect solution at t < t1, but will change the solution after t = t1

Elliptic PDEs The derivatives (ux,uy,vx,vy) can always be uniquely determined at every point in the solution domain No marching or propagation direction ! Boundary conditions needed on all boundaries The solution will be continuous (smooth) in the entire solution domain