MATH 333: Partial Differential Equations
|
|
- Gerard Gordon
- 5 years ago
- Views:
Transcription
1 MATH 333: Partial Differential Equations Problem Set 9, Final version Due Date: Tues., Nov. 29, 2011 Relevant sources: Farlow s book: Lessons 9, MacCluer s book: Chapter 3 44 Show that the Poisson problem with Dirichlet BCs on a bounded domain Ω is stable. Specifically, suppose that Ω R n is open, bounded and connected, and consider the two (nearly identical they both have the same forcing function f ) problems u(x) = f (x), x Ω, subject to u(x) = g 1 (x), for x Ω, (1) and v(x) = f (x), x Ω, subject to v(x) = g 2 (x), for x Ω. (2) If boundary data for the two problems are similar that is, if g 1 g 2 := max { g 1 (x) g 2 (x) : x Ω } is small show that the difference in solutions is small throughout Ω. 45 Set up the BVP corresponding to the thermal equilibrium of a rectangular plate that is insulated on two of its sides, has 0 at its top edge and 100 at the bottom edge. Where do you expect the maximum temperature to be located? What is its value? Can you find a formula for temperature inside the plate? 46 [This problem gives a partial answer to the question of what forms harmonic functions of 2 variables can take.] Find an harmonic function u(x, y) defined on the annulus 1 < x < 1 (where x = (x, y)) subject to the constant Dirichlet BCs u = a on x = and u = b on x = 1. You may assume, as seems physically reasonable given the BCs, that the solution is radially symmetric, so that u = u(r) is a function of r = x 2 + y 2 alone (independent of θ), and so Laplace s equation reduces to u = 2 u r + 1 u 2 r r = 0. Hint: Let v = u r and solve the 1 st order ODE for v. Then find u.
2 MATH 333 Problem Set Find a series solution of the heat problem u t = u xx, 0 < x < 1, t > 0, subject to BCs: u(t, 0) = 0, u(t, 1) = sin t, IC: u(0, x) = Consider the Poisson problem with Dirichlet BCs in a square region of the plane u(x, 0) = sin(πx), u = f (x, y), 0 < x < 1, 0 < y < 1, subject to BCs u(x, 1) = e 2 sin(πx), u(0, y) = 0 = u(1, y). where f (x, y) = (4 π 2 )e 2y sin(πx). (a) Employing a finite difference scheme, solve this problem on a grid that uses N = 4 interior values on the x-axis and M = 4 interior values on the y-axis to obtain the steady-state temperature distribution. Graph the resulting approximate solution as a surface in 3D. I encourage you to rely heavily on my class examples which used scripts like psset1.m, psset2.m, psset3.m, psset4.m, poissonsolver.m and discretelaplacian.m, available to download from scofield/courses/m333/materials/octave/ Pay special attention to how a forcing function is implemented in psset4.m (i.e., the form of its inputs and outputs). Hand in a printout of your code and the graphs you produce, and send me electronic versions as well. (b) Demonstrate that the true solution of the problem is u(x, y) = e 2y sin(πx), and explain why the problem is, therefore, well-posed. (c) Compare the values of the true solution with those from your approximate solution in part (a). Among all interior grid points find both the maximum error and the maximum relative error of the numerical solution. (d) Redo parts (a) and (c), this time doubling N and M so that they are both 8.
3 MATH 333 Problem Set 9 3 Work with a partner and choose one problem pair from 49A-B or 50A-B to do. 49A In class we considered Poisson s equation with Dirichlet BCs u = f, inside R = { (x, y) 0 < x < a, 0 < y < a }, subject to u = g on R. We replaced the PDE with a linear system of difference equations u j 1,m u j+1,m u j,m 1 u j,m+1 + 4u j,m = h 2 f j,m, 1 j N, 1 m M, (3) vectorized the unknowns U = [ u 1,1 u 2,1... u N,1 u 1,2 u 2,2... u N,2... u 1,M u 2,M... u N,M ] T, and then determined the matrix A such that AU corresponds to the left-hand side of Equation (3). (a) Suppose, instead of Dirichlet BCs, we have periodic BCs. That is, as you approach the right edge of the mesh of points (x j, y m ), you wrap around to the left side; the top wraps around to the bottom (like in the classic video game PacMan). For simplicity, assume there are only points (x j, y m ) for j = 1, 2,..., N, m = 1, 2,..., M, so that (x 1, y m ) is to the immediate right of (x N, y m ) and (x j, y 1 ) is immediately above (x j, y M ). Determine the form of the matrix A (give a picture) so that AU is the discrete Laplacian operator on u, taking into account these periodic BCs. [Note: The size of the correct matrix should be NM rows by NM columns.] (b) Write an Octave function that, given inputs N and M, returns the matrix A of part (a). The.m-file I wrote to generate the matrix A for class is discretelaplacian.m, and it actually contains two different ways to construct (my) A. 49B Obtain a small black-and-white image file (say, something on the order of M = 60 pixels by N = 60 pixels, so NM isn t horribly large), and place it in your working directory. I ve prepared the file testim.png which you may use, but you may substitute your own. (Most standard image formats are supported by Octave.) Read this file into Octave, convert it to double precision, and then display it using the commands image = double ( imread ( testim. png ) ) / ; imshow ( image ) Now add some Gaussian noise to the image and display the result via a command like:
4 MATH 333 Problem Set 9 4 nimage = image randn ( s ize ( image ) ) ; figure ( 2 ) imshow ( nimage ) (If you have the clean image displayed, the figure() command creates a 2 nd figure window so the two can be viewed side-by-side.) We will use diffusion to smooth out the noise in the image. discretization of the diffusion equation On a uniform spatial grid with x = y = h, one u t (t, x, y) = u(t, x, y) is or u(t + t, x, y) u(t, x, y) t = u(t, x h, y) + u(t, x + h, y) + u(t, x, y h) + u(t, x, y + h) 4u(t, x, y) h 2, u(t + t, x, y) = u(t, x, y) + t h 2 [ u(t, x h, y) + u(t, x + h, y) + u(t, x, y h) + u(t, x, y + h) 4u(t, x, y) ]. (4) Here, consider t to be the current time (so values at time t refer to the current state of the image, while those at time (t + t) refer to the next state) and, relative to a pixel at position (x, y), positions (x h, y), (x + h, y), (x, y h) and (x, y + h) are neighboring pixels to the left, right, below and above, respectively. You can vectorize the noisy image with a command like U = nimage ( : ) ; and return it to its original M-by-N shape with reshape (U, M, N) Now, if we call the vectorized form of your current image U (k), then Equation (4), which describes the evolution of your image at the local level, can be turned into the global (entire image at a time) scheme (where we have taken h = 1) U (k+1) = U (k) + tau (k). Here, A, the matrix you found for the previous exercise, has periodic BCs built into it. Assuming U (0) is the starting noisy image, you could allow the image intensity values to evolve up until, say, time t = 1 in just one step (so U (1) is the resulting image) by taking t = 1. You could also get to the same time via smaller steps, t = 0.1 for instance, so that the final image is U (10).
5 MATH 333 Problem Set 9 5 (a) Implement this global evolution scheme in Octave or Matlab. Here is pseudocode for what you are to do: 1. Set U to be the vectorized form of the noisy image. 2. Build your matrix A of appropriate size. 3. Set the value of dt, and the number of time steps to carry out. 4. Begin loop Let U = U + dt*a*u Plot the current state of the image (reshaped U). Pause briefly for the viewer to see the result. End loop You should hand in a hardcopy of your code and send me an electronic version as well. (b) Try out your code, independently altering the number of iterations to carry out and the time step dt. Comparing your final image to the original clean image, find a combination of these two settings that best pleases you in terms of removing noise but not blurring the image too badly. The code you send for part (a) should have these settings. (c) Along with a tendency to become blurred in time, what do you notice about the evolving image produced by your routine? Explain these observations. 50A Consider the ordinary differential equation/bvp (a 1-dimensional version of the Poisson problem) y (x) = f (x), subject to BCs y(0) = 0, y (1) = 1. (5) Let us subdivide the interval [0, 1] into n equally-spaced subintervals, 0 = x 0 < x 1 <... < x n = 1, with each x m x m 1 = h := 1 n. (a) If we use a centered difference approximation for y (x) then, writing y m as an approximation to the value of y(x m ), derive the finite difference equation y m 1 + 2y m y m+1 = h 2 f (x m ), (6) holding for m = 1, 2,..., n 1. (b) While equation (6) provides most of the particulars for a finite difference scheme to approximate the solution of Problem (5), we will not have a complete scheme until we specify how to handle the Neumann boundary condition at x = 1. Consider two different schemes, based on the following two ways of dealing with
6 MATH 333 Problem Set 9 6 this Neumann condition. In the simpler of the two schemes S 1, we approximate y (1) using a backward difference: y n y n 1 h = 1. In scheme S 2, we imagine there to be a ghost node x n+1 = 1 + h, outside the right end point (where the Neumann condition occurs) of our interval [0, 1], require equation (6) to hold at the right end point x n (i.e., to hold with m = n, which gives rise to an equation involving the ghost node x n+1 ), and introduce the constraint (approximate equation for y (1) = 1) y n+1 y n 1 2h = 1. Schemes S 1 and S 2 both may be turned into matrix problems A 1 v 1 = b 1 and A 2 v 2 = b 2, where v 1 = (y 1, y 2,..., y n ) and v 2 = (y 1, y 2,..., y n, y n+1 ) represent the vector of unknowns. Determine the entries of the matrices A 1, A 2 and the right-hand sides b 1, b 2. {c} This problem part is optional. Are the matrices A 1, A 2 symmetric? positive definite? (The definition of the matrix A is positive definite is that v T Av 0 for all vectors v of appropriate size, with equality only when v = 0.) If the answers are not obvious, then you may resort to use of numerical evidence. Be clear about what you know to be true (from numerical evidence), and what you only guess at being true. Note: Positive definite matrices are always nonsingular, making a problem like Av = b uniquely solvable. (d) Let f (x) = e x 1, so that the resulting true solution of (5) is y(x) = e 1 (e x 1). Write algorithms to compute finite difference solutions (with this f ) both for scheme S 1 and S 2. Compute the maximum error between true and approximate solutions (i.e, maximum difference y(x m ) y m taken over all grid points x = x m ) using both schemes at several different choices of h (say, h = 0.1 and h = 0.05, at the least). What can you say about the order of convergence for each scheme? 50B (a) Derive an implicit scheme for the following Neumann problem: BCs : u x (t, 0) = α(t), t > 0 u t = u xx, 0 < x < l, t > 0, subj. to u x (t, l) = β(t) IC : u(0, x) = f (x), 0 < x < l (7)
7 MATH 333 Problem Set 9 7 (b) Use your scheme to solve Problem (7) in the case where l = 4, α(t) = 0, β(t) = 100(1 e t ), and f (x) = 100, if 1 x 2, 0, otherwise.
Finite Difference Methods for Boundary Value Problems
Finite Difference Methods for Boundary Value Problems October 2, 2013 () Finite Differences October 2, 2013 1 / 52 Goals Learn steps to approximate BVPs using the Finite Difference Method Start with two-point
More informationNumerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value
Numerical Analysis of Differential Equations 188 5 Numerical Solution of Elliptic Boundary Value Problems 5 Numerical Solution of Elliptic Boundary Value Problems TU Bergakademie Freiberg, SS 2012 Numerical
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationMath 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework
Math 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework Jan Mandel University of Colorado Denver May 12, 2010 1/20/09: Sec. 1.1, 1.2. Hw 1 due 1/27: problems
More informationModule 7: The Laplace Equation
Module 7: The Laplace Equation In this module, we shall study one of the most important partial differential equations in physics known as the Laplace equation 2 u = 0 in Ω R n, (1) where 2 u := n i=1
More informationAM 205: lecture 14. Last time: Boundary value problems Today: Numerical solution of PDEs
AM 205: lecture 14 Last time: Boundary value problems Today: Numerical solution of PDEs ODE BVPs A more general approach is to formulate a coupled system of equations for the BVP based on a finite difference
More informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationMath 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Name Section Math 51 December 14, 5 Answer Key to Final Exam There are 1 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning
More informationSolving linear systems (6 lectures)
Chapter 2 Solving linear systems (6 lectures) 2.1 Solving linear systems: LU factorization (1 lectures) Reference: [Trefethen, Bau III] Lecture 20, 21 How do you solve Ax = b? (2.1.1) In numerical linear
More informationMy signature below certifies that I have complied with the University of Pennsylvania s Code of Academic Integrity in completing this exam.
My signature below certifies that I have complied with the University of Pennsylvania s Code of Academic Integrity in completing this exam. Signature Printed Name Math 241 Exam 1 Jerry Kazdan Feb. 17,
More informationElements of Matlab and Simulink Lecture 7
Elements of Matlab and Simulink Lecture 7 Emanuele Ruffaldi 12th May 2009 Copyright 2009,Emanuele Ruffaldi. This work is licensed under the Creative Commons Attribution-ShareAlike License. PARTIAL DIFFERENTIAL
More informationMath 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question
More informationPDEs in Image Processing, Tutorials
PDEs in Image Processing, Tutorials Markus Grasmair Vienna, Winter Term 2010 2011 Direct Methods Let X be a topological space and R: X R {+ } some functional. following definitions: The mapping R is lower
More informationAMSC/CMSC 466 Problem set 3
AMSC/CMSC 466 Problem set 3 1. Problem 1 of KC, p180, parts (a), (b) and (c). Do part (a) by hand, with and without pivoting. Use MATLAB to check your answer. Use the command A\b to get the solution, and
More informationFundamental Solutions and Green s functions. Simulation Methods in Acoustics
Fundamental Solutions and Green s functions Simulation Methods in Acoustics Definitions Fundamental solution The solution F (x, x 0 ) of the linear PDE L {F (x, x 0 )} = δ(x x 0 ) x R d Is called the fundamental
More informationProject 2: Using linear systems for numerical solution of boundary value problems
LINEAR ALGEBRA, MATH 124 Instructor: Dr. T.I. Lakoba Project 2: Using linear systems for numerical solution of boundary value problems Goal Introduce one of the most important applications of Linear Algebra
More informationMathematics Qualifying Exam Study Material
Mathematics Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering mathematics topics. These topics are listed below for clarification. Not all instructors
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationTMA4220: Programming project - part 1
TMA4220: Programming project - part 1 TMA4220 - Numerical solution of partial differential equations using the finite element method The programming project will be split into two parts. This is the first
More informationClassification of partial differential equations and their solution characteristics
9 TH INDO GERMAN WINTER ACADEMY 2010 Classification of partial differential equations and their solution characteristics By Ankita Bhutani IIT Roorkee Tutors: Prof. V. Buwa Prof. S. V. R. Rao Prof. U.
More information1 Finite difference example: 1D implicit heat equation
1 Finite difference example: 1D implicit heat equation 1.1 Boundary conditions Neumann and Dirichlet We solve the transient heat equation ρc p t = ( k ) (1) on the domain L/2 x L/2 subject to the following
More informationMath 2930 Worksheet Final Exam Review
Math 293 Worksheet Final Exam Review Week 14 November 3th, 217 Question 1. (* Solve the initial value problem y y = 2xe x, y( = 1 Question 2. (* Consider the differential equation: y = y y 3. (a Find the
More informationFinite Difference Methods (FDMs) 1
Finite Difference Methods (FDMs) 1 1 st - order Approxima9on Recall Taylor series expansion: Forward difference: Backward difference: Central difference: 2 nd - order Approxima9on Forward difference: Backward
More informationApplied Mathematics 205. Unit III: Numerical Calculus. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit III: Numerical Calculus Lecturer: Dr. David Knezevic Unit III: Numerical Calculus Chapter III.3: Boundary Value Problems and PDEs 2 / 96 ODE Boundary Value Problems 3 / 96
More informationHOMEWORK 4: Numerical solution of PDEs for Mathmatical Models, Analysis and Simulation, Fall 2011 Report due Mon Nov 12, Maximum score 6.0 pts.
HOMEWORK 4: Numerical solution of PDEs for Mathmatical Models, Analysis and Simulation, Fall 2011 Report due Mon Nov 12, 2012. Maximum score 6.0 pts. Read Strang s (old) book, Sections 3.1-3.4 and 6.4-6.5.
More informationNumerical Solution of Laplace Equation By Gilberto E. Urroz, October 2004
Numerical Solution of Laplace Equation By Gilberto E. Urroz, October 004 Laplace equation governs a variety of equilibrium physical phenomena such as temperature distribution in solids, electrostatics,
More informationMath 3150 Problems Chapter 3
Name Math 15 Problems Chapter Due date: See the internet due date. Problems are collected once a week. Records are locked when the stack is returned. Records are only corrected, never appended. Submitted
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More information6 The SVD Applied to Signal and Image Deblurring
6 The SVD Applied to Signal and Image Deblurring We will discuss the restoration of one-dimensional signals and two-dimensional gray-scale images that have been contaminated by blur and noise. After an
More informationElliptic Problems / Multigrid. PHY 604: Computational Methods for Physics and Astrophysics II
Elliptic Problems / Multigrid Summary of Hyperbolic PDEs We looked at a simple linear and a nonlinear scalar hyperbolic PDE There is a speed associated with the change of the solution Explicit methods
More informationOptimal Interface Conditions for an Arbitrary Decomposition into Subdomains
Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains Martin J. Gander and Felix Kwok Section de mathématiques, Université de Genève, Geneva CH-1211, Switzerland, Martin.Gander@unige.ch;
More information8 The SVD Applied to Signal and Image Deblurring
8 The SVD Applied to Signal and Image Deblurring We will discuss the restoration of one-dimensional signals and two-dimensional gray-scale images that have been contaminated by blur and noise. After an
More information13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs)
13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs) A prototypical problem we will discuss in detail is the 1D diffusion equation u t = Du xx < x < l, t > finite-length rod u(x,
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More information8 The SVD Applied to Signal and Image Deblurring
8 The SVD Applied to Signal and Image Deblurring We will discuss the restoration of one-dimensional signals and two-dimensional gray-scale images that have been contaminated by blur and noise. After an
More informationProblem set 3: Solutions Math 207B, Winter Suppose that u(x) is a non-zero solution of the eigenvalue problem. (u ) 2 dx, u 2 dx.
Problem set 3: Solutions Math 27B, Winter 216 1. Suppose that u(x) is a non-zero solution of the eigenvalue problem u = λu < x < 1, u() =, u(1) =. Show that λ = (u ) 2 dx u2 dx. Deduce that every eigenvalue
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationUNIVERSITY OF MANITOBA
Question Points Score INSTRUCTIONS TO STUDENTS: This is a 6 hour examination. No extra time will be given. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic
More informationLecture 38 Insulated Boundary Conditions
Lecture 38 Insulated Boundary Conditions Insulation In many of the previous sections we have considered fixed boundary conditions, i.e. u(0) = a, u(l) = b. We implemented these simply by assigning u j
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationQualifying Examination
Summer 24 Day. Monday, September 5, 24 You have three hours to complete this exam. Work all problems. Start each problem on a All problems are 2 points. Please email any electronic files in support of
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationQ ( q(m, t 0 ) n) S t.
THE HEAT EQUATION The main equations that we will be dealing with are the heat equation, the wave equation, and the potential equation. We use simple physical principles to show how these equations are
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationFinite Difference Method
Finite Difference Method for BVP ODEs Dec 3, 2014 1 Recall An ordinary differential equation is accompanied by auxiliary conditions. In the analytical method, these conditions are used to evaluate the
More informationHomework 5 Solutions
18-290 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 2018 Homework 5 Solutions. Part One 1. (12 points) Calculate the following convolutions: (a) x[n] δ[n n 0 ] (b) 2 n u[n] u[n] (c) 2 n u[n]
More informationPartial Differential Equations (PDEs) and the Finite Difference Method (FDM). An introduction
Page of 8 Partial Differential Equations (PDEs) and the Finite Difference Method (FDM). An introduction FILE:Chap 3 Partial Differential Equations-V6. Original: May 7, 05 Revised: Dec 9, 06, Feb 0, 07,
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
More informationwhere is the Laplace operator and is a scalar function.
Elliptic PDEs A brief discussion of two important elliptic PDEs. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationMATH 126 FINAL EXAM. Name:
MATH 126 FINAL EXAM Name: Exam policies: Closed book, closed notes, no external resources, individual work. Please write your name on the exam and on each page you detach. Unless stated otherwise, you
More informationThe One Dimensional Heat Equation
The One Dimensional Heat Equation Adam Abrahamsen and David Richards May 22, 2002 Abstract In this document we will study the flow of heat in one dimension through a small thin rod. We will use the derivation
More informationFinite difference method for elliptic problems: I
Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More informationLecture 18 Classical Iterative Methods
Lecture 18 Classical Iterative Methods MIT 18.335J / 6.337J Introduction to Numerical Methods Per-Olof Persson November 14, 2006 1 Iterative Methods for Linear Systems Direct methods for solving Ax = b,
More informationPARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
More informationIntroduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions
More information2 A brief interruption to discuss boundary value/intial value problems
The lecture of 1/9/2013 1 The one dimensional heat equation The punchline from the derivation of the heat equation notes (either the posted file, or equivalently what is in the text) is that given a rod
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationChapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS
Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS 5.1 Introduction When a physical system depends on more than one variable a general
More informationAnalysis III (BAUG) Assignment 3 Prof. Dr. Alessandro Sisto Due 13th October 2017
Analysis III (BAUG Assignment 3 Prof. Dr. Alessandro Sisto Due 13th October 2017 Question 1 et a 0,..., a n be constants. Consider the function. Show that a 0 = 1 0 φ(xdx. φ(x = a 0 + Since the integral
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationSparse Linear Systems. Iterative Methods for Sparse Linear Systems. Motivation for Studying Sparse Linear Systems. Partial Differential Equations
Sparse Linear Systems Iterative Methods for Sparse Linear Systems Matrix Computations and Applications, Lecture C11 Fredrik Bengzon, Robert Söderlund We consider the problem of solving the linear system
More informationQuestion 9: PDEs Given the function f(x, y), consider the problem: = f(x, y) 2 y2 for 0 < x < 1 and 0 < x < 1. x 2 u. u(x, 0) = u(x, 1) = 0 for 0 x 1
Question 9: PDEs Given the function f(x, y), consider the problem: 2 u x 2 u = f(x, y) 2 y2 for 0 < x < 1 and 0 < x < 1 u(x, 0) = u(x, 1) = 0 for 0 x 1 u(0, y) = u(1, y) = 0 for 0 y 1. a. Discuss how you
More informationSecond-Order Linear ODEs (Textbook, Chap 2)
Second-Order Linear ODEs (Textbook, Chap ) Motivation Recall from notes, pp. 58-59, the second example of a DE that we introduced there. d φ 1 1 φ = φ 0 dx λ λ Q w ' (a1) This equation represents conservation
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationLecture 42 Determining Internal Node Values
Lecture 42 Determining Internal Node Values As seen in the previous section, a finite element solution of a boundary value problem boils down to finding the best values of the constants {C j } n, which
More informationMATH20411 PDEs and Vector Calculus B
MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More informationDUE: WEDS MARCH 26TH 2018
HOMEWORK # 2: FINITE DIFFERENCES MAPPING AND TWO DIMENSIONAL PROBLEMS DUE: WEDS MARCH 26TH 2018 NOTE: In this homework, you will choose ONE of the following three questions to perform and hand-in. Each
More informationInverse Problems. Lab 19
Lab 19 Inverse Problems An important concept in mathematics is the idea of a well posed problem. The concept initially came from Jacques Hadamard. A mathematical problem is well posed if 1. a solution
More informationImage Reconstruction And Poisson s equation
Chapter 1, p. 1/58 Image Reconstruction And Poisson s equation School of Engineering Sciences Parallel s for Large-Scale Problems I Chapter 1, p. 2/58 Outline 1 2 3 4 Chapter 1, p. 3/58 Question What have
More informationPreliminary Examination, Numerical Analysis, August 2016
Preliminary Examination, Numerical Analysis, August 2016 Instructions: This exam is closed books and notes. The time allowed is three hours and you need to work on any three out of questions 1-4 and any
More informationWave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationFinite-Elements Method 2
Finite-Elements Method 2 January 29, 2014 2 From Applied Numerical Analysis Gerald-Wheatley (2004), Chapter 9. Finite-Elements Method 3 Introduction Finite-element methods (FEM) are based on some mathematical
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationMcGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA
McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper
More informationIntroduction and some preliminaries
1 Partial differential equations Introduction and some preliminaries A partial differential equation (PDE) is a relationship among partial derivatives of a function (or functions) of more than one variable.
More informationLecture 10: Finite Differences for ODEs & Nonlinear Equations
Lecture 10: Finite Differences for ODEs & Nonlinear Equations J.K. Ryan@tudelft.nl WI3097TU Delft Institute of Applied Mathematics Delft University of Technology 21 November 2012 () Finite Differences
More informationLaboratory #3: Linear Algebra. Contents. Grace Yung
Laboratory #3: Linear Algebra Grace Yung Contents List of Problems. Introduction. Objectives.2 Prerequisites 2. Linear Systems 2. What is a Matrix 2.2 Quick Review 2.3 Gaussian Elimination 2.3. Decomposition
More informationLAPLACE EQUATION. = 2 is call the Laplacian or del-square operator. In two dimensions, the expression of in rectangular and polar coordinates are
LAPLACE EQUATION If a diffusion or wave problem is stationary (time independent), the pde reduces to the Laplace equation u = u =, an archetype of second order elliptic pde. = 2 is call the Laplacian or
More informationBake, shake or break - and other applications for the FEM. 5: Do real-life experimentation using your FEM code
Bake, shake or break - and other applications for the FEM Programming project in TMA4220 - part 2 by Kjetil André Johannessen TMA4220 - Numerical solution of partial differential equations using the finite
More informationLeplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE
Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math EECE 3640 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as
More informationChapter 5. Numerical Methods: Finite Differences
Chapter 5 Numerical Methods: Finite Differences As you know, the differential equations that can be solved by an explicit analytic formula are few and far between. Consequently, the development of accurate
More informationHARMONIC FUNCTIONS. x 2 + 2
HARMONIC FUNCTIONS DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR Contents 1. Harmonic functions 1 1.1. Use of Harmonic mappings 1 1.2. Harmonic functions and holomorphicity 2 1.3. Harmonic
More informationPhysics 342 Lecture 23. Radial Separation. Lecture 23. Physics 342 Quantum Mechanics I
Physics 342 Lecture 23 Radial Separation Lecture 23 Physics 342 Quantum Mechanics I Friday, March 26th, 2010 We begin our spherical solutions with the simplest possible case zero potential. Aside from
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationFinite Difference Methods (FDMs) 2
Finite Difference Methods (FDMs) 2 Time- dependent PDEs A partial differential equation of the form (15.1) where A, B, and C are constants, is called quasilinear. There are three types of quasilinear equations:
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationProblem Set 4 Issued: Wednesday, March 18, 2015 Due: Wednesday, April 8, 2015
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 0139.9 NUMERICAL FLUID MECHANICS SPRING 015 Problem Set 4 Issued: Wednesday, March 18, 015 Due: Wednesday,
More information10.34 Numerical Methods Applied to Chemical Engineering. Quiz 2
10.34 Numerical Methods Applied to Chemical Engineering Quiz 2 This quiz consists of three problems worth 35, 35, and 30 points respectively. There are 4 pages in this quiz (including this cover page).
More informationStudy of Forced and Free convection in Lid driven cavity problem
MIT Study of Forced and Free convection in Lid driven cavity problem 18.086 Project report Divya Panchanathan 5-11-2014 Aim To solve the Navier-stokes momentum equations for a lid driven cavity problem
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
More informationThe method of lines (MOL) for the diffusion equation
Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just
More informationMATH2071: LAB #5: Norms, Errors and Condition Numbers
MATH2071: LAB #5: Norms, Errors and Condition Numbers 1 Introduction Introduction Exercise 1 Vector Norms Exercise 2 Matrix Norms Exercise 3 Compatible Matrix Norms Exercise 4 More on the Spectral Radius
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More information7 Hyperbolic Differential Equations
Numerical Analysis of Differential Equations 243 7 Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena.
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationDOING PHYSICS WITH MATLAB ELECTRIC FIELD AND ELECTRIC POTENTIAL: POISSON S EQUATION
DOING PHYSICS WITH MATLAB ELECTRIC FIELD AND ELECTRIC POTENTIAL: POISSON S EQUATION Ian Cooper School of Physics, University of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS For
More information