Introduction to Partial Differential Equations
|
|
- Elizabeth Thomas
- 6 years ago
- Views:
Transcription
1 Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19
2 Introduction The derivation of the heat equation is based on a more general principle called the conservation law. It is also based on several other experimental laws of physics. We will derive the equation which corresponds to the conservation law. Then, we will state and explain the various relevant experimental laws of physics. Finally, we will derive the one dimensional heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 2 / 19
3 Conservation Law Many PDE models involve the study of how a certain quantity changes with time and space. This change follows a basic law called the conservation law which states that the rate at which a quantity changes in a given domain must equal the rate at which the quantity flows across the boundary of that domain plus the rate at which the quantity is created or destroyed, inside the domain. One can see how this would apply to the study of a certain population in a given area. Similar statements can be made about many other quantities such as heat energy, the mass of a chemical, the number of automobiles on a freeway,... Our study focuses on a quantity which only changes in one direction, say the x-direction. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 3 / 19
4 Conservation Law Figure: Tube with cross-sectional area A Consider this to be our domain. It has a constant cross-sectional area we call A. In it, we study how a certain quantity (mass, energy, species,...) changes. We let u (x, t) denote the density of the quantity. Recall that density is measured in amount of quantity per unit volume or per unit length. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 4 / 19
5 Conservation Law Let us make some remarks and introduce further notation: 1 The amount of the quantity at time t in a small section of width dx will be u (x, t) Adx for each x. The amount of the quantity in an arbitrary section a x b will be b a u (x, t) Adx. 2 φ = φ (x, t) denotes the flux of the quantity. It measures the amount of the quantity crossing the section at x, at time t. Its units are amount of quantity per unit area, per unit time. The amount of the quantity crossing the section at x, at time t is Aφ (x, t). By convention, flux is positive if the flow is to the right. 3 f (x, t) denotes the given rate at which the quantity is created (source) or destroyed (sink) per unit volume within the section at x, at time t. It is measured in amount of quantity per unit volume, per unit time. The amount of the quantity being created in a small section of width dx for each x is f (x, t) Adx per unit time and the amount of the quantity being created in an arbitrary section a x b will be b a f (x, t) Adx. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 5 / 19
6 Conservation Law Write the corresponding equation for the conservation law in an arbitrary section a x b of our domain. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 6 / 19
7 Conservation Law Write the corresponding equation for the conservation law in an arbitrary section a x b of our domain. We should have obtained d b dt a u (x, t) dx = φ (a, t) φ (b, t) + b a f (x, t) dx Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 6 / 19
8 Conservation Law In order to go further, we need some results from mathematics. Theorem (Leibniz Rule) If a (t), b (t), and F (x, t) are continuously differentiable then d dt b(t) a(t) F (y, t) dy = b(t) a(t) F t (y, t) dy+f (b (t), t) b (t) F (a (t), t) a (t) This is often known as "taking the derivative inside the integral". Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 7 / 19
9 Conservation Law Using Leibniz rule, explain how we can obtain that for every a and b, b a What can be concluded? [u t (x, t) + φ x (x, t) f (x, t)] dx = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 8 / 19
10 Conservation Law: Summary We study how a certain quantity changes with time in a given region. We make the following assumptions: 1 u (x, t) denotes the density of the quantity being studied. u is assumed to be continuously differentiable. 2 φ (x, t) is the flux of the quantity at time t at x. It measures the amount of the quantity crossing a cross section of our region at x. φ is assumed to be continuously differentiable. 3 f (x, t) is the rate at which the quantity is created or destroyed within our region. f is assumed to be continuous. 4 We assume the quantity being studied only varies in the x direction. 5 Then, the equation describing how our quantity changes with time in the given region (fundamental conservation law) is u t (x, t) + φ x (x, t) = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 9 / 19
11 Conservation Law: Remarks Remark Let us make a few remarks before looking a specific examples. 1 The above equation is often written as u t + φ x = f for simplicity. 2 The functions φ and f are functions of x and t. That dependence may be through the function u as in f = f (u) or φ = φ (u) which would lead to a nonlinear model. We will see one in the examples. 3 The above equation involves two unknown functions: u and φ, usually the source f is assumed to be given. This means that another equation relating u and φ is needed. We will see various examples. 4 The above equation is in its most general form. As we look at specific models, it will take on different forms. In the next sections we will consider various possibilities including advection and diffusion. Also, the source term can take on different forms. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 10 / 19
12 Advection Equation Advection refers to transport of a certain substance in a fluid (water, any liquid, air,...) such as transport of a pollutant in a river. Definition A model where the flux is proportional to the density is called an advection model. It is easy to understand why. Thinking of the example of the river carrying a pollutant, the amount of pollutant which crosses the boundary of a given region in the river clearly depends on the density of the pollutant. In this case, we have for some constant c: φ = cu The constant c is the speed of the fluid. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 11 / 19
13 Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
14 Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
15 Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Rewrite the fundamental conservation law in the case of an advection model with source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
16 Advection Equation Rewrite the fundamental conservation law in the case of an advection model with no source. We should have obtained u t + cu x = 0 Rewrite the fundamental conservation law in the case of an advection model with source f (x, t). We should have obtained u t + cu x = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 12 / 19
17 Diffusion Diffusion is the transport by molecular motion. From observations and experiments, we know that: Flow is always from more agitated molecules (higher kinetic energy) to less agitated molecules. The steeper the density gradient of the substance being studied, the greater the flow. Hence, with diffusion we have: φ (x, t) = Du x where D > 0 is called the diffusion constant, u is the density of the quantity being studied. What can we say about φ in the case the density u increases from left to right? Does it agree with what we said above? Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 13 / 19
18 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
19 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
20 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
21 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
22 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
23 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
24 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Rewrite the conservation law when there is diffusion and advection with a source f (x, t). Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
25 Diffusion and Advection Equations Rewrite the conservation law when there is only diffusion with no source. We should have obtained u t Du xx = 0 which is known as Fick s law. Rewrite the conservation law when there is only diffusion with a source f (x, t). We should have obtained u t Du xx = f (x, t). Rewrite the conservation law when there is diffusion and advection with no source. We should have obtained u t + cu x Du xx = 0 Rewrite the conservation law when there is diffusion and advection with a source f (x, t). We should have obtained u t + cu x Du xx = f (x, t) Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 14 / 19
26 Heat Equation Consider the heat equation with only diffusion present and no source. Consider a thin rod having a constant density ρ and specific heat C. The specific heat of a substance is the amount of energy needed to raise the temperature of a unit mass of the substance by one degree. Both ρ and C are known for known substances, they can be found in engineering and physics handbooks. If u (x, t) is the energy density and θ the temperature, then u (x, t) = ρcθ (x, t) Also, heat flow follows a diffusion model, hence φ = Kθ x. Combine these two equation with the conservation law to derive the heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 15 / 19
27 Heat Equation Consider the heat equation with only diffusion present and no source. Consider a thin rod having a constant density ρ and specific heat C. The specific heat of a substance is the amount of energy needed to raise the temperature of a unit mass of the substance by one degree. Both ρ and C are known for known substances, they can be found in engineering and physics handbooks. If u (x, t) is the energy density and θ the temperature, then u (x, t) = ρcθ (x, t) Also, heat flow follows a diffusion model, hence φ = Kθ x. Combine these two equation with the conservation law to derive the heat equation. We should have obtained θ t kθ xx = 0 where k = K, it is called ρc the diffusivity or thermal diffusivity. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 15 / 19
28 Heat Equation Remark In deriving this equation, we made some assumptions about the rod. These assumptions lead us to assume that ρ and C and K were constants. If the rod is not homogeneous, K will also depend on x. In addition, if we consider wide ranges of temperatures, K may also depend on θ. So, the heat equation would become This is a non-linear model. ρcθ t (K (x, θ) θ x ) x = 0 Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 16 / 19
29 Steady State Solution Many PDE models, in particular diffusion problems, have the property that after a long time, they approach a steady state, that is a solution which is no longer evolving with time. In other words, for large t, u (x, t) becomes a function of x only and u t = 0. Even if we do not yet know how to solve PDEs, we can, in most cases, find the steady state solution. To do so, we set u t = 0 in the diffusion equation. The resulting equation is a second-order linear ODE, which we know how to solve. Remember that the solution should be a function of x only. The examples below illustrate how. Find the steady state solution for the problem PDE u t = cu xx 0 < x < L 0 < t < BC1 u (0, t) = T 1 0 < t < BC2 u (L, t) = T 2 0 < t < IC u (x, 0) = f (x) 0 x L Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 17 / 19
30 Steady State Solution Many PDE models, in particular diffusion problems, have the property that after a long time, they approach a steady state, that is a solution which is no longer evolving with time. In other words, for large t, u (x, t) becomes a function of x only and u t = 0. Even if we do not yet know how to solve PDEs, we can, in most cases, find the steady state solution. To do so, we set u t = 0 in the diffusion equation. The resulting equation is a second-order linear ODE, which we know how to solve. Remember that the solution should be a function of x only. The examples below illustrate how. Find the steady state solution for the problem PDE u t = cu xx 0 < x < L 0 < t < BC1 u (0, t) = T 1 0 < t < BC2 u (L, t) = T 2 0 < t < IC u (x, 0) = f (x) 0 x L We should have found u (x) = T 2 T 1 x + T 1. L Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 17 / 19
31 Heat Equation:Summary 1 We derived the conservation law formula u t + φ x = f (x, y) 2 The flux, φ, takes on different forms depending on what kind of model corresponds to the situation being studied. 1 Advection model: φ = cu hence the conservation law becomes: u t + cu x = f (x, y) 2 Diffusion model: φ = Du x hence the conservation law becomes: u t Du xx = f (x, y) 3 Advection plus diffusion model: φ = cu Du x hence the conservation law becomes: u t + cu x Du xx = f (x, y) 3 For each equation above, f (x, t) is the source. When there is no source, f (x, t) = 0. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 18 / 19
32 Exercises See the problems at the end of my notes on derivation of the one-dimensional heat equation. Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 19 / 19
Diffusion - The Heat Equation
Chapter 6 Diffusion - The Heat Equation 6.1 Goal Understand how to model a simple diffusion process and apply it to derive the heat equation in one dimension. We begin with the fundamental conservation
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define
More informationReview of Functions. Functions. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Functions Current Semester 1 / 12
Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12 Introduction Students are expected to know the following concepts about functions:
More informationDifferentiation - Quick Review From Calculus
Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13 Introduction In this section,
More informationRepresentation of Functions as Power Series
Representation of Functions as Power Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Functions as Power Series Today / Introduction In this section and the next, we develop several techniques
More informationFunctions of Several Variables
Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend
More informationFinite difference method for solving Advection-Diffusion Problem in 1D
Finite difference method for solving Advection-Diffusion Problem in 1D Author : Osei K. Tweneboah MATH 5370: Final Project Outline 1 Advection-Diffusion Problem Stationary Advection-Diffusion Problem in
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 informationThe Laplace Transform
The Laplace Transform Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Definition of the Laplace Transform Today 1 / 16 Outline General idea behind the Laplace transform and other
More informationTransforming Nonhomogeneous BCs Into Homogeneous Ones
Chapter 10 Transforming Nonhomogeneous s Into Homogeneous Ones 10.1 Goal In the previous chapter, we looked at separation of variables. We saw that this method applies if both the boundary conditions and
More informationThe Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /
The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will
More informationLagrange s Theorem. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10
Lagrange s Theorem Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10 Introduction In this chapter, we develop new tools which will allow us to extend
More informationIntroduction to Vector Functions
Introduction to Vector Functions Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction Until now, the functions we studied took a real number
More information1.061 / 1.61 Transport Processes in the Environment
MIT OpenCourseWare http://ocw.mit.edu 1.061 / 1.61 Transport Processes in the Environment Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Solution
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 informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More informationConsequences of the Completeness Property
Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R
More informationDiffusion of a density in a static fluid
Diffusion of a density in a static fluid u(x, y, z, t), density (M/L 3 ) of a substance (dye). Diffusion: motion of particles from places where the density is higher to places where it is lower, due to
More informationTransforming Nonhomogeneous BCs Into Homogeneous Ones
Chapter 8 Transforming Nonhomogeneous s Into Homogeneous Ones 8.1 Goal In this chapter we look at more challenging example of problems which can be solved by separation of variables. A restriction of the
More informationModeling using conservation laws. Let u(x, t) = density (heat, momentum, probability,...) so that. u dx = amount in region R Ω. R
Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...) so that u dx = amount in region R Ω. R Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...)
More informationTesting Series with Mixed Terms
Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series
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 information1.5 First Order PDEs and Method of Characteristics
1.5. FIRST ORDER PDES AND METHOD OF CHARACTERISTICS 35 1.5 First Order PDEs and Method of Characteristics We finish this introductory chapter by discussing the solutions of some first order PDEs, more
More informationDifferentiation and Integration of Fourier Series
Differentiation and Integration of Fourier Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 12 Introduction When doing manipulations with infinite sums, we must remember
More informationSequences: Limit Theorems
Sequences: Limit Theorems Limit Theorems Philippe B. Laval KSU Today Philippe B. Laval (KSU) Limit Theorems Today 1 / 20 Introduction These limit theorems fall in two categories. 1 The first category deals
More informationFrom the last time, we ended with an expression for the energy equation. u = ρg u + (τ u) q (9.1)
Lecture 9 9. Administration None. 9. Continuation of energy equation From the last time, we ended with an expression for the energy equation ρ D (e + ) u = ρg u + (τ u) q (9.) Where ρg u changes in potential
More informationSome Aspects of Solutions of Partial Differential Equations
Some Aspects of Solutions of Partial Differential Equations K. Sakthivel Department of Mathematics Indian Institute of Space Science & Technology(IIST) Trivandrum - 695 547, Kerala Sakthivel@iist.ac.in
More informationBasic Fluid Mechanics
Basic Fluid Mechanics Chapter 3B: Conservation of Mass C3B: Conservation of Mass 1 3.2 Governing Equations There are two basic types of governing equations that we will encounter in this course Differential
More informationConservation and dissipation principles for PDEs
Conservation and dissipation principles for PDEs Modeling through conservation laws The notion of conservation - of number, energy, mass, momentum - is a fundamental principle that can be used to derive
More informationIntroduction to Vector Functions
Introduction to Vector Functions Differentiation and Integration Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14 Introduction In this section, we study the differentiation
More informationOrdinary Differential Equations
Chapter 10 Ordinary Differential Equations 10.1 Introduction Relationship between rate of change of variables rather than variables themselves gives rise to differential equations. Mathematical formulation
More informationLecture 3: 1. Lecture 3.
Lecture 3: 1 Lecture 3. Lecture 3: 2 Plan for today Summary of the key points of the last lecture. Review of vector and tensor products : the dot product (or inner product ) and the cross product (or vector
More information2. Conservation of Mass
2 Conservation of Mass The equation of mass conservation expresses a budget for the addition and removal of mass from a defined region of fluid Consider a fixed, non-deforming volume of fluid, V, called
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.
More informationNumerical Heat and Mass Transfer
Master degree in Mechanical Engineering Numerical Heat and Mass Transfer 02-Transient Conduction Fausto Arpino f.arpino@unicas.it Outline Introduction Conduction ü Heat conduction equation ü Boundary conditions
More informationFundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering
Fundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering Module No # 05 Lecture No # 25 Mass and Energy Conservation Cartesian Co-ordinates Welcome
More informationRelationship Between Integration and Differentiation
Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined
More informationPartial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
More informationIntroduction to Mass Transfer
Introduction to Mass Transfer Introduction Three fundamental transfer processes: i) Momentum transfer ii) iii) Heat transfer Mass transfer Mass transfer may occur in a gas mixture, a liquid solution or
More informationClass Meeting # 2: The Diffusion (aka Heat) Equation
MATH 8.52 COURSE NOTES - CLASS MEETING # 2 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 2: The Diffusion (aka Heat) Equation The heat equation for a function u(, x (.0.). Introduction
More informationConsequences of Orthogonality
Consequences of Orthogonality Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of Orthogonality Today 1 / 23 Introduction The three kind of examples we did above involved Dirichlet, Neumann
More informationFunctions of Several Variables
Functions of Several Variables Extreme Values Philippe B Laval KSU April 9, 2012 Philippe B Laval (KSU) Functions of Several Variables April 9, 2012 1 / 13 Introduction In Calculus I (differential calculus
More informationLecture 2. Classification of Differential Equations and Method of Integrating Factors
Math 245 - Mathematics of Physics and Engineering I Lecture 2. Classification of Differential Equations and Method of Integrating Factors January 11, 2012 Konstantin Zuev (USC) Math 245, Lecture 2 January
More informationLecture 7. Heat Conduction. BENG 221: Mathematical Methods in Bioengineering. References
BENG 221: Mathematical Methods in Bioengineering Lecture 7 Heat Conduction References Haberman APDE, Ch. 1. http://en.wikipedia.org/wiki/heat_equation Lecture 5 BENG 221 M. Intaglietta Heat conduction
More informationFirst Order Differential Equations
First Order Differential Equations Linear Equations Philippe B. Laval KSU Philippe B. Laval (KSU) 1st Order Linear Equations 1 / 11 Introduction We are still looking at 1st order equations. In today s
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationPartial Differential Equations
Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1
More information1. Differential Equations (ODE and PDE)
1. Differential Equations (ODE and PDE) 1.1. Ordinary Differential Equations (ODE) So far we have dealt with Ordinary Differential Equations (ODE): involve derivatives with respect to only one variable
More informationArc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12
Philippe B. Laval KSU Today Philippe B. Laval (KSU) Arc Length Today 1 / 12 Introduction In this section, we discuss the notion of curve in greater detail and introduce the very important notion of arc
More informationM.Sc. in Meteorology. Numerical Weather Prediction
M.Sc. in Meteorology UCD Numerical Weather Prediction Prof Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin Second Semester, 2005 2006. In this section
More informationDiffusion Processes. Lectures INF2320 p. 1/72
Diffusion Processes Lectures INF2320 p. 1/72 Lectures INF2320 p. 2/72 Diffusion processes Examples of diffusion processes Heat conduction Heat moves from hot to cold places Diffusive (molecular) transport
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationThe Laplace Transform
The Laplace Transform Inverse of the Laplace Transform Philippe B. Laval KSU Today Philippe B. Laval (KSU) Inverse of the Laplace Transform Today 1 / 12 Outline Introduction Inverse of the Laplace Transform
More information5. Coupling of Chemical Kinetics & Thermodynamics
5. Coupling of Chemical Kinetics & Thermodynamics Objectives of this section: Thermodynamics: Initial and final states are considered: - Adiabatic flame temperature - Equilibrium composition of products
More informationOutline. Definition and mechanism Theory of diffusion Molecular diffusion in gases Molecular diffusion in liquid Mass transfer
Diffusion 051333 Unit operation in gro-industry III Department of Biotechnology, Faculty of gro-industry Kasetsart University Lecturer: Kittipong Rattanaporn 1 Outline Definition and mechanism Theory of
More information1D Heat equation and a finite-difference solver
1D Heat equation and a finite-difference solver Guillaume Riflet MARETEC IST 1 The advection-diffusion equation The original concept applied to a property within a control volume from which is derived
More informationMath 5440 Problem Set 2 Solutions
Math Math Problem Set Solutions Aaron Fogelson Fall, 1: (Logan, 1. # 1) How would the derivation of the basic conservation law u t + φ = f change if the tube had variable cross-sectional area A = A() rather
More informationMATH 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
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 informationMIT (Spring 2014)
18.311 MIT (Spring 014) Rodolfo R. Rosales February 13, 014. Problem Set # 01. Due: Mon. February 4. IMPORTANT: Turn in the regular and the special problems stapled in two SEPARATE packages. Print your
More informationBoundary conditions. Diffusion 2: Boundary conditions, long time behavior
Boundary conditions In a domain Ω one has to add boundary conditions to the heat (or diffusion) equation: 1. u(x, t) = φ for x Ω. Temperature given at the boundary. Also density given at the boundary.
More informationMath 46, Applied Math (Spring 2009): Final
Math 46, Applied Math (Spring 2009): Final 3 hours, 80 points total, 9 questions worth varying numbers of points 1. [8 points] Find an approximate solution to the following initial-value problem which
More informationFunctions of Several Variables
Functions of Several Variables Extreme Values Philippe B. Laval KSU Today Philippe B. Laval (KSU) Extreme Values Today 1 / 18 Introduction In Calculus I (differential calculus for functions of one variable),
More informationLecture - 1 Motivation with Few Examples
Numerical Solutions of Ordinary and Partial Differential Equations Prof. G. P. Raja Shekahar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 1 Motivation with Few Examples
More informationFluid Mechanics II Viscosity and shear stresses
Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
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 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 informationsurface area per unit time is w(x, t). Derive the partial differential
1.2 Conduction of Heat in One-Dimension 11 1.2.4. Derive the diffusion equation for a chemical pollutant. (a) Consider the total amount of the chemical in a thin region between x and x + Ax. (b) Consider
More informationIntroduction to Vector Functions
Introduction to Vector Functions Limits and Continuity Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Introduction to Vector Functions Spring 2012 1 / 14 Introduction In this section, we study
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 informationconvection coefficient, h c = 18.1 W m K and the surrounding temperature to be 20 C.) (20 marks) Question 3 [35 marks]
COP 311 June Examination 18 June 005 Duration: 3 hours Starting time: 08:30 Internal examiners: Prof. T. Majozi Mnr. D.J. de Kock Mnr. A.T. Tolmay External examiner: Mnr. B. du Plessis Metallurgists: Questions
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More informationMATH 320, WEEK 4: Exact Differential Equations, Applications
MATH 320, WEEK 4: Exact Differential Equations, Applications 1 Exact Differential Equations We saw that the trick for first-order differential equations was to recognize the general property that the product
More informationSeparation of variables
Separation of variables Idea: Transform a PDE of 2 variables into a pair of ODEs Example : Find the general solution of u x u y = 0 Step. Assume that u(x,y) = G(x)H(y), i.e., u can be written as the product
More informationFinite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
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 information0.2. CONSERVATION LAW FOR FLUID 9
0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is
More informationCHAPTER 7. An Introduction to Numerical Methods for. Linear and Nonlinear ODE s
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 1 A COLLECTION OF HANDOUTS ON FIRST ORDER ORDINARY DIFFERENTIAL
More informationScalar Conservation Laws and First Order Equations Introduction. Consider equations of the form. (1) u t + q(u) x =0, x R, t > 0.
Scalar Conservation Laws and First Order Equations Introduction. Consider equations of the form (1) u t + q(u) x =, x R, t >. In general, u = u(x, t) represents the density or the concentration of a physical
More informationP = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid.
CEE 3310 Thermodynamic Properties, Aug. 27, 2010 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container
More informationPartial Differential Equations - part of EM Waves module (PHY2065)
Partial Differential Equations - part of EM Waves module (PHY2065) Richard Sear February 7, 2013 Recommended textbooks 1. Mathematical methods in the physical sciences, Mary Boas. 2. Essential mathematical
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 15 Heat with a source So far we considered homogeneous wave and heat equations and the associated initial value problems on the whole line, as
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationNotes: Outline. Diffusive flux. Notes: Notes: Advection-diffusion
Outline This lecture Diffusion and advection-diffusion Riemann problem for advection Diagonalization of hyperbolic system, reduction to advection equations Characteristics and Riemann problem for acoustics
More informationNon-linear Scalar Equations
Non-linear Scalar Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro August 24, 2014 1 / 44 Overview Here
More informationThe similarity solutions of concentration dependent diffusion equation
International Journal of Advances in Applied Mathematics and Mechanics Volume 1, Issue 2 : (2013) pp. 80-85 Available online at www.ijaamm.com IJAAMM ISSN: 2347-2529 The similarity solutions of concentration
More informationIntroduction to Fluid Dynamics
Introduction to Fluid Dynamics Roger K. Smith Skript - auf englisch! Umsonst im Internet http://www.meteo.physik.uni-muenchen.de Wählen: Lehre Manuskripte Download User Name: meteo Password: download Aim
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationLecture 19: Heat conduction with distributed sources/sinks
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 ecture 19: Heat conduction
More informationMTH210 DIFFERENTIAL EQUATIONS. Dr. Gizem SEYHAN ÖZTEPE
MTH210 DIFFERENTIAL EQUATIONS Dr. Gizem SEYHAN ÖZTEPE 1 References Logan, J. David. A first course in differential equations. Springer, 2015. Zill, Dennis G. A first course in differential equations with
More informationAnswers to Problem Set # 01, MIT (Winter-Spring 2018)
Answers to Problem Set # 01, 18.306 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Contents 1 Nonlinear solvable ODEs 2 1.1 Statement:
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
More informationRocket Propulsion Prof. K. Ramamurthi Department of Mechanical Engineering Indian Institute of Technology, Madras
Rocket Propulsion Prof. K. Ramamurthi Department of Mechanical Engineering Indian Institute of Technology, Madras Lecture 32 Efficiencies due to Mixture Ratio Distribution and Incomplete Vaporization (Refer
More informationTransport processes. 7. Semester Chemical Engineering Civil Engineering
Transport processes 7. Semester Chemical Engineering Civil Engineering 1 Course plan 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume nalysis 4. Differential nalysis of Fluid Flow
More informationDivergence Theorem and Its Application in Characterizing
Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,
More information